🧠:[<think>Okay, so I need to figure out which of the four people—甲 (Jia), 乙 (Yi), 丙 (Bing), or 丁 (Ding)—is lying. The problem states that each of them chose a different option (A, B, C, D), and only one person is lying. The answer choices give possible pairs of liars, but since only one is lying, I need to check which pairs actually can't both be telling the truth or something? Wait, no. The options are A: 甲 and 乙; B: 乙 and 丙; C: 丙 and 丁; D: 丁 and 甲. But wait, the question says only one person is lying, so the answer must be a single person. But the options are pairs. Hmm, maybe the options are possible pairs where one of them must be the liar? Wait, maybe the question is phrased as "Who could be the one lying?" and the options are possible pairs where either of the two could be the liar? But the problem states that only one person is lying. So each option lists two people, and we need to choose the option where either one of those two could be the liar. But how does that work?Wait, let me read the problem again:Four people are answering a multiple-choice question with options ABCD. The conversation:甲: I choose A;乙: I choose one of BDC;丙: I choose C;丁: I choose D;Each of the four people chose a different option, and only one person is lying. Who could be the one lying? The options are pairs, so maybe the answer is the option where the two people listed are possible candidates for the liar. So we need to check which of the pairs given in the options could include the sole liar.So first, let's list the statements:- Jia says A.- Yi says one of B, D, C.- Bing says C.- Ding says D.Given that each person chose a different option (so all four options are taken), and only one is lying. So three are telling the truth, one is lying. We need to find which individual could be the liar.But the answer choices are pairs (A: Jia and Yi; B: Yi and Bing; etc.), so the correct answer is the pair where either of the two could be the liar. So we need to check for each pair: if either person in the pair could be the sole liar, such that the rest are telling the truth, and all four options are distinct.Let me test each possibility.First, let's suppose Jia is lying. Then Jia didn't choose A. The others (Yi, Bing, Ding) are telling the truth. So Yi chose one of B, D, C; Bing chose C; Ding chose D. But if Bing is telling the truth, he chose C, and Ding is telling the truth, he chose D. Then Yi must choose B, because C and D are already taken. So assignments would be:Jia: not A, so must be the remaining option. Since Yi is B, Bing is C, Ding is D, then Jia must be A? But Jia is lying, so he didn't choose A. Wait, that's a problem. If Jia is lying, he didn't choose A. Then the other three are Yi: B, Bing: C, Ding: D. So Jia must choose the remaining option, which is A. But he can't choose A if he's lying. Contradiction. Therefore, Jia cannot be the liar.Wait, that seems contradictory. Let me check again. If Jia is the liar, he didn't choose A. So the remaining three (Yi, Bing, Ding) must have chosen B, C, D. But Yi says he chose one of B, D, C. If Yi is truthful, he must have chosen one of those. So if Yi is truthful, he could be B, D, or C. But Bing is truthful and chose C, Ding is truthful and chose D. Therefore, if Bing is C and Ding is D, then Yi must choose B. So Yi is B, Bing is C, Ding is D. Then Jia must choose the remaining option, which is A. But Jia is supposed to be lying, so he can't have chosen A. So contradiction. Therefore, Jia cannot be the liar. So option A (Jia and Yi) and D (Ding and Jia) include Jia, but Jia can't be the liar. So maybe the answer is not A or D.Next, check Yi as the liar. Suppose Yi is lying. Then Yi did not choose one of B, D, C. Since each person chooses a different option, and Yi is lying, he must have chosen A. But Jia says he chose A. If Yi also chose A, that's a conflict. But the problem states each person chose a different option, so Yi can't choose A if Jia did. But Jia is telling the truth (since only Yi is lying), so Jia chose A. Therefore, Yi cannot choose A, but since he's lying, he didn't choose B, D, or C either. So Yi must have chosen... but there's no other options. The options are ABCD. If he didn't choose B, D, C, and A is taken by Jia, then Yi cannot choose any. Contradiction. Therefore, Yi cannot be the liar either. Wait, that seems like a problem. So Yi can't be the liar. So options A and B include Yi, but if Yi can't be the liar, then those options can't be correct.Wait, maybe I made a mistake here. Let me re-examine. If Yi is lying, that means his statement "I choose one of B, D, C" is false. Therefore, he must have chosen the only remaining option, which is A. But Jia is telling the truth and already chose A. So Yi can't choose A. Therefore, Yi cannot be lying because it would require him to choose a duplicate. Therefore, Yi can't be the liar. So options A and B (which include Yi) are invalid.Next, check Bing. Suppose Bing is lying. So Bing didn't choose C. The others are telling the truth: Jia chose A, Yi chose one of B, D, C; Ding chose D. Since Ding is telling the truth, he chose D. Therefore, Yi cannot choose D. So Yi must choose either B or C. But Bing is lying, so he didn't choose C. Therefore, the assignments are:Jia: AYi: B or CBing: Not C, so must be the remaining option. If Yi chooses B, then Bing can choose C? Wait, but Bing is lying, so he didn't choose C. If Yi chooses B, then the remaining options are C and D. But Ding is already D. So remaining is C. But Bing must not choose C, so he can't choose C. So he must choose... but all other options are taken. If Jia is A, Yi is B, Ding is D, then Bing must choose C. But Bing is lying, so he can't choose C. Contradiction.Alternatively, if Yi chooses C. Then Jia is A, Yi is C, Ding is D. Then Bing must choose B. But Bing is lying about choosing C, so he actually chose B. Then all options are distinct: A, C, B, D. That works. So in this case, if Bing is lying and chose B instead of C, then the assignments are:Jia: A (truthful)Yi: C (truthful, since Yi said one of B, D, C)Bing: B (lying)Ding: D (truthful)But wait, Yi said he chose one of B, D, C. If Yi actually chose C, that's fine. Then Bing chose B, which was an available option. So in this case, is that possible? Wait, but if Bing is lying and chose B, then Yi is telling the truth by choosing C. Then the options are A, C, B, D. All unique. So that works. So Bing could be the liar. Therefore, Bing is a possible liar. So option B includes Bing and Yi. But we already saw Yi can't be the liar. But in this scenario, Bing is the liar, so option B says "乙 (Yi) and 丙 (Bing)"—so if Bing is the liar, then option B includes Bing, but since Yi isn't lying, does that matter? Wait, the question is "Who could be the one lying?" and the answer options are pairs. So maybe the answer is the option that includes possible liars. Since Bing can be the liar, and if option B includes Bing, then maybe B is a possible answer. But the pair is乙 and 丙 (Yi and Bing). But we already determined Yi can't be the liar. However, in this scenario, Bing is the liar, so option B includes Bing. But the answer options are in pairs. So perhaps the answer is B because Bing could be the liar. But then why is Yi included? Maybe if either of them could be the liar. Wait, but we need to check if both in the pair could be possible liars. Let me check if Bing is the liar (possible) and if Yi could be the liar (we saw it's impossible). So option B is乙 and 丙, but since only Bing could be the liar in this pair, does that make option B correct? Hmm, maybe the problem is structured such that the answer is the pair where at least one of them is the possible liar. But the question is "Who could be the one lying?" So it's asking for which pair includes the possible liar. Since Bing is a possible liar, and option B includes Bing, then B could be the answer. But let me check other possibilities.Next, check Ding as the liar. Suppose Ding is lying. Then Ding didn't choose D. The others are truthful: Jia chose A, Yi chose one of B, D, C; Bing chose C. Since Bing is truthful, he chose C. Ding is lying, so he didn't choose D. Therefore, Ding must choose either B or the remaining option. Let's see:Jia: ABing: CDing: Not D, so must choose B or the remaining option. But Yi is supposed to choose one of B, D, C. Since Bing chose C, and Ding is not choosing D (he's lying), then Yi can choose B or D. But Ding can't choose D, so Yi could choose D. If Yi chooses D, then Ding must choose B. So assignments:Jia: AYi: D (truthful, since D is in B, D, C)Bing: CDing: BBut Ding's lie is not choosing D, but choosing B instead. So all choices are distinct: A, D, C, B. That works. So in this case, Ding is lying. Therefore, Ding could be the liar. So option C is丙 (Bing) and 丁 (Ding). But we already saw Bing can be a liar, and Ding can be a liar. Wait, but in this case, Ding is the liar, not Bing. So option C includes丙 and 丁. So if Ding is the liar, then option C includes Ding. So maybe option C is possible. But in the scenario where Ding is lying, that works. Also, in the previous scenario where Bing is lying, that also works. So there are two possible liars: Bing and Ding. Therefore, the answer options need to include both of these. Looking at the options:A: 甲, 乙B: 乙, 丙C: 丙, 丁D: 丁, 甲So if both Bing and Ding could be liars, then option C includes both. So the answer would be C, since both丙 and 丁 could be the liars. Wait, but the problem states that only one person is lying. So in one scenario, Bing is lying; in another scenario, Ding is lying. So both are possible. Therefore, the answer is option C:丙 and 丁, because either of them could be the liar.Wait, but let me confirm. When we considered Bing as the liar, we had:Jia: AYi: CBing: B (lying)Ding: DBut wait, Yi's choice: Yi said he chose one of B, D, C. If Yi actually chose C, that's allowed. So that works. But if Bing lies and chooses B, then Yi could have chosen C, which is allowed.When we considered Ding as the liar:Jia: AYi: DBing: CDing: B (lying)Here, Yi chose D, which is allowed since he said he chose one of B, D, C.So both scenarios are valid, with either Bing or Ding lying. Therefore, both丙 and 丁 could be the liar. Hence, the answer is C.Wait, but let me check if there's any other possibilities. Could there be another liar?Earlier, we saw that Jia and Yi cannot be liars. Let's confirm again:Jia as liar leads to contradiction.Yi as liar leads to contradiction.Bing as liar is possible.Ding as liar is possible.Therefore, the possible liars are Bing and Ding, which is option C. So the answer should be C.</think>To determine who could be lying, we analyze each person's statement under the constraint that only one person lies and all options are distinct:1. 甲 (Jia) claims A. If Jia lies, he must choose another option. However, if the others (乙, 丙, 丁) tell the truth, they would take B, C, D. This forces Jia back to A, contradicting his lie. Thus, Jia cannot lie.2. 乙 (Yi) claims one of B, D, C. If Yi lies, he must choose A. But Jia (truthful) already took A, so this is impossible. Yi cannot lie.3. 丙 (Bing) claims C. If Bing lies, he chooses another option. Suppose he picks B. Then: - 甲: A (truthful) - 乙: Chooses from B, D, C → since B is taken by Bing, 乙 picks C or D. If 乙 picks C, 丁 must pick D (truthful). All options are distinct: A, C, B, D. This works.4. 丁 (Ding) claims D. If Ding lies, he chooses another option. Suppose he picks B. Then: - 甲: A (truthful) - 乙: Chooses from B, D, C → since B is taken by Ding, 乙 picks D. Then 丙 picks C (truthful). All options are distinct: A, D, C, B. This works.Both 丙 (Bing) and 丁 (Ding) can be the liar in valid scenarios. The correct answer includes these two:Answer: boxed{C},<think>Okay, let's try to figure out this problem. So, we have four people: 甲, 乙, 丙, and 丁. They each answered a multiple-choice question with options A, B, C, D. The key points are that each of them chose a different option, and only one person is lying. We need to find out who could be the liar. The answer choices are pairs of people, so the possible answers are combinations of two people. Hmm, wait, but the problem says only one person is lying. Wait, maybe I misread. Let me check again."Each of the four people chose a different option, and only one person is lying. Who could be the one lying?" The answer options are pairs: A: 甲, 乙; B: 乙, 丙; C: 丙, 丁; D: 丁, 甲. Wait, so even though the question says only one person is lying, the answer choices are pairs. That seems contradictory. Maybe there's a translation issue. Let me read again.Original problem: Four people are answering a multiple-choice question with options ABCD. The conversation: 甲: I choose A; 乙: I choose one of BDC; 丙: I choose C; 丁: I choose D. Each of the four people chose a different option, and only one person is lying. Who could be the one lying? The answer choices are pairs. Wait, that seems conflicting because if only one is lying, then the answer should be a single person, but the options are pairs. Maybe there's a mistake in the problem? Or perhaps the original problem allows for the possibility that either of the two in the pair could be the liar, depending on the scenario. Hmm. Let me think carefully.So, each person made a statement:甲: A乙: one of B, D, C (so either B, D, or C)丙: C丁: DWe know that all four chose different options, meaning each option is taken by exactly one person. Also, only one person is lying. So, three are telling the truth, and one is lying. Our job is to figure out which one could be lying such that the rest can be telling the truth, and all four choices are different.Let's consider each possibility one by one.First, list the options each person claims:甲: A乙: B, D, or C丙: C丁: DNow, if all four are telling the truth, then their choices would be A, (B/D/C), C, D. Wait, but two people would choose C and D. Wait, but 乙 says they chose one of B, D, or C. If 乙 chose B, then we have 甲: A, 乙: B, 丙: C, 丁: D. That's all four options, no duplicates. So in this case, all are telling the truth. But the problem states that only one person is lying. So this scenario where all are telling the truth is impossible, right? Because the problem says only one is lying. Therefore, there must be a lie. So this tells me that my initial thought is wrong. Wait, hold on.Wait, the problem says "each of the four people chose a different option, and only one person is lying." So the scenario where all four are telling the truth is not possible because in that case, there is no liar. But the problem says there is exactly one liar. Therefore, the scenario where everyone tells the truth is invalid. Therefore, the actual answer must involve one liar and three truth-tellers, with all four options chosen. So we need to find which person's lie would allow the other three to tell the truth, and still result in all four options being selected.Let's check each possibility.Case 1: Suppose 甲 is lying. So 甲 didn't choose A. Then, the other three (乙, 丙, 丁) are telling the truth. So 乙 chose one of B, D, C; 丙 chose C; 丁 chose D. If 丙 chose C and 丁 chose D, then 乙 must have chosen B. So then 乙 is B, 丙 is C, 丁 is D. Then 甲 must have chosen the remaining option, which is A. But wait, 甲 is supposed to be lying. So if 甲 is lying about choosing A, but if all others are truthful, then 甲 must have chosen something else. But the remaining options are A, B, C, D. If 乙 is B, 丙 is C, 丁 is D, then the only remaining is A. So 甲 would have to choose A, but that contradicts the assumption that 甲 is lying. Therefore, this case is impossible. Hence, 甲 cannot be the liar.Case 2: Suppose 乙 is lying. So 乙 didn't choose B, D, or C. Therefore, 乙 must have chosen A. But 甲 says they chose A. If 乙 is lying and actually chose A, then 甲 and 乙 both chose A, which conflicts with the requirement that all chose different options. So this is impossible. Therefore, 乙 cannot be the liar.Wait, hold on. If 乙 is lying, that means 乙's statement "I choose one of B, D, C" is false, so 乙 must have chosen the remaining option, which is A. But 甲 says he chose A, so 甲 and 乙 both choose A. That's a conflict. Therefore, 乙 can't be the liar because it would cause duplication. So 乙 can't be the liar. So case 2 is impossible.Case 3: Suppose 丙 is lying. So 丙 didn't choose C. The others are telling the truth. So 甲 chose A, 乙 chose one of B, D, C; 丁 chose D. Now, 乙 can't choose A (since 甲 did), and can't choose C (since 丙 didn't choose C but we don't know if someone else did), but 乙's options are B, D, C. If 乙 chooses B, then 丁 is D, so remaining options are C and maybe others. Wait, but 丙 is lying, so 丙 didn't choose C. So the remaining options for 丙 would be... If 甲 is A, 乙 is B, 丁 is D, then the remaining option is C. But 丙 is lying, so they didn't choose C. Therefore, there's a contradiction. Because the only remaining option is C, but 丙 can't choose C. Therefore, this case is impossible.Alternatively, if 乙 chooses D. Then 丁 is D, but 乙 also chooses D. That's a conflict. Therefore, 乙 can't choose D. If 乙 chooses C, then 丙 is lying and didn't choose C, so 乙 could choose C, but then 丙 needs to choose something else. Let me check again.If 丙 is lying, so 丙 didn't choose C. Then, the assignments would be:甲: A (truth)乙: B, D, or C (truth). So 乙 can choose B, D, or C.丙: Not C.丁: D (truth)So 丁 is D, so 乙 can't choose D. Therefore, 乙 must choose B or C.If 乙 chooses B, then the remaining options are C and something else. Wait, 甲 is A, 乙 is B, 丁 is D. So remaining option is C. But 丙 didn't choose C (because lying), so there's no one left to choose C. Therefore, this is impossible.If 乙 chooses C, then 丙 can't choose C, but 乙 did. So 乙 is C, 丁 is D. Then the remaining options are B. But 丙 must choose something. 甲 is A, 乙 is C, 丁 is D. So remaining option is B. So 丙 must choose B. But 丙 originally said they chose C, which is a lie. So 丙 chooses B. That would work. Wait, let's see:甲: A (truth)乙: C (truth, because 乙 said one of B, D, C, so choosing C is allowed)丙: B (lie, since they said C)丁: D (truth)So in this case, all options are A, C, B, D. All different. Only 丙 is lying. So this works. Therefore, if 丙 is lying, this is possible. Therefore, 丙 could be the liar.Wait, but earlier when I tried 乙 choosing B, it didn't work. But when 乙 chooses C, it works. So this is possible. Therefore, 丙 could be the liar.Case 4: Suppose 丁 is lying. So 丁 didn't choose D. The others are truthful: 甲 chose A, 乙 chose one of B, D, C; 丙 chose C. So if 丙 chose C, 丁 didn't choose D. So 丁 must choose either A, B, or remaining. But 甲 is A, so 丁 can't choose A. 乙's choice is among B, D, C. If 乙 chose B, then remaining options are D and something else. But 丁 didn't choose D, so 丁 must choose... Wait, 甲: A, 乙: B, 丙: C. So the remaining is D. But 丁 is lying, so they didn't choose D. Therefore, there is no option left for 丁. Contradiction.If 乙 chose D, but 丁 is supposed to have not chosen D. But 乙 says they chose one of B, D, C, so choosing D is allowed. Then 乙 is D, 丙 is C, 甲 is A. Then the remaining option is B. But 丁 didn't choose D, so 丁 must choose B. But 丁 said they chose D, which is a lie. So 丁 chooses B. Then the assignments are:甲: A乙: D丙: C丁: BAll different. Only 丁 is lying. That works. So in this case, 丁 is the liar.Alternatively, if 乙 chooses C, then 丙 is already choosing C, which would be a conflict. So 乙 can't choose C. If 乙 chooses D, then as above, 丁 can choose B. That works. If 乙 chooses B, then remaining options are D, but 丁 can't choose D, so no.Wait, if 乙 chooses B, then the assignments are:甲: A, 乙: B, 丙: C. Then remaining is D. But 丁 must not choose D, so there's no option left. So that's impossible. Therefore, only if 乙 chooses D, then 丁 chooses B. Then all options are covered without duplication. Therefore, 丁 can be the liar.So, from case 3 and case 4, both 丙 and 丁 could be liars. Wait, but the answer options are pairs: A: 甲,乙; B:乙,丙; C:丙,丁; D:丁,甲. So the possible answer is C: 丙,丁. Because both 丙 and 丁 could be the liars. Wait, but the question says "only one person is lying". So in each case, only one person is lying. So in case 3, 丙 is the liar; in case 4, 丁 is the liar. Therefore, both 丙 and 丁 could individually be the liar. Therefore, the answer is C: 丙,丁. So both are possible liars.Wait, but the problem says "Who could be the one lying?" So the answer is the pair where each member could individually be the liar. So in the answer options, C is 丙,丁. Since both 丙 and 丁 can be the liar in different scenarios, then answer C is correct.Let me check the answer options again. The options are:A: 甲,乙B:乙,丙C:丙,丁D:丁,甲From our analysis, only 丙 and 丁 can be liars (each in different scenarios), so option C is the answer. So the answer is C.Final Answerboxed{C}</think>Given the problem, four people (甲, 乙, 丙, 丁) answered a multiple-choice question with options A, B, C, D. Each person chose a different option, and only one person is lying. We need to determine who could be the liar.The statements are:- 甲: I choose A- 乙: I choose one of B, D, C- 丙: I choose C- 丁: I choose DAnalyzing each case where one person is lying:1. Case 甲 is lying: If 甲 didn't choose A, the remaining options must be A, B, C, D. However, if the others are truthful, 乙, 丙, 丁 would choose B, C, D respectively, leaving A for 甲, contradicting the lie. Thus, 甲 cannot be lying.2. Case 乙 is lying: If 乙 didn't choose B, D, or C, they must choose A. This conflicts with 甲's choice of A, so 乙 cannot be lying.3. Case 丙 is lying: If 丙 didn't choose C, the remaining options are A, B, D. If 甲, 乙, and 丁 are truthful, 乙 could choose B or C. If 乙 chooses C, 丙 must choose B. This scenario works with all options covered (A, C, D, B), making 丙 the liar.4. Case 丁 is lying: If 丁 didn't choose D, the remaining options are A, B, C. If 甲, 乙, and 丙 are truthful, 乙 could choose D. However, 丁 must choose B. This scenario works with all options covered (A, D, C, B), making 丁 the liar.Both 丙 and 丁 could individually be the liar in different scenarios. Therefore, the answer is the pair 丙, 丁.boxed{C}]

🔑:To solve this problem, we need to analyze the statements made by each person and determine the scenarios under which only one person is lying, given that each person chose a different option.1. Assuming 甲 is lying: - If 甲 is lying about choosing A, then the remaining choices for 乙, 丙, and 丁 are B, C, and D, respectively, based on their statements. - However, 丙 says he chooses C and 丁 says he chooses D, which means 乙 would have to choose B. - This scenario does not contradict the condition directly, but since 甲 did not choose A, and all others chose as they said, there's no option left for 甲, contradicting the premise that each chose a different option. Thus, 甲 cannot be the one lying, ruling out options A and D.2. Assuming 乙 is lying: - If 乙 is lying, then he cannot choose one of B, D, or C. Given the constraint that each person must choose a different option and 甲 chooses A, 乙's lie implies he might choose A. - However, since 甲 already chooses A, and 乙 cannot choose A again, this creates a contradiction, meaning our assumption that 乙 is lying does not hold up without conflict. Therefore, 乙 cannot be the one lying, ruling out option B.3. Considering the remaining options (C and D): - If 甲 chooses A, 乙 has the options B, D, or C. Since 丙 and 丁 claim C and D, respectively, let's examine the possibilities: - If 乙 chooses C, then 丙 must be lying since he claimed C but cannot have it, making 乙's statement true. This means 丙 could choose B, and 丁's choice of D stands. This scenario satisfies all conditions with 丙 as the liar. - If 乙 chooses D, then 丁 must be lying since he claimed D but cannot have it, making 乙's statement true. This means 丁 could choose B, and 丙's choice of C stands. This scenario satisfies all conditions with 丁 as the liar. - Therefore, the only possible liars, given the constraints, are 丙 and 丁.Given the analysis, the correct answer is option C, which suggests that the possible liars are 丙 and 丁.Therefore, the answer is: boxed{C}.

🧠:[<think>Okay, let me try to solve this problem step by step. So, the question is about finding the equation of the trajectory of a point P such that its distance to the point F(4, 0) is 1 less than its distance to the line x + 5 = 0. The options are given, so maybe it's a parabola? Hmm.First, I remember that when dealing with distances from a point and a line, it's often related to the definition of a parabola. A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). But in this case, the distance to the point is 1 less than the distance to the line, so it might not be a standard parabola. Maybe it's a shifted one or has a different coefficient.Let me denote point P as (x, y). Then, the distance from P to F(4, 0) can be calculated using the distance formula. The distance from P to the line x + 5 = 0 is the horizontal distance since the line is vertical. Let me write that down.Distance from P(x, y) to F(4, 0): sqrt[(x - 4)^2 + (y - 0)^2] = sqrt[(x - 4)^2 + y^2]Distance from P(x, y) to the line x + 5 = 0: Since the line is vertical, the distance is |x - (-5)| = |x + 5|. Because the line x + 5 = 0 is equivalent to x = -5, so the horizontal distance from any point (x, y) to this line is |x - (-5)| = |x + 5|.According to the problem, the distance to F is 1 less than the distance to the line. So, the equation would be:sqrt[(x - 4)^2 + y^2] = |x + 5| - 1Wait, but I need to be careful here. The problem says "the distance from P to F is 1 less than its distance to the line". So, distance to F = distance to line - 1. So, mathematically:sqrt[(x - 4)^2 + y^2] = |x + 5| - 1Hmm, but absolute value equations can be tricky because of the cases. Let's consider that the right-hand side must be non-negative because it's a distance minus 1. So, |x + 5| - 1 ≥ 0 → |x + 5| ≥ 1 → x + 5 ≥ 1 or x + 5 ≤ -1 → x ≥ -4 or x ≤ -6. So, the equation is valid only when x is ≥ -4 or ≤ -6. But maybe after squaring both sides, this might resolve.Alternatively, maybe I can set up the equation and square both sides to eliminate the square root. Let's try that.Start with:sqrt[(x - 4)^2 + y^2] = |x + 5| - 1First, to handle the absolute value, perhaps I can square both sides. However, squaring when there's a subtraction can sometimes introduce extraneous solutions, so I need to check later. Let's proceed.Left side squared: (sqrt[(x - 4)^2 + y^2])^2 = (x - 4)^2 + y^2Right side squared: (|x + 5| - 1)^2 = (|x + 5|)^2 - 2|x + 5| + 1 = (x + 5)^2 - 2|x + 5| + 1Therefore, the equation becomes:(x - 4)^2 + y^2 = (x + 5)^2 - 2|x + 5| + 1Let me expand both sides:Left side: (x^2 - 8x + 16) + y^2Right side: (x^2 + 10x + 25) - 2|x + 5| + 1 = x^2 + 10x + 26 - 2|x + 5|So, putting it all together:x^2 - 8x + 16 + y^2 = x^2 + 10x + 26 - 2|x + 5|Subtract x^2 from both sides:-8x + 16 + y^2 = 10x + 26 - 2|x + 5|Bring all terms to the left side:-8x + 16 + y^2 - 10x - 26 + 2|x + 5| = 0Combine like terms:(-18x -10) + y^2 + 2|x + 5| = 0Hmm, so:y^2 = 18x + 10 - 2|x + 5|This seems a bit complicated because of the absolute value. Maybe I should consider cases based on the value of x + 5.Case 1: x + 5 ≥ 0 → x ≥ -5. Then, |x + 5| = x + 5. Plugging into the equation:y^2 = 18x + 10 - 2(x + 5) = 18x + 10 - 2x -10 = 16xSo, y^2 = 16x. But we need to check the domain for this case: x ≥ -5. However, earlier, when we considered |x + 5| ≥ 1, the equation is valid for x ≥ -4 or x ≤ -6. But in this case, x ≥ -5. So, overlapping region would be x ≥ -4 (since x ≥ -5 and x ≥ -4 is more restrictive). So, in this case, for x ≥ -4, the equation is y^2 = 16x.Case 2: x + 5 < 0 → x < -5. Then, |x + 5| = -(x + 5). Plugging into the equation:y^2 = 18x + 10 - 2*(-(x +5)) = 18x +10 + 2x +10 = 20x +20So, y^2 = 20x + 20. But in this case, x < -5. However, from the earlier condition, the original equation is valid only if |x + 5| ≥1, which in this case (x < -5) translates to -(x +5) ≥1 → -x -5 ≥1 → -x ≥6 → x ≤ -6. So, in this case, x ≤ -6. So, the equation here is y^2 = 20x +20 for x ≤ -6.But wait, the problem says "the equation of the trajectory of point P". So, the entire trajectory is the combination of these two cases? But the answer options are single equations. So, perhaps only one of these cases is valid? Or maybe I made a mistake in the algebra.Wait, let me check the problem again. The distance from P to F is 1 less than its distance to the line. So, distance to F = distance to line -1. So, the equation is sqrt[(x -4)^2 + y^2] = |x +5| -1. But in the case where |x +5| -1 is negative, the left side (distance) can't be negative, so those points don't exist. Hence, the equation is valid only when |x +5| -1 ≥0, which is |x +5| ≥1, which gives x ≥ -4 or x ≤ -6 as before.So, in the case x ≥ -4, we have y^2 =16x, and for x ≤ -6, y^2=20x +20. But the answer options are all single equations. So, maybe only one of these is the correct answer? Let's check the options. The options are A: y²= -16x, B: y²= -32x, C:y²=16x, D:y²=32x.So, in case 1 (x ≥ -4), the equation is y²=16x, which is option C. But the answer options include negative coefficients as well. However, the problem is that when x ≥ -4, the parabola y²=16x opens to the right with vertex at (0,0). But if x is allowed to be as low as -4, but the standard parabola y²=16x would have x ≥0. Wait, this is conflicting.Wait, in case 1, we derived y²=16x when x ≥ -4, but the standard parabola y²=16x is defined for x ≥0. So, how come here x can be as low as -4? Maybe there's a mistake in my derivation.Wait, let me check. So, starting with the equation:sqrt[(x -4)^2 + y^2] = |x +5| -1In case 1, where x ≥ -5 (so |x +5|=x +5), then the equation becomes sqrt[(x -4)^2 + y^2] = x +5 -1 = x +4.So, sqrt[(x -4)^2 + y^2] = x +4. But x +4 must be non-negative, so x +4 ≥0 → x ≥-4. So, in this case, x ≥-4.So, squaring both sides:(x -4)^2 + y^2 = (x +4)^2Expand both sides:Left: x² -8x +16 + y²Right: x² +8x +16Subtract x² and 16 from both sides:-8x + y² = 8xThus, y² = 16xSo, that's correct. So, in this case, when x ≥ -4, the equation is y²=16x. However, for the standard parabola y²=16x, the domain is x ≥0 because y² can't be negative. But here, since x can be as low as -4, how does that work? Wait, but in this case, we have x ≥-4, but substituting x=-4 into y²=16x would give y²= -64, which is impossible. Hmm, that's a problem. That suggests that perhaps my earlier reasoning is flawed.Wait, but when x ≥ -4, the right-hand side of the original equation is |x +5| -1 = x +5 -1 = x +4. But when x ≥ -4, x +4 ≥0, so the RHS is non-negative. Then, when we square both sides, we get y²=16x, but when x is negative (from -4 to 0), y²=16x would require y² to be negative, which is impossible. Therefore, in reality, the possible x values are x ≥0, because y² can't be negative. So, even though the original case was x ≥ -4, the resulting equation y²=16x only holds for x ≥0.But then, when x is between -4 and 0, y²=16x would give a negative right-hand side, which is impossible, so there are no points there. Therefore, the actual trajectory in this case is y²=16x with x ≥0.Similarly, in case 2 where x ≤ -6, we have the equation y²=20x +20. Let's see if that's possible. If x ≤ -6, then 20x +20 ≤20*(-6)+20= -120 +20= -100, so y² would be negative, which is impossible. Therefore, in this case, there are no real points.Wait, that can't be. If x ≤ -6, then 20x +20 is 20(x +1). If x ≤ -6, then x +1 ≤ -5, so 20(x +1) ≤ -100. Therefore, y² would be ≤ -100, which is impossible. Therefore, the case 2 gives no real solutions. Therefore, the only possible solutions are in case 1, where x ≥ -4, but due to the equation y²=16x, x must be ≥0. Therefore, the trajectory is y²=16x with x ≥0, which is the standard parabola opening to the right.But wait, the original problem states that the distance to F is 1 less than the distance to the line. Let me check with a point on y²=16x, say the vertex at (0,0). Distance from (0,0) to F(4,0) is 4. Distance from (0,0) to line x=-5 is 5. Then, 5 -1 =4, which matches. So, that's correct. Similarly, take another point on the parabola, say x=1, y=4 or y=-4. Distance to F(4,0): sqrt((1-4)^2 + (4)^2)=sqrt(9+16)=sqrt(25)=5. Distance to line x=-5: |1 +5|=6. Then, 6 -1=5, which matches. So, this works.But in the case x=0, y=0, works. What about x=16, y=16? Wait, y²=16x, so if x=16, y=±16. Distance to F(4,0): sqrt((16-4)^2 +16^2)=sqrt(144 +256)=sqrt(400)=20. Distance to line x=-5: |16 +5|=21. 21 -1=20, which matches. So, that works. So, the entire parabola y²=16x is valid, but when I derived it, the original case was x ≥-4, but because of the equation y²=16x, x must be non-negative, so x ≥0. Therefore, the points on the parabola y²=16x are all valid, and the case x ≤-6 gives no solution. Therefore, the equation is y²=16x, which is option C.But wait, the answer options include C:y²=16x and D:y²=32x. Let me check again. So, according to the problem statement, the correct answer should be C. But let me check with another point. Suppose a point on the parabola y²=16x, say (1,4). Distance to F(4,0) is sqrt((1-4)^2 +16)=sqrt(9+16)=5. Distance to the line x=-5 is 6. Then, 6 -1=5, which matches. So, yes, that works.But why is option C in the choices? Then, the answer should be C:y²=16x. But let me check the other case where x ≤ -6. Wait, if x ≤ -6, then the equation is y²=20x +20. Let's try x=-6. Then, y²=20*(-6) +20= -120 +20= -100, which is impossible. So, there's no solution in that case. Therefore, the only valid equation is y²=16x, which is option C.But wait, the answer given in the options is C:y²=16x. However, according to the options, A is y²=-16x, which opens to the left. But in our problem, the parabola opens to the right because the focus is at (4,0) and the directrix is x=-5, so the parabola should open towards the focus, which is on the right side. Wait, but in the standard parabola, the directrix is opposite the opening direction. For example, if the parabola opens to the right, the directrix is a vertical line to the left of the vertex.In our case, the focus is at (4,0), and the directrix is x=-5. The vertex should be midway between the focus and directrix. The distance between the focus (4,0) and directrix x=-5 is 4 - (-5)=9 units. The midpoint is at ( (-5 +4)/2, (0+0)/2 )= (-0.5, 0). Wait, that seems odd. Wait, the standard parabola's vertex is halfway between the focus and directrix. So, for a parabola opening to the right, the directrix is a vertical line to the left of the focus. The distance between focus and directrix is 4 - (-5)=9, so the vertex is 4 - 9/2=4 -4.5= -0.5. So, vertex at (-0.5, 0). Then, the standard equation would be (y - k)^2 = 4p(x - h), where (h,k) is the vertex, and p is the distance from vertex to focus. Here, p=4 - (-0.5)=4.5=9/2. So, the equation would be y^2=4*(9/2)(x +0.5)= 18(x +0.5). So, y^2=18x +9. Hmm, but that's not one of the options.Wait, this contradicts our previous result. So, maybe my initial approach is incorrect? Wait, according to the problem, the distance to the focus is 1 less than the distance to the directrix. However, in the standard parabola, the distance to the focus equals the distance to the directrix. Here, it's 1 less. So, this is not a standard parabola, but a conic section with some eccentricity?Wait, for a parabola, eccentricity e=1. If the distance to focus is less than the distance to directrix, then e <1, which would make it an ellipse. But here, the distance to focus is equal to distance to directrix minus 1. So, not a constant ratio. Wait, this might be a different conic. Hmm.Alternatively, maybe using the definition of a hyperbola? No, hyperbola is the difference of distances to two foci is constant. Hmm. Wait, but here it's the difference between the distance to a focus and the distance to a line. Not sure.Wait, let's try to approach this again. Let's denote d1 = distance from P(x,y) to F(4,0), and d2 = distance from P(x,y) to the line x=-5. The problem states that d1 = d2 -1.So, d1 +1 = d2. That is, the distance from P to F plus 1 equals the distance from P to the line. If I consider this, maybe it's easier to think of a moving point where the sum of the distance to a focus and a constant equals the distance to a line. But I'm not sure what conic that would be. Alternatively, maybe shifting coordinates.Alternatively, let's use coordinate geometry. Let's start with the equation sqrt[(x -4)^2 + y^2] = |x +5| -1. Let's square both sides again, keeping in mind the restrictions.So:sqrt[(x -4)^2 + y^2] = |x +5| -1Square both sides:(x -4)^2 + y^2 = (|x +5| -1)^2Expand the right-hand side:(|x +5|)^2 - 2|x +5| +1 = (x +5)^2 - 2|x +5| +1So:(x -4)^2 + y^2 = (x +5)^2 - 2|x +5| +1Expand (x -4)^2 and (x +5)^2:Left: x² -8x +16 + y²Right: x² +10x +25 -2|x +5| +1 = x² +10x +26 -2|x +5|Set equal:x² -8x +16 + y² = x² +10x +26 -2|x +5|Subtract x² from both sides:-8x +16 + y² = 10x +26 -2|x +5|Bring all terms to the left:-8x +16 + y² -10x -26 +2|x +5| =0Simplify:-18x -10 + y² +2|x +5|=0So:y² =18x +10 -2|x +5|This is the same equation as before. Then, we need to consider the cases for |x +5|.Case 1: x +5 ≥0 → x ≥ -5. Then |x +5|=x +5.So,y²=18x +10 -2(x +5)=18x +10 -2x -10=16xThus, y²=16x. But the domain here is x ≥-5. However, since we squared the equation, we need to ensure that in the original equation, sqrt[(x -4)^2 + y^2] = |x +5| -1, the right-hand side is non-negative. So, |x +5| -1 ≥0 → |x +5| ≥1.In case 1, x ≥-5, so |x +5| ≥1 → x +5 ≥1 → x ≥-4. Therefore, in case 1, the domain is x ≥-4. But the equation y²=16x requires x ≥0 for y² to be non-negative. Therefore, the overlap is x ≥0. So, the valid points are y²=16x with x ≥0, which is the standard parabola opening to the right.Case 2: x +5 <0 → x < -5. Then |x +5|= -x -5.So,y²=18x +10 -2*(-x -5)=18x +10 +2x +10=20x +20Thus, y²=20x +20. The domain here is x < -5. Also, the original equation requires |x +5| -1 ≥0 → |x +5| ≥1. Since x < -5, |x +5|= -x -5 ≥1 → -x -5 ≥1 → -x ≥6 → x ≤-6.Therefore, in case 2, the domain is x ≤-6. Now, check if y²=20x +20 is valid here. For x ≤-6, 20x +20 ≤20*(-6) +20= -120 +20= -100 <0. But y² can't be negative. Therefore, no solution exists in this case.Therefore, the only valid equation is y²=16x, which is option C. So, the answer should be C.But wait, in the standard parabola with focus at (4,0) and directrix x=-5, the vertex is halfway between the focus and directrix. The distance between focus and directrix is 4 - (-5)=9, so the vertex is at (-0.5,0). Then, the standard equation would be y²=4p(x - h), where p is the distance from vertex to focus, which is 4 - (-0.5)=4.5=9/2. So, the equation should be y²=4*(9/2)(x +0.5)=18(x +0.5)=18x +9. But according to our previous result, it's y²=16x. So, why the discrepancy?Because this is not a standard parabola. In the problem, the distance to the focus is 1 less than the distance to the directrix, not equal. So, it's not a standard parabola. Hence, my initial approach is correct. The equation derived is y²=16x, but this seems to contradict the standard parabola formula.Wait, but when we checked specific points, like (0,0), (1,4), (16,16), it worked. The vertex at (0,0) is 4 units from the focus (4,0) and 5 units from the directrix x=-5. Then, 5 -1=4, which satisfies the condition. However, according to the standard parabola formula, the vertex should be halfway between the focus and directrix, but here, it's shifted. Wait, perhaps this is a different kind of conic.Alternatively, maybe there's a mistake in assuming that the equation y²=16x is valid for all x ≥0. But according to the derivation, after considering the domain restrictions, the only valid solutions are points on y²=16x where x ≥0. However, in reality, the standard parabola with focus (4,0) and directrix x=-5 would have a different equation. But in this problem, since the relation is not the standard definition (distance to focus equals distance to directrix), but instead distance to focus is distance to directrix minus 1, it's a different curve.But the calculation seems correct, and the test points work. So, perhaps the answer is indeed C:y²=16x.But let's check another point. Suppose x=4, then y²=16*4=64, so y=±8. Distance to F(4,0) is sqrt((4-4)^2 +8^2)=8. Distance to line x=-5 is |4 +5|=9. Then, 9 -1=8, which matches. So, that works.Another point: x=9, y=±12. Distance to F: sqrt((9-4)^2 +12^2)=sqrt(25+144)=13. Distance to line: |9 +5|=14. 14 -1=13. Correct.Therefore, the equation y²=16x satisfies the condition for all x ≥0, and the test points check out. So, even though the standard parabola with focus (4,0) and directrix x=-5 would have a different equation, in this specific problem, due to the "1 less" condition, the correct equation is y²=16x, which is option C.But wait, the answer provided in the options is C:y²=16x, so that must be the answer. So, I think the correct choice is C.Final Answerboxed{C}</think>Given the problem, we need to find the equation of the trajectory of point ( P ) such that the distance from ( P ) to point ( F(4, 0) ) is 1 less than its distance to the line ( x + 5 = 0 ).1. Distance Calculations: - The distance from ( P(x, y) ) to ( F(4, 0) ) is ( sqrt{(x - 4)^2 + y^2} ). - The distance from ( P(x, y) ) to the line ( x + 5 = 0 ) is ( |x + 5| ).2. Setting Up the Equation: - According to the problem, the distance to ( F ) is 1 less than the distance to the line: [ sqrt{(x - 4)^2 + y^2} = |x + 5| - 1 ]3. Squaring Both Sides: - Squaring both sides to eliminate the square root: [ (x - 4)^2 + y^2 = (|x + 5| - 1)^2 ] - Expanding both sides: [ (x - 4)^2 + y^2 = (x + 5)^2 - 2|x + 5| + 1 ]4. Simplifying the Equation: - Expanding and simplifying: [ x^2 - 8x + 16 + y^2 = x^2 + 10x + 26 - 2|x + 5| ] [ -18x - 10 + y^2 + 2|x + 5| = 0 ]5. Considering Cases: - Case 1: ( x + 5 geq 0 ) (i.e., ( x geq -5 )): - Here, ( |x + 5| = x + 5 ): [ y^2 = 16x ] - Valid for ( x geq -4 ) (since ( |x + 5| - 1 geq 0 )). - Case 2: ( x + 5 < 0 ) (i.e., ( x < -5 )): - Here, ( |x + 5| = -(x + 5) ): [ y^2 = 20x + 20 ] - This results in negative values for ( y^2 ) when ( x leq -6 ), which is impossible.6. Conclusion: - Only Case 1 provides valid solutions, leading to the equation ( y^2 = 16x ).Thus, the equation of the trajectory of point ( P ) is (boxed{C}).,<think>Okay, let me try to figure out this problem step by step. So, the problem says: If the distance from point P to point F(4, 0) is 1 less than its distance to the line x + 5 = 0, then what is the equation of the trajectory of point P? The options are A to D, all of which are parabolas.First, I need to recall what this kind of problem is about. It seems like it's related to the definition of a conic section. Conic sections can be defined as the set of points where the ratio of the distance to a focus and the distance to a directrix is a constant, called the eccentricity. For a parabola, that ratio is 1. But here, the problem doesn't mention a ratio; instead, it says the distance to F is 1 less than the distance to the line x + 5 = 0. Hmm, so maybe it's not a parabola? Or maybe it is, but with some adjustment?Wait, let me parse the problem again. The distance from P to F(4,0) is 1 less than its distance to the line x + 5 = 0. So, mathematically, if we let P be (x, y), then the distance to F(4,0) is sqrt[(x - 4)^2 + (y)^2], and the distance to the line x + 5 = 0 is |x + 5| (since the line is vertical, the distance is just the horizontal difference). The problem states that the distance to F is 1 less than the distance to the line. So:sqrt[(x - 4)^2 + y^2] = |x + 5| - 1Wait, but distances can't be negative. So, |x + 5| - 1 must be non-negative. So, |x + 5| >= 1. That would mean that x + 5 >= 1 or x + 5 <= -1, so x >= -4 or x <= -6. But maybe we don't need to worry about that right now. Let's set up the equation first.So, sqrt[(x - 4)^2 + y^2] = |x + 5| - 1Hmm, okay. Let's square both sides to eliminate the square root. But before that, let's make sure that both sides are non-negative. Since the left side is a distance, it's always non-negative. The right side is |x + 5| - 1, which as we saw before must be non-negative for the equation to hold. So, when squaring both sides, we need to consider that.But maybe proceed step by step. Let's square both sides:[sqrt((x - 4)^2 + y^2)]^2 = (|x + 5| - 1)^2Simplify left side:(x - 4)^2 + y^2 = (|x + 5| - 1)^2Now, the right side can be expanded. Let's do that. Let me write |x +5| as sqrt((x +5)^2). But maybe it's easier to note that (|a| - b)^2 = a^2 - 2ab + b^2, but only when a is non-negative. Wait, but |x +5| is always non-negative, so (|x +5| -1)^2 = (x +5)^2 - 2*|x +5|*1 +1^2. Wait, but that's not exactly right because |x +5| squared is (x +5)^2. Wait, no, actually:Wait, let's take |x +5| as a variable, say, A. Then (A - 1)^2 = A^2 - 2A + 1. So substituting back, (|x +5| -1)^2 = |x +5|^2 - 2|x +5| +1. But |x +5|^2 is (x +5)^2, so:Right side becomes (x +5)^2 - 2|x +5| +1Therefore, the equation is:(x -4)^2 + y^2 = (x +5)^2 - 2|x +5| +1So, let's expand the left side and the right side:Left side: (x^2 - 8x +16) + y^2Right side: (x^2 +10x +25) -2|x +5| +1 = x^2 +10x +25 -2|x +5| +1 = x^2 +10x +26 -2|x +5|So, putting it all together:x^2 -8x +16 + y^2 = x^2 +10x +26 -2|x +5|Now, subtract x^2 from both sides:-8x +16 + y^2 = 10x +26 -2|x +5|Bring all terms to the left side:-8x +16 + y^2 -10x -26 +2|x +5| =0Simplify:(-18x -10) + y^2 +2|x +5| =0Hmm, so we have:y^2 =18x +10 -2|x +5|This seems complicated because of the absolute value. Let's consider different cases based on the value of x +5, which is inside the absolute value.Case 1: x +5 >=0, which means x >= -5In this case, |x +5| =x +5, so the equation becomes:y^2 =18x +10 -2(x +5) =18x +10 -2x -10 =16xSo, y^2=16xCase 2: x +5 <0, which means x < -5Here, |x +5|= -x -5, so the equation becomes:y^2=18x +10 -2(-x -5)=18x +10 +2x +10=20x +20So, y^2=20x +20But remember from the original problem statement, the distance from P to F(4,0) is 1 less than the distance to the line x +5=0. When x < -5, the point P is to the left of x=-5. Then, the distance to the line x=-5 is |x +5|= -x -5, and the distance to F(4,0) is sqrt[(x -4)^2 + y^2]. According to the problem, sqrt[(x -4)^2 + y^2] = |x +5| -1. However, if x < -5, then |x +5| -1 = (-x -5) -1 = -x -6. But sqrt(...) is non-negative, so -x -6 must be non-negative. Therefore, -x -6 >=0 => -x >=6 => x <= -6. So, in Case 2, x <= -6.But when we derived y^2=20x +20, for x <= -6, let's check if this is possible.If x <= -6, then 20x +20 <=20*(-6) +20= -120 +20= -100. But y^2 cannot be negative, so the equation y^2=20x +20 would have no real solutions when x <=-6 because the right-hand side would be <= -100. Therefore, there are no points in Case 2. So, only Case 1 is valid.Therefore, the equation is y^2=16x, which is option C. Wait, but let me check.Wait, in the options, C is y²=16x and D is y²=32x. Hmm. But according to this, the answer is C. However, let me check my steps again.Starting from the equation sqrt[(x -4)^2 + y^2] = |x +5| -1Then squaring both sides:(x -4)^2 + y^2 = (|x +5| -1)^2Then expanding both sides:Left side: x² -8x +16 + y²Right side: (x +5)^2 -2|x +5| +1 = x² +10x +25 -2|x +5| +1 =x² +10x +26 -2|x +5|So, left side minus right side: (x² -8x +16 + y²) - (x² +10x +26 -2|x +5|) = -18x -10 + y² +2|x +5| =0So, y²=18x +10 -2|x +5|Case 1: x >= -5, so |x +5|=x +5. Then,y²=18x +10 -2(x +5)=18x +10 -2x -10=16xCase 2: x < -5, then |x +5|= -x -5. Then,y²=18x +10 -2(-x -5)=18x +10 +2x +10=20x +20. But as mentioned, when x < -5, especially x <=-6, the RHS is negative, so no solution. But for x between -5 and -6, say x=-5.5, then 20*(-5.5) +20= -110 +20= -90, which is still negative. So no solutions here. Hence, only Case 1 is valid, so y²=16x. Which is option C.Wait, but let me check if there's an error in the calculation. Let's verify with the answer options. The options are:A: y²=-16xB: y²=-32xC: y²=16xD: y²=32xSo according to this, the correct answer is C. But let me verify with a sample point. For instance, take a point on y²=16x. Let's take x=1, then y²=16*1=16, so y=4 or y=-4. So point (1,4). Let's compute the distance to F(4,0): sqrt((1-4)^2 + (4-0)^2)=sqrt(9 +16)=sqrt(25)=5. Distance to the line x +5=0 is |1 +5|=6. Then, 5 is indeed 1 less than 6. So that checks out.Alternatively, take the vertex of the parabola y²=16x. The vertex is at (0,0). Distance to F(4,0) is 4, distance to line x=-5 is 5. 4 is 1 less than 5, which matches. So that seems correct.But wait, the options include both positive and negative coefficients. Since the parabola is opening to the right (since y²=16x is a right-opening parabola), so positive x direction. The focus is at (4,0), which is to the right of the vertex at (0,0), and the directrix should be x=-4 for a standard parabola y²=4ax, where a=4, so directrix x=-4. But in this problem, the line given is x=-5. Wait, that's inconsistent. Wait, perhaps my confusion here.Wait, in the standard parabola definition, the distance from a point to the focus equals the distance to the directrix. But here, the distance to the focus is 1 less than the distance to the line x=-5. So it's not a standard parabola. Wait, but according to our calculation, we arrived at y²=16x, which is a standard parabola with focus at (4,0) and directrix x=-4. But the line given in the problem is x=-5, not x=-4. So, seems conflicting.Wait, maybe I made a mistake here. Let's check again.Wait, in the problem statement, the distance to F(4,0) is 1 less than the distance to x=-5. So, in other words, for any point P on the trajectory, distance(P, F) = distance(P, line) -1.But in the standard parabola, distance(P, F) = distance(P, directrix). Here, it's different by 1. So, perhaps this is a different conic?Alternatively, maybe it's a parabola shifted or scaled. Let's check again.Wait, let's consider another approach. Let’s denote d1 = distance from P(x,y) to F(4,0) = sqrt[(x-4)^2 + y^2]d2 = distance from P(x,y) to line x=-5 is |x +5|Given that d1 = d2 -1So, sqrt[(x-4)^2 + y^2] = |x +5| -1We can rearrange this as sqrt[(x-4)^2 + y^2] +1 = |x +5|Then, square both sides:[sqrt((x-4)^2 + y^2) +1]^2 = (|x +5|)^2Left side: (sqrt((x-4)^2 + y^2))^2 + 2*sqrt((x-4)^2 + y^2) +1 = (x-4)^2 + y^2 + 2*sqrt((x-4)^2 + y^2) +1Right side: (x +5)^2So, equation becomes:(x-4)^2 + y^2 + 2*sqrt((x-4)^2 + y^2) +1 = (x +5)^2Simplify left and right sides:Left: x² -8x +16 + y² + 2*sqrt((x-4)^2 + y²) +1 =x² -8x + y² +17 + 2*sqrt(...)Right: x² +10x +25Bring all terms to the left:x² -8x + y² +17 + 2*sqrt((x-4)^2 + y²) -x² -10x -25 =0Simplify:(-18x -8) + y² +2*sqrt((x-4)^2 + y²)=0Wait, this seems more complicated. Perhaps my initial approach was better. Let's stick with the first method.Wait, but when we first squared both sides, we had:sqrt[(x-4)^2 + y^2] = |x +5| -1Then squaring gives:(x-4)^2 + y² = (|x +5| -1)^2 = |x +5|² -2|x +5| +1 = (x +5)^2 -2|x +5| +1Which leads to:x² -8x +16 + y² = x² +10x +25 -2|x +5| +1Simplify:-18x -10 + y² +2|x +5| =0Then, solving for y² gives:y²=18x +10 -2|x +5|Then, considering cases for x >=-5 and x < -5.For x >=-5, |x +5|=x +5:y²=18x +10 -2x -10=16xFor x < -5:y²=18x +10 -2*(-x -5)=18x +10 +2x +10=20x +20But in this case, x < -5, so 20x +20. Let's see when is 20x +20 >=0:20x +20 >=0 => x >= -1. But x < -5 here, so no solution. Therefore, only valid when x >=-5, leading to y²=16x.But wait, when x >=-5, the equation is y²=16x, which is a parabola opening to the right. However, the standard parabola y²=4ax has directrix x=-a. Here, comparing y²=16x to y²=4ax, we get 4a=16 => a=4. Therefore, the directrix should be x=-4. However, in our problem, the line given is x=-5, which is different.So, how does this reconcile? The problem says the distance to F(4,0) is 1 less than the distance to x=-5. But according to the parabola equation we derived, y²=16x, which has focus at (4,0) and directrix x=-4. Then, according to the standard parabola definition, any point on the parabola is equidistant to the focus and directrix. However, in our problem, the distance to the focus is 1 less than the distance to x=-5, which is different from the standard parabola.But wait, if we have a point on y²=16x, then the distance to F(4,0) equals the distance to directrix x=-4. However, in our problem, the distance to F(4,0) is 1 less than the distance to x=-5, which is further left. So, perhaps the trajectory is a parabola, but not the standard one with directrix x=-4. Wait, but according to our derivation, it is y²=16x. However, the line x=-5 is not the directrix here. There's a discrepancy.Wait, let's take the example point (1,4) on y²=16x. Distance to F(4,0) is 5, distance to x=-5 is 6. So 5 = 6 -1, which matches the problem statement. Similarly, the vertex at (0,0): distance to F(4,0) is 4, distance to x=-5 is 5. 4=5-1, which also matches. So even though x=-5 is not the directrix, the equation still holds. So even though it's not the standard definition, the equation y²=16x satisfies the condition that distance to F(4,0) is 1 less than distance to x=-5. Therefore, the correct answer is C.Alternatively, maybe there's a mistake in the problem options. But given the options, C is the only one that works. Let me check another point. Take x=4, then y²=16*4=64, so y=±8. Point (4,8). Distance to F(4,0) is sqrt(0 +64)=8. Distance to x=-5 is |4 +5|=9. 8=9-1, which works.Another point, x= -5. Wait, x >=-5 in the valid case, so x=-5 is allowed. Then y²=16*(-5)= -80, which is impossible. So x=-5 is not on the curve. Indeed, the parabola y²=16x is defined for x >=0. Wait, no, y²=16x is defined for x >=0, right? Because y²=16x implies x= y²/16, so x is non-negative. But according to our earlier calculation, when x >=-5, but in reality, since x must be such that y²=16x, x must be >=0.Wait, this is confusing. So how come when we had x >=-5 from the case analysis, but the parabola y²=16x only exists for x >=0? Because when we derived y²=16x, it was under the condition that x >=-5 and also that |x +5| -1 >=0. Wait, originally, the right side of the equation sqrt(...)=|x+5| -1 must be non-negative. Therefore, |x +5| -1 >=0 => |x +5| >=1 => x +5 >=1 or x +5 <=-1 => x >=-4 or x <=-6. But in Case 1, we considered x >=-5, but combining with |x +5| >=1, it's x >=-4. Therefore, in reality, the valid domain is x >=-4. But since the parabola y²=16x requires x >=0, there's a conflict here. Wait, no. Wait, in the equation y²=16x, x must be >=0 because y² is non-negative. Therefore, even though in our case analysis x >=-5, the equation y²=16x forces x >=0. Therefore, the actual domain is x >=0. But then, when x >=0, |x +5|=x +5, and we have sqrt((x-4)^2 + y²)= |x +5| -1. Let's check for x=0, y=0. Distance to F(4,0) is 4, distance to x=-5 is 5. 4=5-1, which is correct. For x=1, y=4: distance to F is 5, distance to line is 6: 5=6-1. Correct. For x=16, y=16: distance to F is sqrt((16-4)^2 +16^2)=sqrt(144 +256)=sqrt(400)=20. Distance to line x=-5 is 16 +5=21. 20=21-1. Correct. So all these points satisfy the condition. So even though the case analysis suggested x >=-5, the parabola equation y²=16x restricts x to be >=0, which is a subset of x >=-5, and in that region, the original equation holds. Therefore, the answer should be C:y²=16x.But let me check why the other options are there. For example, option D is y²=32x. If that were the case, then take x=2, y²=64, y=8. Distance to F(4,0): sqrt((2-4)^2 +8^2)=sqrt(4 +64)=sqrt(68)=2*sqrt(17)≈8.246. Distance to line x=-5 is |2 +5|=7. Then 8.246 is not 7 -1=6. So that's not correct. Similarly, option A:y²=-16x. That's a left-opening parabola. Take x=-1, y²=16, y=4. Distance to F(4,0): sqrt(25 +16)=sqrt(41)≈6.403. Distance to line x=-5 is |-1 +5|=4. Then 6.403 is not 4 -1=3. Doesn't work. Option B:y²=-32x. Left-opening. Take x=-1, y²=32, y≈5.656. Distance to F(4,0):sqrt(25 +32)=sqrt(57)≈7.55. Distance to line x=-5 is |-1 +5|=4. 7.55 vs 4 -1=3. Doesn't work. So indeed, only option C works.Therefore, the answer is C:y²=16x.Final Answerboxed{C}</think>Given the problem: If the distance from point ( P ) to point ( F(4, 0) ) is 1 less than its distance to the line ( x + 5 = 0 ), we need to find the equation of the trajectory of point ( P ).1. Define the distances: - The distance from ( P(x, y) ) to ( F(4, 0) ) is ( sqrt{(x - 4)^2 + y^2} ). - The distance from ( P(x, y) ) to the line ( x + 5 = 0 ) is ( |x + 5| ).2. Set up the equation: According to the problem, the distance to ( F ) is 1 less than the distance to the line: [ sqrt{(x - 4)^2 + y^2} = |x + 5| - 1 ]3. Square both sides to eliminate the square root: [ (x - 4)^2 + y^2 = (|x + 5| - 1)^2 ]4. Expand both sides: - Left side: ( (x - 4)^2 + y^2 = x^2 - 8x + 16 + y^2 ) - Right side: ( (|x + 5| - 1)^2 = (x + 5)^2 - 2|x + 5| + 1 )5. Simplify the equation: [ x^2 - 8x + 16 + y^2 = x^2 + 10x + 26 - 2|x + 5| ] [ -18x - 10 + y^2 + 2|x + 5| = 0 ] [ y^2 = 18x + 10 - 2|x + 5| ]6. Consider cases for ( x ): - Case 1: ( x geq -5 ): [ |x + 5| = x + 5 implies y^2 = 18x + 10 - 2(x + 5) = 16x ] - Case 2: ( x < -5 ): [ |x + 5| = -x - 5 implies y^2 = 18x + 10 + 2x + 10 = 20x + 20 ] However, this results in negative values for ( y^2 ) when ( x < -5 ), which is impossible.7. Conclusion: Only Case 1 is valid, leading to the equation ( y^2 = 16x ).The correct answer is (boxed{C}).]

🔑:Since the distance from point P to point (4, 0) is 1 less than its distance to the line x+5=0, move the line x+5=0 one unit to the right to get the line x+4=0, i.e., x=-4. Thus, the distance from point P to the line x=-4 is equal to its distance to point (4, 0). According to the definition of a parabola, the trajectory of point P is a parabola with focus at point (4, 0) and directrix x=-4. Let the equation of the parabola be y^2=2px, then frac{p}{2}=4, so 2p=16, Therefore, the standard equation of the parabola is y^2=16x, which is the trajectory equation of point P. Hence, the correct choice is: boxed{text{C}} Based on the problem, the distance from point P to the line x=-4 is equal to its distance to point (4, 0). By the definition and standard equation of a parabola, it is not difficult to derive the trajectory equation of point P. This problem, which asks for the trajectory equation of a moving point whose distance to a fixed line is 1 more than its distance to a fixed point, focuses on the definition and standard equation of a parabola and the method of finding the trajectory of a moving point, and is considered a basic question.

🧠:[<think>Okay, so I need to solve this problem where increasing the number of sides of a regular polygon by 4 results in an increase of 94 diagonals. I have to find the original number of diagonals. Hmm, let's start by recalling some formulas about polygons and diagonals.First, I remember that the number of diagonals in a regular polygon with n sides is given by the formula: [ frac{n(n - 3)}{2} ]Let me verify that. For each vertex in the polygon, you can draw a diagonal to every other vertex except itself and its two adjacent vertices. So that's n - 3 diagonals per vertex. Since each diagonal is counted twice (once from each end), we divide by 2. Yeah, that makes sense. So the formula is correct.Now, the problem states that if we increase the number of sides by 4, the number of diagonals increases by 94. Let me set up the equations for this. Let the original number of sides be n. Then the new number of sides after the increase is n + 4. The original number of diagonals is:[ frac{n(n - 3)}{2} ]And the new number of diagonals after increasing the sides is:[ frac{(n + 4)(n + 4 - 3)}{2} = frac{(n + 4)(n + 1)}{2} ]According to the problem, the difference between these two quantities is 94. So:[ frac{(n + 4)(n + 1)}{2} - frac{n(n - 3)}{2} = 94 ]Let me simplify this equation step by step. First, since both terms have the same denominator, I can subtract the numerators directly:[ frac{(n + 4)(n + 1) - n(n - 3)}{2} = 94 ]Multiply both sides by 2 to eliminate the denominator:[ (n + 4)(n + 1) - n(n - 3) = 188 ]Now expand each product:First, expand (n + 4)(n + 1):[ n cdot n + n cdot 1 + 4 cdot n + 4 cdot 1 = n^2 + n + 4n + 4 = n^2 + 5n + 4 ]Next, expand n(n - 3):[ n cdot n - n cdot 3 = n^2 - 3n ]Now subtract the second expansion from the first:[ (n^2 + 5n + 4) - (n^2 - 3n) ]Distribute the negative sign:[ n^2 + 5n + 4 - n^2 + 3n ]Combine like terms:n^2 - n^2 = 05n + 3n = 8nSo we have:8n + 4 = 188Subtract 4 from both sides:8n = 184Divide both sides by 8:n = 23Wait, so the original polygon has 23 sides. Now I need to find the number of diagonals it had before the increase. Let's compute that using the original formula:Number of diagonals = (frac{23(23 - 3)}{2} = frac{23 times 20}{2} = frac{460}{2} = 230)So the original number of diagonals is 230. Let me check if increasing the sides by 4 (to 27 sides) gives an increase of 94 diagonals.Number of diagonals for 27 sides:(frac{27 times (27 - 3)}{2} = frac{27 times 24}{2} = frac{648}{2} = 324)Difference: 324 - 230 = 94. Yep, that checks out.Wait, let me just make sure I didn't make a calculation error. Let me compute 23*20. 23*20 is 460, divided by 2 is 230. Then 27*24 is 648, divided by 2 is 324. The difference is 94. That's correct. So n = 23 is the correct original number of sides, leading to 230 diagonals. Therefore, the answer should be 230.But let me go back through the steps again to make sure. So the equations were set up correctly. The formula for diagonals is right. The expansion of the products was done correctly. Let's confirm that part again:(n + 4)(n + 1) = n^2 + 5n + 4n(n - 3) = n^2 - 3nSubtracting: (n^2 + 5n + 4) - (n^2 - 3n) = 8n + 4 = 188Then 8n = 184 => n = 23. Correct.So all steps check out. The answer is 230.Final AnswerThe number of diagonals of the polygon before the increase is boxed{230}.</think>Let ( n ) be the original number of sides of the regular polygon. The number of diagonals in a polygon with ( n ) sides is given by the formula:[ frac{n(n - 3)}{2} ]When the number of sides is increased by 4, the new number of sides becomes ( n + 4 ). The number of diagonals for the new polygon is:[ frac{(n + 4)(n + 1)}{2} ]According to the problem, the increase in the number of diagonals is 94. Therefore, we set up the equation:[ frac{(n + 4)(n + 1)}{2} - frac{n(n - 3)}{2} = 94 ]Multiplying both sides by 2 to clear the denominator:[ (n + 4)(n + 1) - n(n - 3) = 188 ]Expanding and simplifying both sides:[ (n^2 + 5n + 4) - (n^2 - 3n) = 188 ][ n^2 + 5n + 4 - n^2 + 3n = 188 ][ 8n + 4 = 188 ]Solving for ( n ):[ 8n = 184 ][ n = 23 ]The original number of sides is 23. The number of diagonals in the original polygon is:[ frac{23(23 - 3)}{2} = frac{23 times 20}{2} = frac{460}{2} = 230 ]Thus, the number of diagonals of the polygon before the increase is (boxed{230}).,<think>Okay, let's see. The problem says that if we increase the number of sides of a regular polygon by 4, the number of diagonals increases by 94. We need to find the original number of diagonals before the sides were increased. Hmm, right. So first, I need to recall the formula for the number of diagonals in a polygon. Let me think. I remember that for a polygon with n sides, the number of diagonals is given by n(n - 3)/2. Let me verify that. Each vertex connects to n - 3 other vertices via diagonals (excluding itself and its two adjacent vertices), and since each diagonal is counted twice (once from each end), we divide by 2. Yeah, that sounds right. So the formula is (n(n - 3))/2. Okay.So, the original polygon has n sides, and after increasing by 4, it becomes n + 4 sides. The number of diagonals increases by 94. Therefore, the number of diagonals for the original polygon is D = n(n - 3)/2, and after the increase, it's D + 94 = (n + 4)(n + 4 - 3)/2. Let me write that equation out.Original diagonals: D = n(n - 3)/2After increase: D + 94 = (n + 4)(n + 1)/2So, substituting D into the second equation:n(n - 3)/2 + 94 = (n + 4)(n + 1)/2Alright, now I need to solve for n. Let me multiply both sides by 2 to eliminate the denominators:n(n - 3) + 188 = (n + 4)(n + 1)Expanding both sides:Left side: n² - 3n + 188Right side: (n + 4)(n + 1) = n² + 5n + 4So, setting them equal:n² - 3n + 188 = n² + 5n + 4Hmm, subtract n² from both sides:-3n + 188 = 5n + 4Now, let's bring all terms to one side. Subtract 5n and subtract 4 from both sides:-3n - 5n + 188 - 4 = 0Which simplifies to:-8n + 184 = 0Solving for n:-8n = -184Divide both sides by -8:n = (-184)/(-8) = 23So, the original polygon had n = 23 sides. Now, we need to find the number of diagonals before the increase, which is D = 23(23 - 3)/2.Calculating that:23 * 20 = 460Divide by 2:460 / 2 = 230Therefore, the original number of diagonals is 230. Let me check if this makes sense. If we increase the sides by 4 to 27, then the number of diagonals would be 27*24/2 = 27*12 = 324. Then, 324 - 230 = 94, which matches the problem statement. So that checks out.Wait, let me confirm each step again. Starting with n = 23. Original diagonals: 23*20/2 = 230. After adding 4 sides: 27 sides. Diagonals: 27*24/2 = 324. Difference: 324 - 230 = 94. Yep, that's exactly what the problem said. So the answer is 230. I think that's solid. Let me just recap the steps once more to ensure I didn't skip anything. The key was setting up the equation correctly by expressing the number of diagonals before and after, then forming the equation based on the given difference. Then solving for n and plugging back in. All steps check out. Yeah, seems right.Final AnswerThe number of diagonals of the polygon before the increase is boxed{230}.</think>Given a regular polygon with ( n ) sides, the number of diagonals is given by the formula ( frac{n(n - 3)}{2} ). When the number of sides is increased by 4, the number of diagonals increases by 94. We need to determine the original number of diagonals.1. Original number of diagonals: ( frac{n(n - 3)}{2} )2. Number of diagonals after increasing sides by 4: ( frac{(n + 4)(n + 1)}{2} )3. Setting up the equation: [ frac{n(n - 3)}{2} + 94 = frac{(n + 4)(n + 1)}{2} ]4. Multiplying both sides by 2 to clear the denominators: [ n(n - 3) + 188 = (n + 4)(n + 1) ]5. Expanding both sides: [ n^2 - 3n + 188 = n^2 + 5n + 4 ]6. Simplifying by subtracting ( n^2 ) from both sides: [ -3n + 188 = 5n + 4 ]7. Solving for ( n ): [ -8n + 184 = 0 implies 8n = 184 implies n = 23 ]8. Calculating the original number of diagonals with ( n = 23 ): [ frac{23(23 - 3)}{2} = frac{23 times 20}{2} = 230 ]Verification:- Number of diagonals with ( 27 ) sides (after increase): ( frac{27 times 24}{2} = 324 )- Difference: ( 324 - 230 = 94 ), which matches the problem statement.Thus, the number of diagonals of the polygon before the increase is (boxed{230}).]

🔑:## Solution.Let n be the number of sides of the polygon, and d be the number of its diagonals. The formula for calculating the number of diagonals of a regular polygon is d=frac{n(n-3)}{2}.According to the condition of the problem, we get the equation:frac{n(n-3)}{2}+94=frac{(n+4)(n+1)}{2}After rearranging, we have in sequence:begin{aligned}n(n-3)+188 & =(n+4)(n+1) n^{2}-3 n+188 & =n^{2}+5 n+4 8 n & =184end{aligned}It follows that n=23.The total number of diagonals is d=frac{23 cdot(23-3)}{2}=230.

🧠:[<think>Okay, so I have this problem here where I need to calculate the sum of k times p_k from k=0 to n, where p_k is the number of permutations of the set {1, ..., n} with exactly k fixed points. Hmm, fixed points in permutations are also called "fixed elements" or "fixed numbers," right? Like, if a permutation maps a number to itself, that's a fixed point. So, for example, in the permutation (1,3,2) of the set {1,2,3}, the number 1 is a fixed point because it stays in the first position, while 2 and 3 are swapped. So p_k counts how many permutations have exactly k fixed points.Alright, the question is asking for the sum of k multiplied by p_k for all k from 0 to n. Let me write that down to make it clearer:Sum = Σ (k * p_k) from k=0 to n.I need to figure out what this sum equals. Let me recall some concepts related to permutations and fixed points. There's something called derangements, which are permutations with no fixed points, right? So derangements would be p_0 in this notation.But how does that help here? Hmm. Maybe there's a formula related to the number of fixed points across all permutations. Wait, maybe this sum is related to the expected number of fixed points in a random permutation?Wait, actually, if you consider all permutations of n elements, each permutation contributes its number of fixed points to this sum. So the sum Σ k * p_k is equal to the total number of fixed points across all permutations. Because for each k, there are p_k permutations each contributing k fixed points, so multiplying k by p_k and summing over k gives the total fixed points.So maybe the answer is equal to the total number of fixed points across all permutations of n elements. But how do I compute that?Alternatively, maybe we can compute this sum by linearity of expectation. If we consider the expected number of fixed points in a permutation, that's equal to the average number of fixed points, which is Σ k * p_k divided by n! (since there are n! total permutations). If I recall correctly, the expected number of fixed points in a random permutation is 1. Is that right?Wait, let me verify. The expectation E[X] where X is the number of fixed points can be computed by the linearity of expectation. For each element i in {1, ..., n}, let X_i be an indicator random variable that is 1 if the permutation fixes i, and 0 otherwise. Then X = X_1 + X_2 + ... + X_n. Then, E[X] = E[X_1] + E[X_2] + ... + E[X_n].Now, E[X_i] is the probability that a random permutation fixes i. For any i, the probability that a permutation fixes i is 1/n, since there are (n-1)! permutations that fix i (you fix i and permute the remaining n-1 elements), and (n-1)! / n! = 1/n. Therefore, each E[X_i] = 1/n, so the expectation E[X] = n * (1/n) = 1.Therefore, the expected number of fixed points is 1. But expectation is Σ k * (p_k / n!), so Σ k * p_k / n! = 1. Therefore, Σ k * p_k = n! * 1 = n!.Wait, so does that mean the sum is equal to n!? That seems too straightforward, but let me check with a small n.Take n=1. Then the set {1} has only one permutation, which has 1 fixed point. So p_0 = 0, p_1 = 1. The sum is 0*0 + 1*1 = 1, and 1! = 1. So that works.Take n=2. The permutations are (1,2) and (2,1). The first has 2 fixed points, the second has 0. So p_0 =1, p_2=1. The sum is 0*1 + 2*1 = 2. And 2! = 2. So that works.n=3. Let's list all permutations:1. (1,2,3) - 3 fixed points2. (1,3,2) - 1 fixed point3. (2,1,3) - 1 fixed point4. (2,3,1) - 0 fixed points5. (3,1,2) - 0 fixed points6. (3,2,1) - 1 fixed pointWait, wait, actually, let's check:Wait, no. For n=3:1. (1,2,3): fixed points are 1,2,3 → 3 fixed points.2. (1,3,2): fixed point is 1 → 1 fixed point.3. (2,1,3): fixed point is 3 → 1 fixed point.4. (2,3,1): no fixed points.5. (3,1,2): no fixed points.6. (3,2,1): fixed point is 2 → 1 fixed point.So p_0 = 2 (permutations 4 and 5), p_1 = 3 (permutations 2,3,6), p_3 =1 (permutation 1). So the sum is 0*2 +1*3 +3*1= 3 +3=6. Which is 3! =6. So that works.So for n=1,2,3, the sum Σ k p_k equals n!. Therefore, perhaps this is a general formula. So the answer should be n!.But wait, let me make sure. So according to the expectation approach, since E[X] =1, then total fixed points across all permutations is E[X] * n! =1 *n! =n!.Therefore, the sum is n!.But let me think if there's another way to get this result, just to confirm.Alternative approach: For each element i in {1,...,n}, count how many permutations fix i. Then sum over all i.Since the total number of fixed points across all permutations is equal to the number of permutations that fix 1, plus the number that fix 2, ..., plus the number that fix n. But for each i, the number of permutations that fix i is (n-1)! (since you fix i and permute the rest). Therefore, total fixed points would be n*(n-1)! =n!.Hence, the sum Σ k p_k is equal to n!.Therefore, regardless of the approach, the answer is n!.But let me check with n=4 to be thorough. Maybe this will help solidify.For n=4, total permutations are 24. Let's compute total fixed points.But computing this manually would be tedious, but maybe we can use the formula. If the total fixed points is 4! =24. Let's see.Alternatively, we can compute Σ k p_k.But what is p_k for n=4?p_0 is the number of derangements of 4 elements. Derangements for n=4 is 9.p_1: number of permutations with exactly 1 fixed point. To compute this, we choose 1 element to fix, and derange the remaining 3. So C(4,1)*D_3 =4*2=8. So p_1=8*2=8*?Wait, hold on, derangements for n=3 is 2. So p_1=4*2=8.p_2: number of permutations with exactly 2 fixed points. Choose 2 elements to fix, and derange the remaining 2. So C(4,2)*D_2=6*1=6. Because derangements for n=2 is 1.p_3: choose 3 elements to fix, derange the remaining 1. But derangements for n=1 is 0, because the only permutation of 1 element is itself, which is a fixed point. So C(4,3)*D_1=4*0=0.p_4: only 1 permutation, the identity, so 1.Therefore, p_0=9, p_1=8, p_2=6, p_3=0, p_4=1.Then Σ k p_k =0*9 +1*8 +2*6 +3*0 +4*1=0 +8 +12 +0 +4=24, which is 4! =24. So that works.Therefore, seems the formula holds. So regardless of n, the sum is n!.Hence, the answer is n!.Final AnswerThe sum is equal to boxed{n!}.</think>To calculate the sum (sum_{i=0}^{n} k p_{k}), where (p_k) is the number of permutations of the set ({1, ldots, n}) with exactly (k) fixed points, we can use the concept of expected value and linearity of expectation.1. Understanding Fixed Points in Permutations: - A fixed point in a permutation is an element that remains in its original position. - The number of permutations with exactly (k) fixed points is denoted by (p_k).2. Total Fixed Points Across All Permutations: - The sum (sum_{i=0}^{n} k p_{k}) represents the total number of fixed points across all permutations of the set ({1, ldots, n}).3. Using Linearity of Expectation: - Consider the expected number of fixed points in a random permutation. For each element (i) in the set, define an indicator random variable (X_i) which is 1 if (i) is a fixed point and 0 otherwise. - The expected number of fixed points (E[X]) is the sum of the expectations of these indicator variables: (E[X] = E[X_1] + E[X_2] + cdots + E[X_n]). - The probability that a permutation fixes any specific element (i) is (frac{1}{n}). Therefore, (E[X_i] = frac{1}{n}) for each (i). - Summing these expectations, (E[X] = n cdot frac{1}{n} = 1).4. Relating Expectation to Total Fixed Points: - The expected number of fixed points is the average number of fixed points per permutation, which is (frac{sum_{k=0}^{n} k p_k}{n!}). - Setting this equal to 1, we get (sum_{k=0}^{n} k p_k = n!).5. Verification with Examples: - For small values of (n) (such as (n = 1, 2, 3, 4)), manual verification confirms that the sum (sum_{k=0}^{n} k p_k) equals (n!).Thus, the sum (sum_{i=0}^{n} k p_{k}) is equal to (boxed{n!}).,<think>Okay, so I need to calculate the sum (sum_{k=0}^{n} k p_{k}), where (p_k) is the number of permutations of the set ({1, ldots, n}) that have exactly (k) fixed points. Hmm, fixed points in permutations are elements that stay in their original position, right? So a permutation with (k) fixed points means exactly (k) elements are in their original places, and the other (n - k) elements are deranged, i.e., not in their original positions. First, let me recall what (p_k) is. If I remember correctly, the number of permutations of (n) elements with exactly (k) fixed points is given by the formula (binom{n}{k} cdot D_{n - k}), where (D_{n - k}) is the number of derangements (permutations with no fixed points) of (n - k) elements. That makes sense because we choose (k) elements to fix, and then derange the remaining (n - k) elements. So, (p_k = binom{n}{k} D_{n - k}). Now, the sum we need is (sum_{k=0}^{n} k p_{k}). Let me substitute the expression for (p_k) into this sum:[sum_{k=0}^{n} k binom{n}{k} D_{n - k}]Hmm, I wonder if there's a smarter way to compute this sum rather than dealing directly with derangements. Let me think about the properties of permutations. Another approach might be to consider the linearity of expectation. Wait, maybe we can model this problem using expected value. Let me recall that the expected number of fixed points in a random permutation of (n) elements is 1. Is that true? Let me verify.If we define an indicator variable (X_i) for each element (i) in the permutation, where (X_i = 1) if the element is a fixed point and 0 otherwise. Then, the expected value (E[X_i]) is the probability that element (i) is a fixed point, which is (frac{1}{n}). Since there are (n) such variables, the expected total number of fixed points is (n cdot frac{1}{n} = 1). Yes, that's correct. So the expected number of fixed points is 1. But how does this relate to our sum? Wait, the expected value can also be calculated as the average over all permutations of the number of fixed points. Since there are (n!) total permutations, the average number of fixed points is (frac{1}{n!} sum_{k=0}^{n} k p_k). But we already know that this average is 1, so:[frac{1}{n!} sum_{k=0}^{n} k p_k = 1 implies sum_{k=0}^{n} k p_k = n!]Is that really the case? Let me check with small values of (n) to see if this holds. Take (n = 1). Then the permutations are just [1], which has 1 fixed point. So (p_1 = 1), and all other (p_k = 0). The sum is (1 cdot 1 = 1), and (1! = 1). So that works.For (n = 2). The permutations are [1,2] and [2,1]. The first has 2 fixed points, the second has 0. So (p_0 = 1), (p_2 = 1). The sum is (0 cdot 1 + 2 cdot 1 = 2). But (2! = 2), so that checks out.Wait, but according to the expected value, the average is 1, so total sum is (2! times 1 = 2), which matches. Another check: (n = 3). The permutations are:1. [1,2,3] – 3 fixed points2. [1,3,2] – 1 fixed point3. [2,1,3] – 1 fixed point4. [2,3,1] – 0 fixed points5. [3,1,2] – 0 fixed points6. [3,2,1] – 1 fixed pointSo (p_0 = 2), (p_1 = 3), (p_3 = 1). The sum is (0 cdot 2 + 1 cdot 3 + 3 cdot 1 = 6). And (3! = 6). That works too. So it seems that the sum (sum_{k=0}^{n} k p_k) is indeed equal to (n!). Therefore, the answer is (boxed{n!}). But let me try to derive this without relying on expectation, just to confirm. Maybe using combinatorial arguments. So, let's consider the sum (sum_{k=0}^{n} k p_k). Each term (k p_k) counts the total number of fixed points across all permutations with exactly (k) fixed points. Therefore, the entire sum is equal to the total number of fixed points in all permutations of (n) elements. But how many total fixed points are there across all permutations? Let's calculate that. For each element (i) from 1 to (n), we can count how many permutations fix (i). For a specific element (i), the number of permutations where (i) is fixed is ((n - 1)!), since the remaining (n - 1) elements can permute freely. Therefore, for each of the (n) elements, there are ((n - 1)!) permutations where that element is fixed. Thus, the total number of fixed points across all permutations is (n times (n - 1)! = n!). Therefore, the sum (sum_{k=0}^{n} k p_k = n!). This matches our earlier conclusion from the expectation approach. Another way to see this is by noting that when we sum over all permutations the number of fixed points, we are essentially counting the number of pairs ((f, sigma)) where (sigma) is a permutation and (f) is a fixed point of (sigma). Alternatively, using generating functions. Let me recall that the generating function for the number of permutations by fixed points is:[sum_{k=0}^{n} p_k x^k = sum_{k=0}^{n} binom{n}{k} D_{n - k} x^k]But maybe differentiating this generating function and evaluating at (x = 1) would give the sum (sum_{k=0}^{n} k p_k). Let's compute that. Let (G(x) = sum_{k=0}^{n} p_k x^k = sum_{k=0}^{n} binom{n}{k} D_{n - k} x^k). Then, the derivative (G'(x) = sum_{k=1}^{n} k p_k x^{k - 1}). Evaluating at (x = 1), we get (G'(1) = sum_{k=1}^{n} k p_k = sum_{k=0}^{n} k p_k). So, if we can compute (G'(1)), we can find the desired sum. But also, we know that (G(x) = sum_{k=0}^{n} binom{n}{k} D_{n - k} x^k). But the generating function for derangements is known. The exponential generating function for derangements is (e^{-x}/(1 - x)), but here we are dealing with ordinary generating functions. Alternatively, recall that (D_m = m! sum_{i=0}^m frac{(-1)^i}{i!}). But perhaps we can relate this generating function to something else. Alternatively, notice that the generating function (G(x)) can be rewritten as:[G(x) = sum_{k=0}^n binom{n}{k} D_{n - k} x^k = sum_{k=0}^n binom{n}{k} x^k cdot D_{n - k}]But maybe this is the same as the generating function for derangements multiplied by something. Alternatively, consider that (D_{n - k} = (n - k)! sum_{i=0}^{n - k} frac{(-1)^i}{i!}). Alternatively, think about the relation between derangements and permutations. For each permutation, the number of fixed points is the sum over each element being fixed or not. But perhaps this is getting too convoluted. Alternatively, recall that (G(1) = sum_{k=0}^n p_k = n!), since the total number of permutations is (n!). But if we take the derivative (G'(x)), which gives the sum (sum_{k=0}^n k p_k x^{k-1}), and evaluate at (x = 1), we get the total number of fixed points across all permutations, which we already know is (n!). Therefore, (G'(1) = n!), hence (sum_{k=0}^n k p_k = n!). So, all methods point to the same conclusion. Therefore, regardless of the approach—combinatorial counting, expectation, generating functions—the sum is equal to (n!). Final Answerboxed{n!}</think>To calculate the sum (sum_{k=0}^{n} k p_k), where (p_k) is the number of permutations of the set ({1, ldots, n}) with exactly (k) fixed points, we can use the concept of expected value in probability theory.1. Expected Value Approach: - The expected number of fixed points in a random permutation of (n) elements is 1. This is calculated by considering indicator variables (X_i) for each element (i), where (X_i = 1) if (i) is a fixed point and 0 otherwise. The expected value (E[X_i] = frac{1}{n}) for each (i), and summing over all (n) elements gives an expected total of (1). - The expected value can also be interpreted as the average number of fixed points per permutation. Since there are (n!) permutations, the total number of fixed points across all permutations is (n! times 1 = n!).2. Combinatorial Argument: - Each element (i) from 1 to (n) is fixed in ((n - 1)!) permutations. Summing over all (n) elements, the total number of fixed points is (n times (n - 1)! = n!).3. Generating Function Approach: - The generating function (G(x) = sum_{k=0}^{n} p_k x^k) has a derivative (G'(x)) which, when evaluated at (x = 1), gives the sum (sum_{k=0}^{n} k p_k). This derivative evaluates to (n!), confirming the result.Thus, the sum (sum_{k=0}^{n} k p_k) is equal to (n!).[boxed{n!}]]

🔑:It is about calculating the total number of fixed points for all permutations of {1, ldots, n}. There are (n-1)! permutations that fix 1, (n-1)! permutations that fix 2, etc. We deduce that the total number of fixed points is equal to n times (n-1)! = n!.

🧠:[<think>Okay, let me try to solve this problem step by step. So, we have α = -1910°, and we need to do two things with it. First, express α in the form β + k·360°, where k is an integer and β is between 0° and 360°, and determine which quadrant β is in. Second, find an angle θ that has the same terminal side as α but is between -720° and 0°. Alright, let's start with part (1).When angles are given in degrees, adding or subtracting multiples of 360° doesn't change the terminal side because a full circle is 360°. So, to express α = -1910° as β + k·360°, we need to find how many full 360° rotations are in -1910°, and then the remaining angle will be β. Since α is negative, we can think of it as rotating clockwise from the positive x-axis. To make it positive, we can add multiples of 360° until we get a value between 0° and 360°.But let's see, maybe an easier way is to compute β by taking -1910° modulo 360°, right? The modulo operation will give the equivalent angle. Let me check.So, to compute -1910° modulo 360°, first, divide 1910 by 360 to see how many times 360 goes into 1910. Let's do 360*5 = 1800, 360*5 = 1800, subtract 1800 from 1910: 1910 - 1800 = 110. So, 1910° is 5*360° + 110°, which would mean that -1910° is -5*360° -110°. But we need a positive β, so perhaps adding 6*360°? Let's see:-1910° + 6*360° = -1910 + 2160 = 250°. Hmm, so β would be 250°, and k would be -6? Wait, because -1910 = 250 + (-6)*360. Let me check: 250 - 6*360 = 250 - 2160 = -1910. Yes, that works. So, part (1) would be β = 250°, k = -6. But wait, the question says to write α in the form β + k·360° where β is between 0° and 360°, so β is 250°, k is -6. That seems correct.Alternatively, maybe I can use division. Let's compute how many times 360° fits into 1910°, but since it's negative, maybe we can compute how many times we need to add 360° to -1910° to get into 0° to 360°.But let me verify the modulo approach. The formula for β is α mod 360°, which is the remainder when α is divided by 360°, but since α is negative, different programming languages handle it differently, but in mathematics, modulo is usually defined such that the remainder is positive. So, to compute -1910° mod 360°, we can find the smallest non-negative integer such that -1910° + k·360° is in [0°, 360°).So, solving for k:0° ≤ -1910° + k·360° < 360°Adding 1910° to all parts:1910° ≤ k·360° < 1910° + 360° = 2270°Divide all parts by 360°:1910 / 360 ≈ 5.305555...2270 / 360 ≈ 6.305555...Therefore, k must be 6, since 5.305555... < k < 6.305555..., so k=6. Then β = -1910° + 6*360° = -1910 + 2160 = 250°. Yes, that's the same as before. So, β=250°, k=6. Wait, but in the problem statement, it says α = β + k·360°, so if we rearrange, β = α - k·360°, but here we have β = α + k·360°. So, maybe k is negative? Let's check.Wait, α = β + k·360°, so β = α - k·360°. Since α is negative, to make β positive, we need to add multiples of 360°, which is equivalent to subtracting negative multiples. So, in this case, β = -1910° + 6*360°, so α = β + (-6)*360°, so k would be -6? Wait, but in the problem statement, k is an integer, and β is between 0° and 360°, so whether k is positive or negative doesn't matter as long as β is in the correct range. The problem says "express α in the form β + k·360°", so α = β + k·360°, so here β is 250°, k is -6. Alternatively, since k can be any integer, there are multiple correct answers, but the problem specifies 0° ≤ β < 360°, so the standard way is to have k such that β is in that range. So, as per the calculation, β=250°, and k=-6. But I can also write 250° - 6*360° = -1910°, so that's correct.Now, which quadrant is 250° in? Let's recall the quadrants:- 0° to 90°: Quadrant I- 90° to 180°: Quadrant II- 180° to 270°: Quadrant III- 270° to 360°: Quadrant IVSo, 250° is between 180° and 270°, so that's Quadrant III. Wait, no: 250° is between 180° (which is III) and 270° (which is III), wait, 250° is between 180° and 270°, which is Quadrant III. Wait, 180° to 270° is Quadrant III, right? Because Quadrant I is 0-90, II is 90-180, III is 180-270, IV is 270-360. So, 250° is in Quadrant III. Wait, but 250° is 180° + 70°, so yes, in the third quadrant. So part (1) answer: β=250°, which is in Quadrant III.Moving on to part (2): Find θ such that the terminal side of θ is the same as α, and θ is between -720° and 0°. So, θ must be coterminal with α = -1910°, but θ must be in the range -720° ≤ θ < 0°. So, θ is a negative angle, more than -720°, but not less than that. Since coterminal angles differ by multiples of 360°, we need to add or subtract 360° enough times to bring θ into that range.Given that α = -1910°, we can add 360° multiple times until θ is within -720° to 0°. Let's see how many times we need to add 360° to -1910° to get into that range.First, let's calculate how many multiples of 360° are between -1910° and -720°. Let's compute the difference between -1910° and -720°: -1910 - (-720) = -1910 + 720 = -1190°. So, the difference is -1190°, which is equivalent to saying we need to add 360° enough times to cover 1190°.So, 1190 divided by 360 is approximately 3.305. So, we need to add 4 times 360°, which is 1440°, to -1910° to get θ. Let me check:-1910° + 4*360° = -1910 + 1440 = -470°. Hmm, -470° is still less than -720°? Wait, no: -470° is greater than -720°, right? Because -470 is to the right of -720 on the number line. Wait, but -470 is still less than 0°, so if we add 4*360°, θ = -470°, which is within -720° ≤ θ < 0°, because -720° ≤ -470° < 0°? Wait, no: -470° is greater than -720°, so it is within the desired range. Wait, but hold on:Wait, -720° is less than -470°, which is less than 0°, so -470° is in the interval. So, θ = -470°.But let me verify. Alternatively, maybe another multiple. Let me think.Alternatively, we can compute θ = α + k·360°, and find k such that θ is in [-720°, 0°). So:-720° ≤ -1910° + k·360° < 0°Let's solve for k:Adding 1910° to all parts:1910° - 720° ≤ k·360° < 1910° + 0°Which is:1190° ≤ k·360° < 1910°Divide all parts by 360°:1190 / 360 ≈ 3.305...1910 / 360 ≈ 5.305...Therefore, k must be integers 4, 5.Wait, let me check:If k=4: θ = -1910 + 4*360 = -1910 + 1440 = -470°, which is in the interval.If k=5: θ = -1910 + 5*360 = -1910 + 1800 = -110°, which is also in the interval.Wait, but the problem says "find θ such that...", so there might be multiple solutions. Wait, but the problem says "-720° ≤ θ < 0°", so θ can be -470° or -110°, but we need to check if both are in the range.Wait, -470° is between -720° and 0°, as -720 ≤ -470 < 0. Similarly, -110° is between -720 and 0. So, there are two possible solutions?But the problem says "Find θ", implying a single answer. Maybe the question expects the angle within that range, but there could be two. Wait, but let's check.Wait, coterminal angles repeat every 360°, so if θ is in -720° to 0°, which is a range of 720°, there can be two coterminal angles. Wait, but 720° is two full circles. So, for example, -1910°, when adding 360°, we can get -1550°, -1190°, -830°, -470°, -110°, etc. So, to get within -720° to 0°, we need to see which of these fall in that interval.Wait, -1910° + 5*360° = -1910 + 1800 = -110°, which is in the interval.But if we add 4*360°, we get -470°, which is also in the interval.Wait, so there are two angles in that interval: -470° and -110°. But the problem says "Find θ such that...", maybe both? But maybe the question expects the one closest to 0°, which is -110°, or perhaps the most straightforward one. Wait, let's check again.Wait, -1910° divided by 360° is approximately -5.3055... So, to get into the range of -720° (which is -2*360°) to 0°, we can add 6*360°, which is 2160°, to -1910°, giving 250°, but that's positive. Alternatively, adding 5*360°: 5*360=1800, so -1910 + 1800 = -110°, which is in the range. Alternatively, adding 4*360°: 1440, so -1910 + 1440 = -470°, which is also in the range. So both -470° and -110° are coterminal with -1910°, and in -720° to 0°.But the problem says "Find θ such that...", so maybe both? But the problem might expect the answer to be the one within that range. Wait, but sometimes such problems expect the angle closest to 0°, but let me check.Wait, but the problem says "-720° ≤ θ < 0°". Since θ is negative, we need to find all θ in that interval. However, θ must be coterminal with α. Since coterminal angles differ by 360°, there can be two angles in -720° to 0° that are coterminal with α. Let me check with examples.Given that α = -1910°, which is coterminal with 250°, as we saw. So, 250° - 360° = -110°, and -110° - 360° = -470°, and -470° - 360° = -830°, which is outside the range. So, within -720° to 0°, -470° and -110° are the two angles. But -720° is equal to -2*360°, so the interval from -720° to 0° includes two full circles. Therefore, there are two angles: -470° and -110°. Wait, but -470° is more than -720°, so it's -470°, which is greater than -720°, so it's in the interval. Similarly, -110° is also in the interval. So both are valid.But the problem says "Find θ", maybe both? Let me check the problem statement again. It says: "Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°." So, there can be multiple θs. But in the instructions, it says "put your final answer within boxed{}", which suggests a single answer. Hmm. Maybe the question expects the angle closest to zero, which would be -110°, or the most negative one within the range? Let me check with another approach.Alternatively, we can calculate how many times we need to add 360° to α to get into the desired range. Let's see:Starting with α = -1910°.Add 360°: -1910 + 360 = -1550° (still less than -720°)Add another 360°: -1550 + 360 = -1190° (still less than -720°)Add another 360°: -1190 + 360 = -830° (still less than -720°)Add another 360°: -830 + 360 = -470° (now within -720° to 0°)Add another 360°: -470 + 360 = -110° (still within the range)Add another 360°: -110 + 360 = 250° (outside the range)So, in the range -720° to 0°, θ can be -470° and -110°. However, the problem says "Find θ", maybe the one with the smallest absolute value? Or both? Wait, but the question is probably expecting one answer, so maybe the user made a mistake here. Wait, but let me check with modulo.Alternatively, compute θ = α modulo 720°, but since θ has to be negative, perhaps?Wait, θ needs to be between -720° and 0°, so we can take α and add 720° until it's in that range. Let's see.α = -1910°.Add 720°: -1910 + 720 = -1190° (still less than -720°)Add another 720°: -1190 + 720 = -470° (now between -720° and 0°)So θ = -470°, but also, we can add 360° to that to get another angle: -470 + 360 = -110°, which is also in the interval.So, the two angles are -470° and -110°. But if the problem expects both, maybe we need to list both? But the problem says "Find θ", so maybe the answer is -470° and -110°, but let's check if the question allows multiple answers or expects one. Since the original angle is -1910°, which is equivalent to 250°, the negative coterminal angles would be 250° - 360° = -110°, and -110° - 360° = -470°, and -470° - 360° = -830°, which is outside the range. So, within -720° to 0°, we have -470° and -110°. But if the problem expects both, but the instructions say to put the final answer in a box, maybe as a list? But maybe I should check with the calculation.Wait, perhaps there's an error here. Let me verify.Wait, -1910° divided by 360° is approximately -5.3055... So, the integer division would give -6, which is k in the first part. But when looking for θ between -720° and 0°, which is -2*360° to 0°, we can think of θ as α + k*360°, where k is chosen such that θ is in that interval. So, solving for k:-720° ≤ -1910° + k*360° < 0°Adding 1910°:1910° - 720° ≤ k*360° < 1910°1190° ≤ k*360° < 1910°Divide by 360°:≈3.3055 ≤ k < ≈5.3055Thus, integer k = 4, 5So for k=4:θ = -1910 + 4*360 = -1910 + 1440 = -470°For k=5:θ = -1910 + 5*360 = -1910 + 1800 = -110°Therefore, both θ = -470° and θ = -110° satisfy the condition. However, the problem says "Find θ", using the singular, which is a bit confusing. Maybe in some contexts, the principal value is expected, which would be the angle closest to zero, which is -110°, but I need to check the mathematical convention. Alternatively, maybe the question expects the angle within the range that has the same terminal side as α, which is equivalent to 250°, so expressing θ as a negative angle, you can subtract 360° until you get into the desired range.Wait, 250° - 360° = -110°, which is within -720° to 0°, and also -110° - 360° = -470°, which is also in the range. Therefore, both answers are correct. However, the problem might expect both answers. But looking back at the problem statement:"(2) Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°."Since the interval -720° to 0° is two full circles, there can be two angles coterminal with α. Therefore, both -470° and -110° are valid. However, the problem says "Find θ", not "Find all θ", so perhaps it's expecting both? But the user instruction says "put your final answer within boxed{}", which usually denotes a single answer. Maybe there's a mistake in my calculation?Wait, but let's check:-1910° + 5*360° = -1910 + 1800 = -110°-1910° + 4*360° = -470°But -470° is equal to -720° + 250°, right? Because 720° - 250° = 470°, so -720° + 250° = -470°. Wait, that might not be helpful.Alternatively, since θ must be in -720° to 0°, we can represent θ as α + 2*360°, which is -1910 + 720 = -1190°, which is outside the range. Wait, no, that's not. Wait, 2*360 is 720. So, -1910 + 720 = -1190, still less than -720. Then, adding another 360°, so 3*360°=1080, -1910 + 1080 = -830°, still less than -720. Then 4*360°=1440, -1910 + 1440 = -470°, which is in the range. Then 5*360°=1800, -1910 + 1800=-110°, which is in the range. Then 6*360=2160, which would be positive. So, yes, θ can be -470° and -110°.But the problem might want both answers. However, given the way the question is phrased, maybe it's expecting the most recent one, which is closer to zero, which is -110°, but I need to verify.Wait, the terminal side is the same, so if you have two angles in the given interval, both are correct. Since the problem didn't specify to find all possible θ, but just "Find θ", maybe either is acceptable? But the problem may have a unique answer. Wait, maybe there is a mistake here.Wait, in part (1), we found β = 250°, which is in Quadrant III. So, θ needs to be coterminal with 250°, but in the range -720° to 0°, so 250° - 360° = -110°, which is in the range. Then, -110° - 360° = -470°, which is also in the range. So both -110° and -470° are correct. Therefore, perhaps the answer is both, but given that the problem is from a math textbook, maybe the expected answer is -110°, because when you subtract 360° once from 250°, you get -110°, but subtracting again gives -470°, which is also acceptable. However, depending on how the problem is structured, sometimes they expect the angle within -360° to 0°, but here it's -720° to 0°, which allows two rotations. Therefore, both answers are correct. But since the problem says "Find θ", not "Find all θ", maybe there's a specific answer they are looking for. Let me check with calculations.Alternatively, since -1910° is the given angle, to find θ in -720° to 0°, we can compute how many times 360° fits into 1910°.But 1910 divided by 360 is 5.305... So, 5*360=1800, 1910-1800=110. So, -1910°= -5*360° -110°, which is = - (5*360 + 110)°. To get θ between -720° and 0°, we can add 6*360° to α: -1910 +6*360= 250°, which is positive. Alternatively, subtract multiples of 360° from 250°, which is positive, to get into the negative range.250° - 360°= -110°, which is in the range.-110° -360°= -470°, which is also in the range.So, θ can be -110° or -470°. Both correct. Therefore, since the problem allows θ in -720° to 0°, both answers are valid. But in some cases, maybe the question expects the larger angle (closer to 0°), which is -110°, but I need to check.Alternatively, perhaps the problem expects only one answer. Maybe my calculation is wrong. Let me check with another approach.If we want θ to be between -720° and 0°, then θ = α + k*360°, find k such that θ is in that interval.So, let's solve for k:-720 ≤ -1910 + 360k < 0Adding 1910 to all parts:1190 ≤ 360k < 1910Divide by 360:≈3.3055 ≤ k < ≈5.3055Thus, k=4,5.Therefore, θ = -1910 + 4*360= -470°, and θ= -1910 +5*360= -110°. So, both correct. Hence, the answer should be both, but since the problem uses "Find θ", maybe both are required. However, the user instruction says to put the final answer in a box. Maybe the problem expects both answers. So, perhaps writing both -470° and -110°?But in the problem statement, part (2) says "Find θ such that...", so maybe the answer is two angles. However, I need to check.Alternatively, perhaps the problem expects θ to be in the range -720° ≤ θ < 0°, and we can also note that θ can be expressed as β - 360°*n, where n is a positive integer, such that θ is in that range. Since β=250°, θ=250 - 360= -110°, which is in the range. θ=250 - 720= -470°, which is also in the range. θ=250 - 1080= -830°, which is outside. So, only -110° and -470° are in the range. Therefore, two answers. But since the problem is in Chinese, maybe the translation is accurate, but the user instruction here says to "find θ", so perhaps the answer is two angles. But the user might expect boxed{-470} and boxed{-110} or both in one box. But the original problem might have different expectations.Alternatively, maybe the problem has a typo, but given the information, I can only proceed with the two correct answers. However, since the original problem is in Chinese, and the user has translated it, maybe there's a misunderstanding. Alternatively, maybe in part (1), they need β and the quadrant, and part (2) θ. So, perhaps θ is -110°, but I need to check.Wait, perhaps the problem expects the answer to part (2) to be -470°, but let me see.If I consider that -720° is equivalent to 0° in terms of coterminal angles, but no. Wait, -720° is two full rotations clockwise. So, adding 720° to any angle brings it back. So, θ is in the range -720° to 0°, which is two full circles. So, any angle in that range is unique modulo 720°, but since coterminal angles are modulo 360°, there can be two angles in that interval. Therefore, both -470° and -110° are correct. Therefore, the answer should include both. However, given the problem's phrasing, perhaps they expect both answers. But given the instruction here, the assistant is to write the final answer in boxes. Maybe two separate boxed answers. But the user instruction says "put your final answer within boxed{}", implying a single box. Therefore, maybe there is a mistake in my thinking.Wait, let's recast the problem. Given α = -1910°, which is coterminal with 250°, which is in Quadrant III. The problem asks in part (2) for θ between -720° and 0° which is coterminal with α. So, θ = 250° - 360°*n, where n is a positive integer such that θ is in [-720°, 0°). Let's solve for n:250 - 360n ≥ -720250 + 720 ≥ 360n970 ≥ 360nn ≤ 970/360 ≈ 2.694, so n=1,2.Thus, θ = 250 - 360*1 = -110°, and θ=250 -360*2= -470°, both in the interval. So, yes, two answers. Therefore, the answer to part (2) is θ = -110° and θ = -470°, but given the problem's wording, maybe the answer is two angles. However, the user's initial problem statement may differ. If forced to pick one, perhaps -470°, but I need to confirm.Wait, if we consider that -1910° is equivalent to -1910° + 6*360°=250°, and to get into the range of -720° to 0°, we subtract 360° twice from 250°, getting -470°, which is equivalent to 250° - 720°, but that might not be necessary. Alternatively, the original angle is -1910°, so adding 5*360°=1800°, we get -110°, and adding 4*360°=1440°, we get -470°. Both correct. So, the answer is two angles: -470° and -110°. But since the problem says "Find θ", it's possible that they expect both. However, the user instructions here specify to put the final answer in a box. If the original problem is in Chinese, maybe the comma is indicating multiple answers. In Chinese, sometimes they use semicolons to separate answers. If the user expects two answers, perhaps we need to box both. But as per the instructions here, maybe each part's answer is boxed separately. Let me check the original problem structure:The problem has two parts, (1) and (2). For each part, we need to provide the answer. For part (1), we have β=250°, which is in Quadrant III, so we need to write β and the quadrant. For part (2), we need to find θ, which is two angles. But the user instructions say "put your final answer within boxed{}". In Chinese textbooks, sometimes answers are presented as comma-separated in one box. So, maybe for part (2), the answer is two angles: -470° and -110°, written in a box.But since the problem says "-720° ≤ θ < 0°", θ is in that range, and coterminal angles differ by 360°, so two solutions. Therefore, the answer for part (2) is θ = -470° and θ = -110°. However, since the problem might expect a single answer, perhaps there is a mistake in my previous steps.Wait, let's check again. Given that θ is in -720° ≤ θ < 0°, and θ is coterminal with α=-1910°, which is equivalent to β=250°, then θ can be 250° - 360°*2 = 250 - 720 = -470°, and 250° - 360°*1 = -110°. Both are in the range. So, two answers. Therefore, if the problem allows multiple answers, they should both be given. If not, maybe there's a different approach.Alternatively, maybe the problem expects θ to be expressed as a negative angle with the smallest magnitude, which is -110°, because -110° is closer to 0°, but I need to verify. However, in math problems like these, usually, all possible solutions within the given interval are required. So, given that, both -470° and -110° are correct. Therefore, the answer to part (2) is both angles. Since the user instruction mentions putting the final answer in boxes, maybe each part is boxed. So, part (1): β=250°, Quadrant III; part (2): θ=-470° and θ=-110°, both boxed.But given the initial problem statement by the user, it's possible that part (2) only expects one answer. Let me check once more with calculations.If we take α = -1910°, which is coterminal with 250°, then θ must be equal to 250° + k*360°, and in the range -720° ≤ θ < 0°. Solving 250° +k*360° >= -720°, and <0°:For θ = 250 + k*360,-720 ≤ 250 +k*360 < 0Subtract 250:-970 ≤ k*360 < -250Divide by 360:-970/360 ≈ -2.694 ≤ k < -250/360 ≈ -0.694Thus, integer k = -2, -1Thus, θ = 250 + (-2)*360 = 250 - 720 = -470°θ = 250 + (-1)*360 = 250 - 360 = -110°Thus, two solutions. Therefore, the problem's part (2) answer is both -470° and -110°, but the user may expect them listed.However, since the problem says "Find θ such that...", and θ is in that interval, but θ is not unique. So, maybe the answer is both. But in many standard problems, even if there are multiple answers, they might expect both. So, in conclusion:Part (1): β = 250°, Quadrant IIIPart (2): θ = -470° and θ = -110°But since the user instruction says "put your final answer within boxed{}", maybe each part is boxed. So, for part (1), boxed as 250° and Quadrant III, and for part (2), boxed as -470° and -110°, but formatting might be an issue.Alternatively, maybe list them as separate boxed answers. But since the user hasn't specified, and given the original problem's structure, the assistant should present both answers clearly.But in the original problem statement, in Chinese, maybe the answer for part (2) is a single angle. However, based on the mathematics, there are two correct answers. Therefore, in this case, I will provide both angles as the answer for part (2).Final Answer(1) boxed{250^circ} in Quadrant boxed{text{III}}; (2) boxed{-470^circ} and boxed{-110^circ}.</think>Given (alpha = -1910^circ):(1) To express (alpha) in the form (beta + k cdot 360^circ) where (0^circ leq beta < 360^circ), we calculate (beta) by finding the equivalent angle modulo (360^circ). [-1910^circ mod 360^circ = -1910^circ + 6 cdot 360^circ = -1910^circ + 2160^circ = 250^circ]Thus, (alpha = 250^circ + (-6) cdot 360^circ). The angle (250^circ) is in Quadrant III because it lies between (180^circ) and (270^circ).(2) To find (theta) such that the terminal side of (theta) is the same as (alpha) and (-720^circ leq theta < 0^circ), we add multiples of (360^circ) to (alpha) until it falls within the desired range. [-1910^circ + 4 cdot 360^circ = -1910^circ + 1440^circ = -470^circ][-1910^circ + 5 cdot 360^circ = -1910^circ + 1800^circ = -110^circ]Both (-470^circ) and (-110^circ) are within the range (-720^circ leq theta < 0^circ). Final Answer(1) (boxed{250^circ}) in Quadrant (boxed{text{III}}); (2) (boxed{-470^circ}) and (boxed{-110^circ}).,<think>Okay, so I need to solve these two parts about the angle α = -1910°. Let me start with part (1). The problem says to express α in the form β + k·360°, where k is an integer and β is between 0° and 360°. Then I have to figure out which quadrant β is in. Alright, let's start by understanding what this means.Negative angles in trigonometry mean rotation clockwise from the positive x-axis. So, -1910° would be rotating clockwise 1910 degrees. But angles are periodic every 360°, so I need to find how many full rotations (360°) are in 1910°, then find the remaining angle after subtracting those rotations. Since it's negative, adding multiples of 360° should bring it into the range [0°, 360°).Wait, but maybe a better approach is to add 360° repeatedly until the angle is positive and within 0° to 360°. Let me think. So, if I have α = -1910°, and I want to add 360° enough times until the result is between 0° and 360°. Let's calculate how many times 360° goes into 1910°, that'll give me the number of times I need to add 360° to make it positive.First, divide 1910 by 360 to find how many full circles that is. Let me do 1910 ÷ 360. Let's see, 360*5=1800, which is less than 1910, and 360*6=2160, which is more than 1910. So 5 full circles would be 1800°, and 1910 - 1800 = 110°. So, 1910° is equivalent to 5*360° + 110°, right? But since α is -1910°, that would be equivalent to rotating clockwise 1910°, which is the same as rotating counterclockwise -1910°. Hmm, maybe another way: to convert a negative angle to a positive one, we can add multiples of 360° until it's positive. So:Start with α = -1910°. Let's add 360° repeatedly. How many times do I need to add 360° to get into 0° to 360°? Let's compute:-1910° + 360° = -1550° -1550° + 360° = -1190° -1190° + 360° = -830° -830° + 360° = -470° -470° + 360° = -110° -110° + 360° = 250°Wait, so after adding 360° six times, we get 250°. Let me check: 6*360 = 2160, so -1910 + 2160 = 250°. Yes, that's right. So β is 250°, and k is 6? Wait, but the original problem says to write α = β + k·360°, with k ∈ ℤ and 0° ≤ β < 360°. But here, we have α = -1910°, which is equal to β - k·360°, right? Wait, perhaps I need to represent α as β plus k·360°, where β is between 0° and 360°, and k is an integer. So in this case, since α is negative, we need to find β such that β = α + k·360°, where k is chosen so that β is in [0°, 360°). So:β = α + k·360° So, β = -1910° + k·360° We need to find integer k such that 0° ≤ β < 360°.Let's solve for k:0° ≤ -1910° + k·360° < 360° Add 1910° to all parts:1910° ≤ k·360° < 1910° + 360° 1910° ≤ k·360° < 2270° Divide all parts by 360°:1910 / 360 ≈ 5.305555... 2270 / 360 ≈ 6.305555... So k must be an integer between approximately 5.3055 and 6.3055, so k=6.Therefore, β = -1910° + 6*360° = -1910° + 2160° = 250°, which is in [0°, 360°). So part (1) answer is β=250°, k=6, and then quadrant.Now, which quadrant is 250° in? Let's recall that quadrants are divided as follows:0° to 90°: Quadrant I 90° to 180°: Quadrant II 180° to 270°: Quadrant III 270° to 360°: Quadrant IV 250° is between 180° and 270°, so that's Quadrant III. Wait, 250° is 180° + 70°, so yes, in Quadrant III.Wait, but hold on, 250° is 180° to 270°, which is Quadrant III. Correct.So part (1) is β=250°, k=6, Quadrant III.Now part (2): Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°. So θ must be co-terminal with α, which is -1910°, but θ has to be between -720° and 0°. So θ = α + k·360°, find k such that θ is in that interval.Alternatively, since we already know that α is equivalent to 250°, which is the same as -110°, but that might not be directly helpful. Wait, let's think. θ must be co-terminal with α, so θ = α + k·360°, for some integer k. We need θ to be between -720° and 0°.So starting from α = -1910°, we can add 360° multiple times until θ is in the desired range. Let's compute how many times we need to add 360°.We have α = -1910°, and we want θ = -1910° + k·360° such that -720° ≤ θ < 0°.Let me solve for k:-720° ≤ -1910° + k·360° < 0° Add 1910° to all parts:1910° - 720° ≤ k·360° < 1910° + 0° 1190° ≤ k·360° < 1910° Divide by 360°:1190 / 360 ≈ 3.3055... 1910 / 360 ≈ 5.3055... So k must be integers between 3.3055 and 5.3055, so k=4 or k=5.Let me check for k=4:θ = -1910° + 4*360° = -1910° + 1440° = -470°But -470° is less than -720°? Wait, no. Wait, -470° is greater than -720°, right? Because -470 is to the right of -720 on the number line. So -720° ≤ -470° < 0°? Wait, -470° is between -720° and 0°, yes. So that's a valid θ. But wait, let me check:Wait, -1910° + 4*360° is -1910 + 1440 = -470°, which is between -720 and 0, right? Because -720 is less than -470, and -470 is less than 0. So θ = -470° is a possible answer.But maybe there's another θ with k=5:θ = -1910° + 5*360° = -1910 + 1800 = -110°Which is also between -720° and 0°, so that's another valid angle.Wait, so both k=4 and k=5 give θ in the desired range. But since we need θ such that -720° ≤ θ < 0°, we have two angles: -470° and -110°. Wait, but maybe there are more? Let's check with k=6:θ = -1910 +6*360 = -1910 +2160 = 250°, which is positive, so not in the desired range.k=3: θ = -1910 +3*360 = -1910 +1080 = -830°, which is less than -720°, so outside the range.So the valid θs are -470° and -110°. But the question says "Find θ", implying a single answer. Wait, perhaps the question expects the angle within the specified range that is co-terminal. So maybe we need to find the angle within -720° to 0° that is equivalent to -1910°, so there could be two, but maybe the one closest to 0°, or maybe both?Wait, but let me check the problem statement again: "Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°." So there might be two such angles, but maybe we need to find both?Wait, but the problem says "Find θ", which might suggest one answer. Wait, let's check. Let's see: starting from α = -1910°, adding 360° once gives -1910 + 360 = -1550°, which is less than -720°, so not in the range. Adding 360° again: -1550 +360 = -1190°, still less than -720. Adding again: -1190 +360 = -830°, still less. Then again: -830 +360 = -470°, which is within -720° and 0°. Then again: -470 +360 = -110°, still within. Then again: -110 +360 = 250°, which is outside. So the angles in the range are -470° and -110°. But the problem says "-720° ≤ θ < 0°". So both are valid. However, the problem says "Find θ", which might expect the answer, maybe both? But let me check the exact problem statement.The problem says: "Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°." So technically, there can be two angles: -470° and -110°, because adding 360° once to -470° gives -110°, which is still within the range. Wait, no, wait. If you start from -470°, adding 360° gives -110°, which is still in the range. But if you add another 360°, you get 250°, which is out. So in the range from -720° to 0°, there are two co-terminal angles: -470° and -110°. Wait, but maybe not, because -470° minus 360° is -830°, which is outside the range, so within -720° to 0°, there are two angles? Wait, but θ must be between -720° and 0°, inclusive of -720° but exclusive of 0°. Let's check:-470° is greater than -720°, so okay. -110° is also greater than -720°, so okay. So both are in the range. Therefore, the answer would be two angles. But perhaps the problem expects the one that is closest to 0°, which is -110°, but maybe not. Let me check if there's a standard here. Usually, when asked for an angle within a specific range, especially negative, you might need to find all such angles. But let me check the calculation again.Wait, perhaps I made a mistake here. Let's recast α = -1910°, which we found is equivalent to 250°, which is in Quadrant III. To find θ in the range -720° to 0°, which is two full rotations clockwise. So co-terminal angles would be 250° - 360° = -110°, and -110° - 360° = -470°, and -470° - 360° = -830°, but -830° is less than -720°, so out of range. Therefore, within -720° to 0°, the co-terminal angles are -470° and -110°. Therefore, two answers. However, the problem says "Find θ", so maybe both? But the problem might expect the one that's in the range. Let's check again.Alternatively, perhaps the question expects θ such that when you add multiples of 360° to θ, you get α. So θ must satisfy θ ≡ α mod 360°, and θ is in [-720°, 0°). Since 720° is 2*360°, the angles co-terminal to α in that range would be α + k*360°, where k is chosen such that θ is in that interval. Let's compute:We can compute θ = α mod 360°, but adjusted to the range -720° to 0°. Alternatively, since we know α ≡ 250°, which is equivalent to -110°, as 250° - 360° = -110°, and -110° - 360° = -470°, which is still within -720°, so θ can be -110° or -470°. Since -470° is within -720° and 0°, and -110° is also in that range. Therefore, both are valid. However, maybe the problem expects the angle closest to zero, which is -110°, but I need to check if there's a specific instruction. The problem says "Find θ", so perhaps both? Wait, but maybe the range is -720° ≤ θ < 0°, so θ can be -720° up to but not including 0°, so θ can be two full rotations. Therefore, possible two answers. But in the problem statement for part (2), it's possible they expect one answer. Wait, let me check with calculation.Alternatively, maybe I can compute how many times 360° fits into 1910° when going backwards. Let me think. Since θ must be between -720° and 0°, which is the same as 0° to -720°, which is two full rotations. So, starting from α = -1910°, which is a large negative angle, we can add 360° multiple times to bring it into the range. Let me compute how many times to add 360°.Start with -1910°.Add 360° once: -1910 + 360 = -1550° Still less than -720°, so add another 360°: -1550 +360 = -1190° Still less, add 360°: -1190 +360 = -830° Still less, add 360°: -830 +360 = -470° Now -470° is greater than -720°, so within range. Add another 360°: -470 +360 = -110°, still within range. Add another 360°: -110 +360 = 250°, outside the range.So in total, two angles: -470° and -110°. But the problem says "Find θ", so maybe both? But the problem might expect the answer to list both. But looking back at the problem statement for part (2): "Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°." The wording is singular, "θ", but maybe there are two solutions. Wait, let's check if -470° and -110° are both within the range.Yes: -720° ≤ -470° < 0°, and -720° ≤ -110° < 0°. So both are valid. Therefore, the answer should include both. But in the problem statement's part (2), maybe the user expects two answers? However, looking back at how the problem is structured, maybe in the original problem, they expect a single answer. Hmm. Alternatively, maybe there is a unique solution. Let me verify with another approach.Alternatively, since we know that α = -1910° is co-terminal with β = 250°, then all co-terminal angles can be written as 250° + k*360°, where k is any integer. So to find θ in the range -720° ≤ θ < 0°, which is equivalent to finding k such that 250° + k*360° is between -720° and 0°.Let's solve for k:-720° ≤ 250° + k*360° < 0° Subtract 250°:-720° - 250° ≤ k*360° < 0° - 250° -970° ≤ k*360° < -250° Divide by 360°:-970 / 360 ≈ -2.694 -250 / 360 ≈ -0.694 So k must be integers between -2.694 and -0.694, so k = -2, -1.Therefore, θ = 250° + (-2)*360° = 250° - 720° = -470° θ = 250° + (-1)*360° = 250° - 360° = -110° So both -470° and -110° are solutions. Therefore, part (2) has two answers: -470° and -110°. However, the problem says "Find θ", so maybe they expect both? But in the instructions, the user said "put your final answer within boxed{}". Usually, boxed answers are single, but maybe separated by commas. Alternatively, perhaps I made a mistake here.Wait, but let's cross-verify with another method. Let's take the positive equivalent angle 250°, which is in Quadrant III. To find the negative angles co-terminal to it within -720° to 0°, subtract 360° each time:250° - 360° = -110° 250° - 720° = -470° 250° - 1080° = -830° (which is less than -720°, so invalid)So within -720° to 0°, we have -470° and -110°. Hence, two answers. Therefore, part (2) has two solutions. But the problem might expect both. Alternatively, maybe there's an error in my calculation. Wait, but when I added 360° to -1910°, I found θ = -470° and -110°, both in the desired range. So yes, two answers. However, the problem's part (2) says "Find θ such that...", which might expect both. But given the way the question is structured, maybe it's expecting just one, perhaps the one closest to zero. But given that both are within the range, perhaps both. However, given that the original angle is -1910°, when adding 360° six times gives 250°, as in part (1), but in part (2) we need θ in -720° to 0°, which is two full rotations negative. Since -1910° is equivalent to 250°, subtracting 360° gives -110°, subtracting another 360° gives -470°, which is within -720°, but subtracting another 360° would be -830°, which is outside. Therefore, two answers: -110° and -470°. But let's check if these are indeed co-terminal with α. For θ = -110°, adding 360° gives 250°, which is co-terminal. For θ = -470°, adding 720° gives 250°, so yes, co-terminal. Therefore, both are correct. But since the problem says "Find θ", I need to check if the question expects multiple answers. If the user's initial problem statement says "put your final answer within boxed{}", which is typically for a single answer. Maybe I made a mistake. Wait, let's check again.Wait, -1910° divided by 360°, how many times does 360° go into 1910°? As calculated earlier, 5*360=1800, 1910-1800=110, so -1910° = -1800° -110° = -5*360° -110°, so another way to write this is -110° -5*360°, which is θ = -110° -5*360°, but if we want θ in the range -720° to 0°, then perhaps:Wait, another approach: The angle -1910° can be expressed as θ + k*360°, where θ is in [-720°, 0°). So θ = -1910° + k*360°, find k such that θ is in that range.So:θ = -1910° + k*360°We need θ ≥ -720°, so:-1910° + k*360° ≥ -720° k*360° ≥ 1910° -720° = 1190° k ≥ 1190/360 ≈ 3.3055, so k ≥ 4Also, θ < 0°:-1910° + k*360° < 0° k*360° < 1910° k < 1910/360 ≈ 5.3055, so k ≤5Therefore, integer k can be 4 or 5.For k=4:θ = -1910 +4*360 = -1910 +1440 = -470°For k=5:θ = -1910 +5*360 = -1910 +1800 = -110°So two solutions: -470° and -110°. Therefore, the correct answers are both. But since the problem says "Find θ", maybe the answer should list both. However, in the initial problem statement part (2), it's possible that the answer is expecting two angles. But looking at the problem statement again:"(2) Find θ such that the terminal side of θ is the same as that of α, and -720° ≤ θ < 0°."Since co-terminal angles repeat every 360°, within -720° to 0°, there can be two such angles. Therefore, the answer should include both θ = -470° and θ = -110°. But given the way the problem is presented, maybe the assistant is expected to list both. However, the user instruction says "put your final answer within boxed{}", which usually indicates a single answer. Maybe the problem expects the angle closest to zero, which is -110°, but the calculation shows there are two. Hmm.Wait, maybe I need to check the total number of rotations. Since -720° is two full rotations, so θ can be -470° and -110°, which are both within two full rotations. So yes, two answers. But since the problem might expect a single answer, perhaps I made a mistake. Wait, let me check with modulo operations.Compute -1910° modulo 360°, but adjusted to the range -720° to 0°.Alternatively, compute how many times 360° fits into 1910° to make it negative.Wait, let's see: 1910 divided by 360 is 5.305..., so 5*360=1800, remainder 110. Therefore, -1910° = -5*360° -110°, which is - (5*360° +110°) = -1800° -110° = -1910°. So, to express this as a positive angle, it's 250°, but as a negative angle, it's -110° -5*360°. If we want θ between -720° (which is -2*360°) and 0°, then we can take -110° -2*360° = -110° -720° = -830°, which is too low. Alternatively, take -110°, which is within the range. Wait, but perhaps this is confusing.Alternatively, since we know that -1910° ≡ 250°, then in the range -720° to 0°, we subtract 360° enough times from 250° to get into that range. 250° - 360° = -110°, which is in the range. 250° - 720° = -470°, also in the range. 250° - 1080° = -830°, which is outside. So the two angles are -110° and -470°. Therefore, two answers.But given the problem's phrasing, I think the correct answers are both -470° and -110°, but the problem may expect both. However, in many standard problems, when asked for an angle within a range like -720° to 0°, they might expect all possible solutions. But since the user hasn't specified, and given the initial problem might be from a textbook, it's possible that the answer is -470° and -110°, so two angles. But the user might expect them both.But looking back at part (1), the problem had a single answer (β=250°, Quadrant III). So maybe part (2) also has a single answer. Let me double-check my earlier calculation where I added 360° six times to get to 250°, which was for part (1). For part (2), θ must be in -720° to 0°, so I need to go the other way, subtracting 360° from 250° until I'm in that range. So 250° - 360° = -110°, and -110° - 360° = -470°, which is still within -720°. Then subtracting another 360° would give -830°, which is less than -720°, so out of range. Therefore, θ can be -110° or -470°, both within the specified range. Therefore, two answers.But given the instructions say "put your final answer within boxed{}", which usually boxes one answer, but perhaps for two answers, separate boxes. But the original problem might have a different intention. Maybe there's a mistake in my calculation. Wait, let me check with another method.Let me convert α = -1910° to radians first to see if that helps, but maybe not necessary. Alternatively, compute how many full rotations are in 1910°, which we did earlier: 5 rotations (1800°) with a remainder of 110°, so -1910° = -5*360° -110°, which is equivalent to -110° -5*360°, but if we want θ in -720° ≤ θ < 0°, which is -2*360° ≤ θ < 0°, then we need to adjust the number of rotations. Let's see:-110° -5*360° = -110° -1800° = -1910°, which is the original angle. But to get θ in -720° to 0°, we can add 360° multiple times to -1910° until it's in that range. So:-1910° + 360°*4 = -1910 + 1440 = -470° -1910° + 360°*5 = -1910 + 1800 = -110° -1910° + 360°*6 = 250°, which is outside.So θ can be -470° or -110°, both within -720° to 0°, so both correct.However, given that the problem asks for θ in the range -720° ≤ θ < 0°, there are two valid answers. But since the user might expect one answer, perhaps I need to check the question's source or standard approach. In many textbooks, when asked for an angle within a certain range, especially negative, the answer is often the angle closest to zero, which would be -110°, but technically, both -470° and -110° are correct. So maybe the problem expects both answers.But since the user's instructions mention putting the final answer within boxed{}, which is typically for a single answer, perhaps there is a mistake in my earlier reasoning. Let me check once more.Wait, in part (1), we found that α = -1910° is equivalent to 250°, which is in Quadrant III. So for part (2), to find a co-terminal angle between -720° and 0°, we can subtract 360° until we're in that range. 250° - 360° = -110°, which is within the desired range. If we subtract another 360°, we get -470°, which is also within the range because -720° ≤ -470° < 0°. Subtracting another 360° would give -830°, which is less than -720°, so out of range. Therefore, there are two co-terminal angles within that range: -110° and -470°. Therefore, both answers are correct.But the problem's part (2) says "Find θ such that...", and if the original angle is -1910°, adding 5*360° gives 250°, which is part (1). To get into the range -720° to 0°, we have to subtract 360° from 250° once to get -110°, and twice to get -470°. Both are within the range. Therefore, two correct answers. Hence, I think the problem expects both angles, but given the format, maybe listed as two separate boxed answers. However, since the user's instruction says "put your final answer within boxed{}", which may imply one answer, perhaps there's an error. Alternatively, maybe the question expects the most straightforward answer, which is -110°, but that's speculative.Alternatively, maybe I made a mistake in calculating the number of times to add 360°. Wait, let me recalculate θ for k=4 and k=5:θ = -1910 +4*360 = -1910 + 1440 = -470° θ = -1910 +5*360 = -1910 + 1800 = -110° Yes, both correct. Therefore, the answer is two angles: -470° and -110°, both co-terminal with α and within the specified range.Therefore, after careful consideration, part (1) is β=250°, Quadrant III, and part (2) has two solutions: -470° and -110°. However, given the problem's phrasing, I think it's safe to include both answers for part (2).Final Answer(1) boxed{250^circ} in Quadrant boxed{text{III}}; (2) boxed{-470^circ} and boxed{-110^circ}.</think>Given (alpha = -1910^circ):(1) To express (alpha) in the form (beta + k cdot 360^circ) where (k in mathbb{Z}) and (0^circ leq beta < 360^circ):We start by converting (-1910^circ) to a positive angle by adding multiples of (360^circ):[-1910^circ + 6 cdot 360^circ = -1910^circ + 2160^circ = 250^circ]Thus, (alpha = 250^circ + 6 cdot 360^circ). Here, (beta = 250^circ) and (k = 6). The angle (250^circ) is in Quadrant III because it lies between (180^circ) and (270^circ).(2) To find (theta) such that the terminal side of (theta) is the same as that of (alpha) and (-720^circ leq theta < 0^circ):We need to find co-terminal angles of (-1910^circ) within the range (-720^circ leq theta < 0^circ). By adding multiples of (360^circ) to (-1910^circ):[-1910^circ + 4 cdot 360^circ = -1910^circ + 1440^circ = -470^circ][-1910^circ + 5 cdot 360^circ = -1910^circ + 1800^circ = -110^circ]Both (-470^circ) and (-110^circ) are within the range (-720^circ leq theta < 0^circ). Final Answer(1) (boxed{250^circ}) in Quadrant (boxed{text{III}}); (2) (boxed{-470^circ}) and (boxed{-110^circ}).]

🔑:(1) The angle belongs to the boxed{text{third quadrant}}. (2) theta = boxed{-110^circ text{ or } -470^circ}.

🧠:[<think>Alright, let me try to figure out this problem. So, we have an irreducible fraction. Let me call the numerator of this fraction 'a' and the denominator 'b'. So, the original fraction is a/b. The problem says that the denominator was cubed, so the new denominator is b³. Then, the numerator was increased by 2, making the new numerator a + 2. So, the new fraction after these changes is (a + 2)/b³. Now, it says that as a result, the fraction became three times smaller. That means the new fraction is one-third of the original fraction. So, mathematically, this can be written as:(a + 2)/b³ = (1/3) * (a/b)Hmm, okay. Let me write that equation down again to make sure:(a + 2)/b³ = (a)/(3b)Now, let's solve for a and b. Since they are natural numbers and the original fraction is irreducible, a and b should have no common factors other than 1.First, cross-multiplying both sides to eliminate the denominators:3b(a + 2) = a * b³Simplify both sides:3ab + 6b = a b³Hmm, maybe let's bring all terms to one side:a b³ - 3ab - 6b = 0Factor out a b from the first two terms:b(a b² - 3a) - 6b = 0Wait, that might not be helpful. Let's see. Maybe factor out a b from the entire equation:b(a b² - 3a - 6) = 0But since b is a natural number, it can't be zero. So, we have:a b² - 3a - 6 = 0Let me write that again:a b² - 3a - 6 = 0Hmm, perhaps factor out an a from the first two terms:a(b² - 3) - 6 = 0So,a(b² - 3) = 6Therefore,a = 6 / (b² - 3)Since a and b are natural numbers, and the original fraction a/b is irreducible, we need to find natural numbers b such that (b² - 3) divides 6, and the resulting a is a natural number. Additionally, a and b must be coprime.So, let's find possible values of b. Let's list the divisors of 6 first. The divisors of 6 are 1, 2, 3, 6. But since b² - 3 must be one of these, let's set up equations:Case 1: b² - 3 = 1Then, b² = 4, so b = 2 (since b is natural). Then, a = 6 / 1 = 6. So, a = 6 and b = 2. But check if they are coprime: gcd(6, 2) is 2, which is greater than 1. But the original fraction is supposed to be irreducible. So this is invalid. So, discard this case.Case 2: b² - 3 = 2Then, b² = 5. But 5 is not a perfect square. So, b is not a natural number here. Disregard.Case 3: b² - 3 = 3Then, b² = 6. Again, not a perfect square. Disregard.Case 4: b² - 3 = 6So, b² = 9. Therefore, b = 3. Then, a = 6 / 6 = 1. So, a = 1 and b = 3. Check if they are coprime: gcd(1, 3) = 1. Yes, that's good. So, this is a possible solution.Case 5: b² - 3 might be a negative divisor of 6. Wait, but divisors can also be negative, but since b is a natural number, b² - 3 must be positive. Because if b² - 3 is negative, then a would be negative, but a is a natural number. So, b² must be greater than 3. Therefore, only positive divisors considered above. So, only cases where b² - 3 is 1, 2, 3, 6. But only 1 and 6 led to integer b, but 1 gave b=2 which was invalid, and 6 gave b=3 which is valid.Wait, but 6 is not a divisor of 6? Wait, no. The equation was a = 6 / (b² - 3). So, (b² -3) must divide 6. So the possible (b² -3) are positive divisors of 6, which are 1, 2, 3, 6. So, as above.But wait, 6 divided by (b² -3) must be a natural number. So, (b² -3) must be a positive divisor of 6, i.e., 1, 2, 3, 6, leading to b² -3 =1,2,3,6. But as we saw, only for 1 and 6 do we get integer b's. For 1, b=2, but a=6, which is not coprime with 2. For 6, b=3, a=1, which is coprime. So, this seems like the only solution.But let me check if there are other possible divisors. Wait, the problem says "natural numbers", so only positive integers. Therefore, the only possible positive divisors of 6 are 1,2,3,6, so only those four cases.Therefore, the only solution is a=1, b=3. Therefore, the original fraction is 1/3.Wait, let me verify this. Original fraction is 1/3. Then, denominator is cubed: 3³=27. Numerator is increased by 2: 1 + 2 = 3. So, the new fraction is 3/27 = 1/9. Which is indeed three times smaller than 1/3 (since 1/3 divided by 3 is 1/9). So, that works.But let me double-check if there are other possible values. Maybe if we consider negative divisors? But since both a and b are natural numbers, (b² -3) must be positive, so b² >3. So, b must be at least 2. But for b=2, we saw it's invalid. For b=3, it works. For b=4, let's compute a:If b=4, then b² -3=16-3=13. Then a=6/13, which is not a natural number.b=5: b² -3=25-3=22. a=6/22=3/11, not natural.b=1: b² -3=1-3=-2, which would give a negative a. Not allowed.So, the only possible b is 3, leading to a=1.Therefore, the original fraction is 1/3.But just to make sure, let me check if the problem mentions that the original fraction is in its simplest form (irreducible). Yes, the problem states it's an irreducible fraction. Since 1 and 3 are coprime, that's valid.So, the answer should be 1/3. Thus, boxed, boxed{dfrac{1}{3}}.Final AnswerThe original fraction is boxed{dfrac{1}{3}}.</think>Let the original fraction be (frac{a}{b}) where (a) and (b) are natural numbers and the fraction is irreducible. The problem states that when the denominator is cubed and the numerator is increased by 2, the new fraction becomes three times smaller. This can be written as:[frac{a + 2}{b^3} = frac{1}{3} cdot frac{a}{b}]Cross-multiplying to eliminate the denominators, we get:[3b(a + 2) = a b^3]Simplifying both sides:[3ab + 6b = ab^3]Rearranging terms:[ab^3 - 3ab - 6b = 0]Factoring out (b):[b(ab^2 - 3a - 6) = 0]Since (b) is a natural number, it cannot be zero. Therefore, we focus on the equation:[ab^2 - 3a - 6 = 0]Factoring out (a):[a(b^2 - 3) = 6]Thus, (a = frac{6}{b^2 - 3}). We need (b^2 - 3) to be a positive divisor of 6. The positive divisors of 6 are 1, 2, 3, and 6. We check each case:1. (b^2 - 3 = 1 implies b^2 = 4 implies b = 2). Then (a = frac{6}{1} = 6). However, (gcd(6, 2) = 2), so the fraction is not irreducible.2. (b^2 - 3 = 2 implies b^2 = 5). Not a perfect square.3. (b^2 - 3 = 3 implies b^2 = 6). Not a perfect square.4. (b^2 - 3 = 6 implies b^2 = 9 implies b = 3). Then (a = frac{6}{6} = 1). (gcd(1, 3) = 1), so the fraction is irreducible.The only valid solution is (a = 1) and (b = 3). The original fraction is (frac{1}{3}).Verifying, the new fraction after changes is (frac{1 + 2}{3^3} = frac{3}{27} = frac{1}{9}), which is indeed three times smaller than (frac{1}{3}).Thus, the original fraction is (boxed{dfrac{1}{3}}).,<think>Okay, let me try to figure out this problem. So, the problem says that there's an irreducible fraction. When the denominator is cubed and the numerator is increased by 2, the resulting fraction is three times smaller than the original one. I need to find the original fraction where both the numerator and denominator are natural numbers. Hmm, let's break this down step by step.First, let me define the original fraction. Let me call the numerator "a" and the denominator "b". Since the fraction is irreducible, that means the greatest common divisor (GCD) of a and b is 1, right? So, GCD(a, b) = 1. That's important because it might come into play later when solving for possible values.Now, according to the problem, if we cube the denominator (so the denominator becomes b³) and increase the numerator by 2 (so the numerator becomes a + 2), the new fraction (a + 2)/b³ is three times smaller than the original fraction a/b. Wait, "three times smaller" means it's 1/3 of the original, right? So, (a + 2)/b³ = (1/3) * (a/b). Let me write that equation out:(a + 2)/b³ = (1/3)*(a/b)Okay, now I can solve this equation for a and b. Let's cross-multiply to eliminate the fractions. Multiplying both sides by 3b³ gives:3(a + 2) = a*b²Let me write that as:3a + 6 = a*b²Hmm, let's rearrange the terms to group the a terms together. Subtract 3a from both sides:6 = a*b² - 3aFactor out the a:6 = a*(b² - 3)So, a = 6 / (b² - 3)Since a and b are natural numbers, the denominator (b² - 3) must be a positive divisor of 6. Let's note that b² - 3 must divide 6, and since b is a natural number, b² must be greater than 3 to make the denominator positive. So, b² > 3 => b > sqrt(3) ≈ 1.732. Since b is a natural number, the smallest possible value for b is 2.Let's check possible values of b starting from 2:1. b = 2: b² - 3 = 4 - 3 = 1 So, a = 6 / 1 = 6 Then, check if the original fraction 6/2 is irreducible. GCD(6, 2) is 2, which is greater than 1. But the problem states the original fraction is irreducible. So, this is invalid. Therefore, b=2 is not acceptable.2. b = 3: b² - 3 = 9 - 3 = 6 So, a = 6 / 6 = 1 Check if 1/3 is irreducible. GCD(1, 3)=1. Yes, that's irreducible. Let's verify the condition: Original fraction: 1/3 Modified numerator: 1 + 2 = 3 Modified denominator: 3³ = 27 New fraction: 3/27 = 1/9 Original fraction divided by 3: (1/3)/3 = 1/9 So, yes, that works. So, b=3 and a=1 is a possible solution.3. b = 4: b² - 3 = 16 - 3 = 13 a = 6 /13. But 6/13 is not a natural number. So, invalid.4. b = 5: b² - 3 = 25 - 3 = 22 a = 6 /22 = 3/11. Not a natural number. Invalid.5. b = 6: b² -3 = 36 -3=33 a=6/33=2/11. Not natural. Invalid.Continuing, but since b² - 3 increases as b increases, and 6 divided by a larger number will result in fractions less than 1, which aren't natural numbers. So, no solutions for b >=4.Wait, let's check if there are any divisors of 6 that could be equal to b² -3. The positive divisors of 6 are 1,2,3,6. So, b² -3 must be one of these. Let's see:For each divisor d of 6, set b² -3 = d, so b² = d +3. Then check if d +3 is a perfect square.Divisors of 6: 1,2,3,6.For d=1: b²=4 => b=2. Which we checked earlier, but a=6/1=6 gives a reducible fraction. So invalid.For d=2: b²=5. Not a perfect square. Disregard.For d=3: b²=6. Not a perfect square. Disregard.For d=6: b²=9 => b=3. Then a=6/6=1. Which gives a=1 and b=3, which is valid. So that's the only possible solution.Therefore, the original fraction is 1/3. Let me check again to make sure I didn't make a mistake. Original fraction is 1/3, numerator increased by 2 is 3, denominator cubed is 27. The new fraction is 3/27=1/9, which is indeed three times smaller than 1/3. And the original fraction is irreducible. So that checks out.Wait, but the problem says "the denominator was cubed, and the numerator was increased by 2". So, the operations are applied to the original numerator and denominator. So, yes, that's exactly what we did. So, seems correct.But let me check if there are other divisors. Wait, 6 can have positive divisors 1,2,3,6. But maybe negative divisors? But since b is a natural number, b² -3 must be positive. So, d has to be positive. So, no negative divisors considered here.Therefore, the only possible value is a=1 and b=3, so the original fraction is 1/3.Is there any other way this could be solved? Maybe I should check if there are other possible solutions where b² -3 divides 6. As we saw, only when b=2 and b=3, but b=2 gives a reducible fraction. So, only b=3 works. Therefore, the answer is 1/3.Final AnswerThe original fraction is boxed{dfrac{1}{3}}.</think>Let the original fraction be (frac{a}{b}) where (a) and (b) are natural numbers and the fraction is irreducible (i.e., (gcd(a, b) = 1)). According to the problem, cubing the denominator and increasing the numerator by 2 results in a fraction that is three times smaller. This gives us the equation:[frac{a + 2}{b^3} = frac{1}{3} cdot frac{a}{b}]Cross-multiplying to eliminate the fractions, we get:[3(a + 2) = a b^2]Rearranging the terms, we have:[3a + 6 = a b^2 implies 6 = a(b^2 - 3) implies a = frac{6}{b^2 - 3}]Since (a) and (b) are natural numbers, (b^2 - 3) must be a positive divisor of 6. The positive divisors of 6 are 1, 2, 3, and 6. We check each divisor to see if (b^2 - 3) results in a perfect square:- For (b^2 - 3 = 1), (b^2 = 4) gives (b = 2). This results in (a = 6), but (gcd(6, 2) = 2), so the fraction is not irreducible.- For (b^2 - 3 = 6), (b^2 = 9) gives (b = 3). This results in (a = 1), and (gcd(1, 3) = 1), so the fraction is irreducible.Verifying the solution, the original fraction (frac{1}{3}) when modified becomes (frac{1 + 2}{3^3} = frac{3}{27} = frac{1}{9}), which is indeed three times smaller than (frac{1}{3}).Thus, the original fraction is (boxed{frac{1}{3}}).]

🔑:Answer: frac{1}{3} approx 0.33.