🧠:[<think>Okay, let me try to figure out this problem. So, the problem says there's a division equation where the sum of the dividend and divisor is 136, and the quotient is 7. We need to find the divisor. Hmm, let's start by recalling some division terminology. In a division equation, we have dividend ÷ divisor = quotient. So if I denote the dividend as D and the divisor as d, then the equation is D ÷ d = 7. That means D = 7d, right? Because if you multiply both sides by d, you get D = 7d.Now, the problem also states that the sum of the dividend and divisor is 136. So that would be D + d = 136. But since we already know that D is equal to 7d, we can substitute that into the equation. So substituting D with 7d, we have 7d + d = 136. Let me write that out: 7d + d = 136. Combining those terms on the left side gives us 8d = 136.Alright, now to solve for d, we just divide both sides by 8. So 8d = 136 means d = 136 ÷ 8. Let me compute that. 136 divided by 8. 8 times 16 is 128, and 136 minus 128 is 8, so that's 16 with a remainder of 8, but wait, 8 goes into 136 exactly 17 times because 8 times 17 is 136. Wait, hold on, 8 times 10 is 80, 8 times 7 is 56, so 80 + 56 = 136. Yeah, that's right. So 8 times 17 is 136. Therefore, d = 17.Let me double-check that. If the divisor is 17, then the dividend would be 7 times 17, which is 119. Then the sum of 119 and 17 is 136, which matches the given condition. And the quotient when 119 is divided by 17 is indeed 7. So that seems correct. Hmm, did I miss anything here?Wait, just to make sure, let's verify all steps again. The problem states:1. Dividend + Divisor = 136.2. Quotient = 7.We know that Dividend ÷ Divisor = Quotient, so Dividend = Quotient × Divisor. Therefore, substituting Dividend with 7 × Divisor, the equation becomes 7d + d = 136. That simplifies to 8d = 136, leading to d = 17. Checks out. Dividend is 119, 119 + 17 = 136, and 119 ÷ 17 = 7. Yep, that all works. I don't think there's any other possible solution here. Since this is a linear equation with one variable, there's only one possible answer. So the divisor must be 17. I think that's solid. No mistakes in the calculations. The key was setting up the equations correctly using the relationship between dividend, divisor, and quotient. Once substituted, solving was straightforward. Alright, confident that 17 is the correct answer.Final AnswerThe divisor is boxed{17}.</think>Given a division equation where the sum of the dividend and divisor is 136, and the quotient is 7, we need to find the divisor.1. Let ( D ) be the dividend and ( d ) be the divisor.2. According to the problem, the quotient is 7, so we have ( frac{D}{d} = 7 ), which implies ( D = 7d ).3. The sum of the dividend and divisor is given as 136, so we have ( D + d = 136 ).4. Substituting ( D ) with ( 7d ) in the equation ( D + d = 136 ), we get: [ 7d + d = 136 ]5. Simplifying the left side, we get: [ 8d = 136 ]6. Solving for ( d ), we divide both sides by 8: [ d = frac{136}{8} = 17 ]Verification:- The divisor ( d ) is 17.- The dividend ( D ) is ( 7 times 17 = 119 ).- The sum of the dividend and divisor is ( 119 + 17 = 136 ), which matches the given condition.- The quotient ( frac{119}{17} = 7 ), which is correct.Thus, the divisor is (boxed{17}).,<think>Okay, so I have this math problem here. Let me read it again to make sure I understand. It says there's a division equation where the sum of the dividend and the divisor is 136, and the quotient is 7. Then I need to find the divisor. Hmm, division equation terms... Let me recall. In division, we have dividend divided by divisor equals quotient. So, like, dividend ÷ divisor = quotient. And they told us that the sum of the dividend and the divisor is 136, and the quotient is 7. So, I need to find the divisor. Alright.Let me break this down. Let me denote the dividend as D and the divisor as d. According to the problem, D ÷ d = 7, which is the quotient. So that means D divided by d is 7. Therefore, D = 7d. That makes sense because if you divide the dividend by the divisor and get 7, then the dividend is seven times the divisor.Also, the problem states that the sum of the dividend and the divisor is 136. So, D + d = 136. But since we already know that D = 7d from the quotient, we can substitute that into the equation. So instead of D + d, it would be 7d + d = 136. Let me write that out:7d + d = 136Combining like terms, that's 8d = 136. Then, to find d, which is the divisor, I just need to divide both sides by 8. Let's do that:d = 136 ÷ 8Hmm, calculating 136 divided by 8. Let me see. 8 times 10 is 80, subtract that from 136, we get 56 left. 8 times 7 is 56. So 10 + 7 is 17. So, 8 times 17 is 136. Therefore, d = 17.Wait, let me check that again. If the divisor is 17, then the dividend would be 7 times 17, which is 119. Then, adding them together, 119 + 17 = 136. That matches the given sum. And 119 divided by 17 is 7, which matches the quotient. So that seems correct. But just to make sure I didn't make any mistakes in the algebra. Starting with D + d = 136 and D/d = 7. So D = 7d. Substitute into the first equation: 7d + d = 8d = 136, so d = 136/8 = 17. Yep, that's straightforward. Is there any other way to approach this problem? Maybe using different variables or another method? Let's see. Suppose I didn't use algebra. If the quotient is 7, that means when you divide the dividend by the divisor, you get 7. So the dividend is 7 times the divisor. So if I imagine the divisor as a certain number, the dividend is seven of those. Together, they add up to 136. So, divisor plus seven divisors equals 136. Which is eight divisors, so each divisor is 136 divided by 8, which is 17. That's the same result. So regardless of the approach, the answer seems consistent.Another way to check could be to list multiples of 7 and see which pair adds up to 136. For example, if the divisor is 10, dividend is 70, sum is 80. If divisor is 15, dividend is 105, sum is 120. Divisor 16, dividend 112, sum 128. Divisor 17, dividend 119, sum 136. There we go. So that's another way to verify. So 17 is the correct divisor.I think that's solid. All methods lead to the divisor being 17, and checking by substitution works. So, I don't see any errors here.Final AnswerThe divisor is boxed{17}.</think>Given a division equation where the sum of the dividend and the divisor is 136, and the quotient is 7, we need to find the divisor.Let ( D ) be the dividend and ( d ) be the divisor. According to the problem, we have:[frac{D}{d} = 7 implies D = 7d]Additionally, the sum of the dividend and the divisor is given by:[D + d = 136]Substituting ( D = 7d ) into the equation ( D + d = 136 ):[7d + d = 136 implies 8d = 136 implies d = frac{136}{8} = 17]Verification:- If the divisor ( d ) is 17, then the dividend ( D ) is ( 7 times 17 = 119 ).- The sum ( D + d = 119 + 17 = 136 ), which matches the given condition.- The quotient ( frac{119}{17} = 7 ), which also matches the given condition.Thus, the divisor is (boxed{17}).]

🔑:2. (3 points) There is a division equation, the sum of the dividend and divisor is 136, the quotient is 7, then the divisor is qquad 17.【Analysis】Method one: The sum of the dividend and divisor is 136, the quotient is 7, which means the dividend is 7 times the divisor. The sum of the dividend and divisor is (7+1) times the divisor. Dividing 136 by (7+1) will give the divisor, thus solving the problem; Method two: According to the dividend = quotient times divisor, let the divisor be x, then the dividend is 7x. Based on "the sum of the dividend and divisor is 136", set up an equation and solve it.【Solution】Solution: Method one:begin{array}{l}136 div(7+1) =136 div 8 =17end{array}Answer: The divisor is 17.Method two:Let the divisor be x, the dividend is 7x, according to the problem:begin{aligned}7 x+x & =136 8 x & =136 x & =17end{aligned}Answer: The divisor is 17.Therefore, the answer is: 17.【Comment】To solve this problem, it can be treated as a sum-multiples problem: the sum of two numbers div the sum of multiples =1 times the number; or set an unknown, find the equal relationship based on the relationship between the dividend, divisor, and quotient, and solve the equation.

🧠:[<think>Okay, let me try to figure out this problem step by step. So, we have a team of lumberjacks cutting trees on two plots: a large one and a small one. The small plot is three times less in area than the large one, which means the large plot is three times bigger than the small one. So, if the small plot has area A, the large one has area 3A. Got that.Then, it says that in the part of the team working on the large plot, there were 8 more lumberjacks than in the part working on the small plot. Let me denote the number of lumberjacks on the small plot as S. Then, the number on the large plot would be S + 8. So total team size is S + (S + 8) = 2S + 8. But we need to find the maximum possible number of lumberjacks, so maybe there's a constraint here?The key part seems to be when they mention that when the tree harvesting on the large plot was completed, the part of the team on the small plot was still working. That means the time taken to finish the large plot is less than the time needed to finish the small plot. Wait, but the large plot is bigger, so how come they finished it faster even with more lumberjacks? Hmm, maybe the rate at which they cut trees affects this. Let me think.Let’s model the work rates. The time taken to finish a plot depends on the area divided by the number of lumberjacks, assuming each lumberjack works at the same rate. Wait, actually, if each lumberjack cuts at a certain rate (say, area per unit time), then the time to finish a plot would be area divided by (number of lumberjacks multiplied by their rate). But since the rate is the same for all lumberjacks, maybe we can ignore the actual rate and just consider the ratio between the areas and the number of lumberjacks.Let me formalize this. Let’s denote:- Area of small plot = A- Area of large plot = 3A- Number of lumberjacks on small plot = S- Number of lumberjacks on large plot = S + 8Time to finish large plot: T_large = (3A) / (S + 8)Time to finish small plot: T_small = A / SAccording to the problem, when the large plot was completed, the small plot was still being worked on. That means T_large < T_small.So, 3A / (S + 8) < A / SWe can simplify this inequality. Let's divide both sides by A (assuming A > 0, which it is since it's an area):3 / (S + 8) < 1 / SCross-multiplying (since S and S + 8 are positive):3S < S + 8Subtract S from both sides:2S < 8Divide both sides by 2:S < 4So, the number of lumberjacks on the small plot must be less than 4. Since the number of lumberjacks has to be a positive integer, S can be 1, 2, or 3.Therefore, the possible values of S are 1, 2, 3. Then, the total number of lumberjacks would be 2S + 8. Let's compute that for each S:- If S = 1: total = 2(1) + 8 = 10- If S = 2: total = 2(2) + 8 = 12- If S = 3: total = 2(3) + 8 = 14But we need the maximum number of lumberjacks possible, so the answer would be 14? Wait, but let me check if S = 3 is allowed. The inequality was S < 4, so S = 3 is allowed (since 3 < 4). So yes, 14 is possible.Wait, but hold on. Let me verify with S = 3:Compute T_large = 3A / (3 + 8) = 3A / 11T_small = A / 3So, comparing 3A/11 and A/3:Multiply both by 33/A to eliminate denominators:3A/11 * 33/A = 9A/3 * 33/A = 11So 9 < 11, which is true. So T_large = 9 units, T_small = 11 units. Hence, when the large plot is done at 9 units, the small plot still has 2 units left. So that works.Similarly, check S = 4. Wait, S must be less than 4, so S = 4 is not allowed. If S = 4, the inequality would be 3/(4 + 8) = 3/12 = 1/4, and 1/4 is equal to 1/4 (since S = 4, 1/S = 1/4). So 1/4 is not less than 1/4, so S = 4 would make T_large equal to T_small, which is not allowed because the problem says when the large plot was completed, the small plot was still working. So they must not have finished yet. So S must be strictly less than 4.Therefore, the maximum S is 3, leading to a total of 14 lumberjacks. So the answer is 14? But let me check again.Wait, but the problem is in the translation. Let me check the original problem again. It says "the area of the small plot being 3 times less than that of the large plot." Sometimes "3 times less" can be ambiguous. Does it mean the small plot is 1/3 the area of the large one, or the large is 3 times the small? Usually, "3 times less" is interpreted as 1/3, which is what I did. So large is 3A, small is A. So that seems correct.So, if S is 3, the large plot has 11 lumberjacks. Time for large: 3A / 11, time for small: A / 3. So 3A/11 compared to A/3: 3/11 ≈ 0.2727, A/3 ≈ 0.3333. So yes, 0.2727 < 0.3333, so large plot finishes first. Therefore, correct.Therefore, the maximum number is 14. But let me check with S=3, 3 lumberjacks on small, 11 on large. Total team size is 14. So that's the answer?Wait, but is there another constraint? Like, when the large plot is finished, the small plot is still working, but maybe the remaining time requires that the number of lumberjacks on the small plot can't be too low. Wait, no, the problem doesn't specify any other conditions. So the only constraint is that T_large < T_small, leading to S < 4, so maximum S is 3, total team is 14. Therefore, the answer is 14. I think that's it.But to make sure, let me think if there's another way to interpret the problem. Maybe when the large plot is completed, some lumberjacks from the large plot could move to the small plot? But the problem says "the part of the team that was on the small plot was still working." So it implies that only the original small plot team is still working. So they didn't get any additional help. Therefore, the initial assignment is S and S + 8, and after the large plot is done, the S lumberjacks are still working on the small plot. So the time it took to finish the large plot is less than the total time needed for the small plot. So our original reasoning holds.Therefore, 14 is the maximum number.Final AnswerThe maximum number of lumberjacks that could have been in the team is boxed{14}.</think>Given a team of lumberjacks cutting trees on two plots, a large plot and a small plot, where the area of the small plot is three times less than that of the large plot. The number of lumberjacks on the large plot is 8 more than on the small plot. When the large plot was completed, the small plot was still being worked on. We need to find the maximum number of lumberjacks in the team.1. Let the area of the small plot be ( A ). Therefore, the area of the large plot is ( 3A ).2. Let ( S ) be the number of lumberjacks on the small plot. The number of lumberjacks on the large plot is ( S + 8 ).3. The time to complete the large plot is ( frac{3A}{S + 8} ).4. The time to complete the small plot is ( frac{A}{S} ).5. Since the large plot was completed before the small plot, we have the inequality: [ frac{3A}{S + 8} < frac{A}{S} ]6. Simplifying the inequality: [ frac{3}{S + 8} < frac{1}{S} ] [ 3S < S + 8 ] [ 2S < 8 ] [ S < 4 ]7. The possible integer values for ( S ) are 1, 2, or 3. The corresponding total number of lumberjacks is ( 2S + 8 ).8. Calculating the total number of lumberjacks for each ( S ): - If ( S = 1 ), total = ( 2(1) + 8 = 10 ) - If ( S = 2 ), total = ( 2(2) + 8 = 12 ) - If ( S = 3 ), total = ( 2(3) + 8 = 14 )The maximum number of lumberjacks is when ( S = 3 ), leading to a total of ( 14 ).[boxed{14}],<think>Okay, let's see. I need to figure out the maximum number of lumberjacks that could have been in the team. The problem states that there's a large plot and a small plot, with the small plot being 3 times less in area than the large one. So, if the large plot is area A, the small one is A/3. There were 8 more lumberjacks working on the large plot than on the small one. Let me denote the number of lumberjacks on the small plot as x. Then, the large plot has x + 8 lumberjacks. When the large plot was finished, the small plot was still being worked on. That implies that the time taken to finish the large plot is less than the time needed to finish the small plot. Wait, no. If they started at the same time, and the large plot was completed first, then the remaining lumberjacks on the small plot were still working. Hmm. So, the key here is that the amount of work done (area) divided by the number of workers and their time should be considered.Let me think in terms of work rates. Let’s assume that all lumberjacks work at the same rate. Let’s say each lumberjack can cut 1 unit of area per day (or per some time unit). Then, the time taken to finish the large plot would be (A) / (x + 8), and the time taken to finish the small plot would be (A/3) / x. But the problem states that when the large plot was completed, the small plot was still being worked on. So, the time taken to finish the large plot is equal to the time that the small plot had been worked on up to that point, but the small plot wasn't finished yet. That means that the amount of work done on the small plot in that time is less than A/3. So, the time taken to finish the large plot is T = A / (x + 8). In that same time, the small plot had x lumberjacks working for T days, so the work done on the small plot is x * T = x * (A / (x + 8)). This must be less than A/3. So, x * (A / (x + 8)) < A/3. We can cancel A from both sides (assuming A is positive, which it is since it's an area). So, x / (x + 8) < 1/3. Multiplying both sides by (x + 8) gives x < (x + 8)/3. Multiply both sides by 3: 3x < x + 8. Subtract x from both sides: 2x < 8. Therefore, x < 4. Since the number of lumberjacks must be a positive integer, x can be 1, 2, or 3. But we need the maximum number of lumberjacks in the team. The total team size is x (small plot) + x + 8 (large plot) = 2x + 8. So, to maximize 2x + 8, we need to take the maximum possible x, which is 3 (since x < 4). Therefore, x = 3. Total team size: 2*3 + 8 = 6 + 8 = 14. Wait, but hold on. Let me check this again. If x = 3, then the large plot has 11 lumberjacks. Let me verify the time taken. Time to finish large plot: A / 11. Work done on small plot in that time: 3 * (A / 11) = 3A/11. But the small plot has area A/3 ≈ 0.333A. 3A/11 ≈ 0.2727A, which is less than A/3 ≈ 0.333A. So, yes, that checks out. So, after the large plot is done, the small plot still has A/3 - 3A/11 = (11A - 9A)/33 = 2A/33 ≈ 0.0606A left. Therefore, x=3 is acceptable. But wait, let me check x=4. If x=4, then the large plot has 12 lumberjacks. Time to finish large plot: A/12. Work done on small plot: 4*(A/12) = A/3. But the small plot is exactly A/3, so that would mean they finished at the same time. But the problem says that when the large plot was completed, the small plot was still working. So, they can't have finished at the same time. Therefore, x=4 is invalid. Hence, x must be less than 4, so 3 is the maximum. Therefore, the total number is 14. But let me check if there's another way to model this. Maybe different work rates or different interpretations. Alternatively, perhaps the problem is not assuming they start at the same time? Wait, no, the way it's phrased: "the part of the team that worked on the large plot... when the tree harvesting on the large plot was completed, the part of the team that was on the small plot was still working." So, it's two parts of the team working simultaneously on different plots. When the large plot is done, the small plot is still not done. So, my original reasoning holds. Therefore, the maximum number of lumberjacks is 14. So, the answer should be boxed{14}. But wait, another thought: maybe the areas are such that the large plot is 3 times the small plot. The problem says "the area of the small plot being 3 times less than that of the large plot". Hmm, "3 times less" is sometimes ambiguous. Some people interpret it as 1/3, others might think it's original minus 3 times, but in standard terms, "n times less" usually means 1/n times. So, the small plot is 1/3 the area of the large plot. So, if the large plot is A, the small is A/3. That's how I interpreted it. Alternatively, if someone misreads it as the large plot is 3 times less, meaning the large plot is 1/3 of the small, but that seems contradictory. But the problem says "the area of the small plot being 3 times less than that of the large plot". So, small = large / 3. So, my original interpretation was correct. Therefore, x < 4, x=3, total team 14. Is there a way to get a larger team? Suppose if x is fractional, but the number of lumberjacks must be integer. So, x must be integer. So, x=3 is the maximum. Therefore, the answer is 14. Final Answerboxed{14}</think>Given a large plot and a small plot, where the area of the small plot is 3 times less than that of the large plot, we need to determine the maximum number of lumberjacks in the team. The number of lumberjacks on the large plot is 8 more than on the small plot. When the large plot was completed, the small plot was still being worked on.1. Let the number of lumberjacks on the small plot be ( x ). Therefore, the number of lumberjacks on the large plot is ( x + 8 ).2. Let the area of the large plot be ( A ). The area of the small plot is ( frac{A}{3} ).3. The time taken to complete the large plot is ( frac{A}{x + 8} ).4. In this time, the work done on the small plot is ( x times frac{A}{x + 8} ).5. This work done on the small plot must be less than its total area ( frac{A}{3} ): [ x times frac{A}{x + 8} < frac{A}{3} ]6. Simplifying the inequality by canceling ( A ): [ frac{x}{x + 8} < frac{1}{3} ]7. Solving the inequality: [ 3x < x + 8 implies 2x < 8 implies x < 4 ]8. Since ( x ) must be a positive integer, the maximum value for ( x ) is 3.9. The total number of lumberjacks is ( x + (x + 8) = 2x + 8 ). Substituting ( x = 3 ): [ 2 times 3 + 8 = 6 + 8 = 14 ]Thus, the maximum number of lumberjacks that could have been in the team is (boxed{14}).]

🔑:Answer: 14.Solution. Let the number of workers on the smaller plot be denoted as n, then the number of workers on the larger plot is n+8, and the total number of workers in the team is 2n+8. The problem "implicitly assumes" that the productivity of each worker is the same, denoted as a. Therefore, the productivity of each part of the team is an and a(n+8). If the area of the smaller plot is S, then the area of the larger plot is 3S. The time spent on completing the entire work by each part of the team is frac{S}{an} and frac{3S}{a(n+8)}, respectively. According to the problem, frac{S}{an} > frac{3S}{a(n+8)}. Given the positivity of all variables, this inequality is equivalent to n+8 > 3n, or n < 4. Therefore, n leqslant 3, and 2n+8 leqslant 14. The equality situation is clearly possible: just take any positive S and a.

🧠:[<think>Okay, let's see. I need to find the coordinates of point C that lies on the bisector of angle AOB, where A is (0,1) and B is (-3,4). Also, the magnitude of vector OC is 2. Hmm, angle bisector... So, first, maybe I should figure out what the angle AOB is. Points A, O, and B. O is the origin, right? So OA is from O(0,0) to A(0,1), which is along the y-axis. OB is from O(0,0) to B(-3,4). So angle AOB is the angle between OA and OB. Wait, OA is along the positive y-axis, and OB is in the second quadrant because x is negative and y is positive. So angle AOB is between the positive y-axis and the vector OB. The bisector of that angle would be a line that splits that angle into two equal parts. So point C lies somewhere on that bisector, and OC has a magnitude of 2. So I need to find the coordinates of C such that it's on the bisector and 2 units away from the origin.First, maybe I should find the angle bisector. How do I find the angle bisector between two vectors? I remember that there's a formula or method involving unit vectors. If I can find the direction of the angle bisector, then the vector OC would be in that direction with magnitude 2.Let me recall: The angle bisector between two vectors is in the direction of the sum of the unit vectors in those directions. So if I take the unit vectors of OA and OB, add them together, the resulting vector points in the direction of the angle bisector. Then, I can normalize that vector and scale it to length 2 to get OC.Let me test this. First, OA is from O to A(0,1), so vector OA is (0,1). The unit vector in the direction of OA is just (0,1) since its magnitude is 1. Then vector OB is (-3,4). Let's compute its magnitude. The magnitude of OB is sqrt((-3)^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. So the unit vector in the direction of OB is (-3/5, 4/5).So the sum of these unit vectors is (0 + (-3/5), 1 + 4/5) = (-3/5, 9/5). That gives the direction of the angle bisector. Then, to get the vector OC, we need a vector in this direction with magnitude 2. But first, let me check if this direction is correct.Alternatively, another method: The angle bisector theorem states that the bisector divides the opposite side in the ratio of the adjacent sides. But since we are dealing with coordinates, maybe the vector approach is better.Wait, but OA has length 1, and OB has length 5. If we use the angle bisector theorem, the point C on the bisector would divide the opposite side in the ratio OA:OB = 1:5. But I'm not sure how this applies here because we are in coordinate system. Maybe not directly applicable. The vector approach seems better.So, back to adding the unit vectors. The direction vector is (-3/5, 9/5). Let's find the unit vector in that direction. First, compute its magnitude. The magnitude is sqrt( (-3/5)^2 + (9/5)^2 ) = sqrt(9/25 + 81/25) = sqrt(90/25) = sqrt(18/5) = (3 sqrt(10))/5 ≈ 0.94868. Hmm, so the unit vector in the bisector direction is (-3/5)/( (3 sqrt(10))/5 ) , (9/5)/( (3 sqrt(10))/5 ) ) = (-3/(3 sqrt(10)), 9/(3 sqrt(10))) = (-1/sqrt(10), 3/sqrt(10)).Therefore, the unit vector is (-1/sqrt(10), 3/sqrt(10)). So the vector OC with magnitude 2 would be 2 times this unit vector: (-2/sqrt(10), 6/sqrt(10)). But we can rationalize the denominators: (-2/sqrt(10) = -2*sqrt(10)/10 = -sqrt(10)/5, and 6/sqrt(10) = 6*sqrt(10)/10 = 3*sqrt(10)/5. So vector OC is (-sqrt(10)/5, 3*sqrt(10)/5). Hmm, that's one possibility.Wait, but angle bisectors can have two directions, right? Because the angle bisector can go into two opposite directions. But in this case, since both OA and OB are in the upper half-plane (since A is (0,1) and B is (-3,4), both have positive y-coordinates), so the angle between them is less than 180 degrees, and the bisector should also lie within that angle. Therefore, the direction should be between OA and OB, which would be in the second quadrant as well. So the vector (-sqrt(10)/5, 3*sqrt(10)/5) has x negative and y positive, which is in the second quadrant, so that makes sense.Alternatively, if we take the other direction, it would be (sqrt(10)/5, -3*sqrt(10)/5), but that's in the fourth quadrant, which is opposite and not between OA and OB. So that's not the case. So the correct vector should be (-sqrt(10)/5, 3*sqrt(10)/5). But let me verify this result. Let's check the coordinates. If C is (-sqrt(10)/5, 3*sqrt(10)/5), then the magnitude of OC is sqrt( ( (-sqrt(10)/5)^2 + (3*sqrt(10)/5)^2 )) = sqrt( (10/25 + 90/25 )) = sqrt(100/25) = sqrt(4) = 2. Okay, that works.But let me check if this point is indeed on the angle bisector. To verify, we can check that the angle between OC and OA is equal to the angle between OC and OB. Since OA is (0,1), the angle between OC and OA can be found using the dot product. Similarly for the angle between OC and OB.The formula for the angle θ between vectors u and v is cosθ = (u·v)/( |u||v| )First, compute the angle between OC and OA. Vector OA is (0,1), OC is (-sqrt(10)/5, 3*sqrt(10)/5). The dot product is 0*(-sqrt(10)/5) + 1*(3*sqrt(10)/5) = 3*sqrt(10)/5. The magnitude of OA is 1, magnitude of OC is 2. So cosθ1 = (3*sqrt(10)/5)/(1*2) )= (3*sqrt(10)/5)/2 = 3*sqrt(10)/10 ≈ 0.94868 / 2 ≈ 0.4743, so θ1 ≈ arccos(0.4743) ≈ 61.5 degrees.Now compute the angle between OC and OB. Vector OB is (-3,4), OC is (-sqrt(10)/5, 3*sqrt(10)/5). The dot product is (-3)(-sqrt(10)/5) + 4*(3*sqrt(10)/5) = (3 sqrt(10)/5 + 12 sqrt(10)/5 ) = 15 sqrt(10)/5 = 3 sqrt(10). The magnitude of OB is 5, magnitude of OC is 2. So cosθ2 = (3 sqrt(10))/(5*2) = 3 sqrt(10)/10 ≈ same as before, 0.4743. Therefore, θ2 is also arccos(0.4743) ≈ 61.5 degrees. So the angles are equal, which confirms that OC is indeed the angle bisector.Therefore, the answer should be (-sqrt(10)/5, 3 sqrt(10)/5). But let me check if there's another possible point. Because sometimes angle bisectors can be in two different directions, but in this case, since the angle is between OA and OB in the upper half-plane, the bisector is unique in that region. However, technically, the angle bisector is a line, so if we consider the line extending in both directions, but since OC has magnitude 2, the other direction would be in the fourth quadrant, but that point would not lie within the angle AOB. So the correct answer is the one in the second quadrant.Alternatively, maybe there's another approach. Let's consider coordinates. The angle bisector can also be found by finding a point C such that the distance from C to OA and to OB are equal. Since OA and OB are lines, maybe we can use the formula for the distance from a point to a line.But OA is the positive y-axis. The distance from any point (x,y) to OA (the y-axis) is |x|. Similarly, OB is the line from O(0,0) to B(-3,4). Let's find the equation of line OB. The slope of OB is (4-0)/(-3 -0) = 4/(-3) = -4/3. Wait, but actually, the line passes through O(0,0) and B(-3,4), so the equation is y = (-4/3)x. Wait, check that: when x = -3, y = (-4/3)(-3) = 4. Correct. So the equation is 4x + 3y = 0. So the line OB is 4x + 3y = 0.So the distance from point C(x,y) to line OA (y-axis) is |x|, and the distance from C(x,y) to line OB is |4x + 3y| / sqrt(4^2 + 3^2) = |4x + 3y| / 5. Since C is on the angle bisector, these distances should be equal. So |x| = |4x + 3y| / 5.But also, the point C is on the angle bisector of ∠AOB. However, angle bisectors can be internal or external. But since we are dealing with the angle between OA and OB, which is less than 180 degrees, the internal bisector is the one we need. Therefore, the signs of the distances should be considered. Since OA is the y-axis (x=0), and OB is the line 4x + 3y = 0, which is in the second and fourth quadrants. Since point C is in the region of the angle between OA and OB, which is the second quadrant, so x < 0, y > 0.Therefore, the distance to OA (y-axis) is |x| = -x (since x is negative). The distance to line OB (4x + 3y = 0) is |4x + 3y| /5. But since C is in the angle between OA and OB, which is the region where 4x + 3y < 0 (since for points in the second quadrant above the line OB, which goes from O to B(-3,4), let's check a point like (-1, 2): 4*(-1) + 3*2 = -4 + 6 = 2 >0, so actually, maybe the sign is different. Wait, let me check.Wait, the line OB is 4x + 3y = 0. If we take a point in the second quadrant not on the line, say (-1, 1): 4*(-1) + 3*1 = -4 + 3 = -1 <0. But point B(-3,4): 4*(-3) + 3*4 = -12 +12 = 0. So the line divides the plane. Points above the line in the second quadrant would satisfy 4x + 3y >0 or <0? Let's test point (-1, 2): 4*(-1) +3*2= -4 +6=2>0. So points above the line OB in the second quadrant have 4x +3y >0, and points below have 4x +3y <0. Since OA is the positive y-axis, the angle between OA and OB is the region from OA (positive y-axis) to OB (line going to second quadrant). So the angle between them is the area where points are above the line OB in the second quadrant. Wait, but point A is (0,1). Plugging into 4x +3y: 0 +3*1=3>0. So point A is above the line OB. Therefore, the angle between OA and OB is the region above the line OB and along OA. Wait, this is confusing.Wait, maybe better to use the distance formula with signs. Since point C is on the angle bisector, it should be equidistant to both sides OA and OB. But OA is the y-axis, so distance is |x|. OB is the line 4x +3y=0, distance is |4x +3y| /5. Since C is in the angle between OA and OB, which is the region where x <0 and above the line OB (since OA is the positive y-axis and OB is going to (-3,4)). So in that region, 4x +3y >0 (as tested with point (-1,2)), so |4x +3y| = 4x +3y (since positive). Therefore, the equation becomes:Distance to OA: |x| = -x (since x <0)Distance to OB: (4x +3y)/5Set equal: -x = (4x +3y)/5Multiply both sides by 5: -5x =4x +3yBring all terms to left: -5x -4x -3y =0 => -9x -3y=0 => 9x +3y=0 => 3x + y=0.But wait, this gives 3x + y =0. But that's another line. However, point C lies on this line and also has magnitude 2. But this seems conflicting with the previous result.Wait, hold on. There must be a mistake here. Let's re-examine.If point C is in the angle between OA and OB, then in that region, x <0 and y >0. The distance from C to OA is |x|, which is -x because x <0. The distance from C to OB is |4x +3y| /5. Now, since C is in the angle between OA and OB, which is above the line OB. Wait, line OB is from the origin to B(-3,4). Let me check whether a point in that angle would be on which side of OB.Take point A(0,1): plug into 4x +3y =0 +3(1)=3>0. So the region containing OA (the positive y-axis) is where 4x +3y >0. Therefore, the angle between OA and OB is in the region where 4x +3y >0. Therefore, the distance from C to OB is (4x +3y)/5, because 4x +3y is positive. So the equation is:-x = (4x +3y)/5Multiply both sides by 5: -5x =4x +3yBring variables to left: -5x -4x -3y =0 => -9x -3y =0 => 3x + y =0.So point C lies on the line 3x + y =0 and has magnitude 2. Wait, but 3x + y =0 is a line in the fourth and second quadrants. But since point C is in the second quadrant (x <0, y >0), so on the line 3x + y =0 in the second quadrant.But if we solve 3x + y =0 and x^2 + y^2 =4 (since OC has magnitude 2). Let's substitute y = -3x into x^2 + y^2 =4:x^2 + (-3x)^2 =4 => x^2 +9x^2=4 =>10x^2=4 =>x^2=4/10=2/5 =>x=±sqrt(2/5). But since we are in the second quadrant, x is negative, so x= -sqrt(2/5), and y= -3x=3 sqrt(2/5).Therefore, point C is (-sqrt(2/5), 3 sqrt(2/5)). Let's rationalize sqrt(2/5)=sqrt(10)/5, so C is (-sqrt(10)/5, 3 sqrt(10)/5). Which matches the earlier result. So that's the same point as before. Therefore, the answer is (-sqrt(10)/5, 3 sqrt(10)/5). Therefore, vector OC is equal to this.Wait, but earlier when I used the vector addition method, I got the same result. So both methods agree. Therefore, this must be the correct answer.But just to be thorough, let me confirm once more. Suppose point C is (-sqrt(10)/5, 3 sqrt(10)/5). Check if it's on the angle bisector. Using the distance method, the distance from C to OA is |x| = sqrt(10)/5. The distance from C to OB is |4*(-sqrt(10)/5) +3*(3 sqrt(10)/5)| /5 = | (-4 sqrt(10)/5 +9 sqrt(10)/5 ) | /5 = |5 sqrt(10)/5| /5 = |sqrt(10)| /5 = sqrt(10)/5. So both distances are equal, which confirms that C is indeed on the angle bisector.Additionally, the magnitude of OC is sqrt( ( -sqrt(10)/5 )^2 + ( 3 sqrt(10)/5 )^2 ) = sqrt(10/25 + 90/25)=sqrt(100/25)=sqrt(4)=2. Perfect.So everything checks out. Therefore, the answer is (-sqrt(10)/5, 3 sqrt(10)/5). So in boxed form, that's boxed{left( -dfrac{sqrt{10}}{5}, dfrac{3sqrt{10}}{5} right)}.Final AnswerThe vector overrightarrow {OC} is equal to boxed{left( -dfrac{sqrt{10}}{5}, dfrac{3sqrt{10}}{5} right)}.</think>Given points ( A(0, 1) ) and ( B(-3, 4) ) in the Cartesian coordinate system ( xOy ), we need to find the vector ( overrightarrow{OC} ) where point ( C ) lies on the bisector of ( angle AOB ) and the magnitude of ( overrightarrow{OC} ) is 2.1. Vectors and Unit Vectors: - Vector ( overrightarrow{OA} ) is ( (0, 1) ) with magnitude 1. - Vector ( overrightarrow{OB} ) is ( (-3, 4) ) with magnitude 5. The unit vector in the direction of ( overrightarrow{OB} ) is ( left( -frac{3}{5}, frac{4}{5} right) ).2. Angle Bisector Direction: - The angle bisector direction is found by adding the unit vectors of ( overrightarrow{OA} ) and ( overrightarrow{OB} ): [ left( 0, 1 right) + left( -frac{3}{5}, frac{4}{5} right) = left( -frac{3}{5}, frac{9}{5} right) ] - The magnitude of this resultant vector is: [ sqrt{left( -frac{3}{5} right)^2 + left( frac{9}{5} right)^2} = sqrt{frac{9}{25} + frac{81}{25}} = sqrt{frac{90}{25}} = frac{3sqrt{10}}{5} ] - The unit vector in the direction of the bisector is: [ left( -frac{1}{sqrt{10}}, frac{3}{sqrt{10}} right) ]3. Scaling to Magnitude 2: - The vector ( overrightarrow{OC} ) with magnitude 2 is: [ 2 left( -frac{1}{sqrt{10}}, frac{3}{sqrt{10}} right) = left( -frac{2}{sqrt{10}}, frac{6}{sqrt{10}} right) ] - Rationalizing the denominators: [ left( -frac{sqrt{10}}{5}, frac{3sqrt{10}}{5} right) ]4. Verification: - Check if the point ( C ) lies on the angle bisector and the distances to OA and OB are equal. - The line equation for the angle bisector is found to be ( 3x + y = 0 ). - Solving ( 3x + y = 0 ) with ( x^2 + y^2 = 4 ) confirms the coordinates of ( C ).Thus, the vector ( overrightarrow{OC} ) is (boxed{left( -dfrac{sqrt{10}}{5}, dfrac{3sqrt{10}}{5} right)}).,<think>Okay, so I need to find the vector OC where point C is on the angle bisector of angle AOB, and the magnitude of OC is 2. The given points are A(0,1) and B(-3,4). Let me try to visualize this first.First, let me recall that in the Cartesian coordinate system, the angle bisector of an angle between two vectors is a line that divides the angle into two equal parts. So, here, angle AOB is formed by vectors OA and OB. Point C lies on this bisector, and OC has a length of 2. Therefore, I need to find the coordinates of point C such that it's on the bisector of angle AOB and is 2 units away from the origin.Let me write down the coordinates of points A, O, and B:- O is the origin (0,0).- A is (0,1), so vector OA is (0,1).- B is (-3,4), so vector OB is (-3,4).First, I should probably find the angle bisector of angle AOB. The angle bisector theorem might be helpful here. The angle bisector theorem states that the bisector of an angle in a triangle divides the opposite side into segments proportional to the adjacent sides. However, in this case, since we are dealing with vectors, maybe there's a different approach.Alternatively, there is a formula for the angle bisector between two vectors. If I have two vectors, the angle bisector can be found by normalizing the vectors and then adding them together. Let me confirm that.Yes, the angle bisector direction vector is given by the sum of the unit vectors in the directions of OA and OB. So, if I find the unit vectors in the directions of OA and OB, add them together, the resulting vector will point in the direction of the angle bisector.First, let's find the unit vectors of OA and OB.Vector OA is (0,1). Its magnitude is sqrt(0^2 + 1^2) = 1. So, the unit vector in the direction of OA is (0,1).Vector OB is (-3,4). Its magnitude is sqrt((-3)^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. Therefore, the unit vector in the direction of OB is (-3/5, 4/5).Now, adding these two unit vectors gives the direction of the angle bisector:(0,1) + (-3/5, 4/5) = (-3/5, 1 + 4/5) = (-3/5, 9/5).So, the direction vector of the bisector is (-3/5, 9/5). To make this a direction vector, we can also write it as (-3, 9) when scaled by 5, but maybe we should keep it as (-3/5, 9/5) for now.However, since the direction vector is (-3/5, 9/5), any point on the bisector can be written as t*(-3/5, 9/5) where t is a scalar parameter. But we need the point C such that the magnitude of OC is 2. So, the vector OC is t*(-3/5, 9/5), and its magnitude is 2.Let's compute the magnitude of the direction vector first. The magnitude is sqrt( (-3/5)^2 + (9/5)^2 ) = sqrt(9/25 + 81/25) = sqrt(90/25) = sqrt(18/5) = (3*sqrt(10))/5 ≈ 3*3.16/5 ≈ 9.48/5 ≈ 1.896.Wait, sqrt(90/25) is sqrt(90)/5 = 3*sqrt(10)/5. So, approximately 3*3.162/5 ≈ 9.486/5 ≈ 1.897. So, the magnitude of the direction vector is 3*sqrt(10)/5.Therefore, if we want OC to have magnitude 2, then t must satisfy |t| * (3*sqrt(10)/5) = 2. So, |t| = 2 * 5 / (3*sqrt(10)) = 10 / (3*sqrt(10)) = (10*sqrt(10)) / (3*10) ) = sqrt(10)/3 ≈ 3.162/3 ≈ 1.054. Therefore, t = sqrt(10)/3 or t = -sqrt(10)/3.But since the angle bisector is in a certain direction, we need to check whether t should be positive or negative. Since points A and B are in the first and second quadrants, respectively, the angle between OA and OB is between the positive y-axis (since OA is (0,1)) and the vector OB which is in the second quadrant. Therefore, the angle bisector should also lie in the second quadrant? Wait, point A is (0,1), which is on the positive y-axis, and point B is (-3,4), which is in the second quadrant. The angle AOB is the angle between OA (positive y-axis) and OB (second quadrant vector). So, the bisector would lie somewhere in the second quadrant as well. Therefore, the direction vector we found is (-3/5, 9/5). If we scale this by a positive t, then the x-coordinate is negative and the y-coordinate is positive, which is in the second quadrant, which makes sense. Therefore, t should be positive. So, t = sqrt(10)/3.Therefore, vector OC is t*(-3/5, 9/5) = (sqrt(10)/3)*(-3/5, 9/5) = (-3*sqrt(10)/15, 9*sqrt(10)/15) = (-sqrt(10)/5, 3*sqrt(10)/5).Therefore, the coordinates of point C are (-sqrt(10)/5, 3*sqrt(10)/5). So, the vector OC is (-sqrt(10)/5, 3*sqrt(10)/5).But let me check if this is correct.Alternatively, another way to approach this problem is to parametrize the angle bisector and then find the point at distance 2 from the origin.Alternatively, we can use the formula for the internal angle bisector in coordinates.Wait, perhaps using the angle bisector theorem. The angle bisector theorem states that the bisector divides the opposite side in the ratio of the adjacent sides.But in this case, triangle AOB is formed by points O, A, B. The angle at O is being bisected by OC. Then, according to the angle bisector theorem, the ratio of OA to OB is equal to the ratio of AC to CB, where C is the point where the bisector meets AB. But in this case, C is on the bisector but not necessarily on AB. Wait, actually, in our problem, point C is on the bisector of angle AOB but is not necessarily on AB. Wait, AB is a line segment between A(0,1) and B(-3,4). The bisector of angle AOB is a ray starting at O and going through some point in the plane. So, point C is on that ray, at a distance of 2 from O. Therefore, the angle bisector theorem might not directly apply here unless we consider the bisector intersecting AB, but the problem states that C is on the bisector, not necessarily on AB. Therefore, maybe the first method is better.Wait, let me confirm my first approach. The direction vector of the bisector is the sum of the unit vectors of OA and OB. Then scaling that direction vector to have length 2. But that gives the coordinates as I found above.Alternatively, another method: the angle bisector can be found by finding a point C such that the distances from C to both OA and OB are equal. Wait, no, that's for the bisector in terms of distance from the sides. But in this case, OA and OB are vectors from the origin. The angle bisector is the set of points such that the angle between OC and OA is equal to the angle between OC and OB. Alternatively, using the formula for the angle bisector between two lines.Wait, maybe I need to find the equation of the angle bisector. Let's consider the lines OA and OB. OA is along the positive y-axis, since point A is (0,1). OB is the line from the origin to (-3,4). Let me find the equations of these lines.Line OA is the positive y-axis, so its equation is x = 0. Line OB goes from (0,0) to (-3,4), so its slope is (4 - 0)/(-3 - 0) = 4/(-3) = -4/3. Therefore, the equation of line OB is y = (-4/3)x.The angle between the positive y-axis (OA) and line OB (y = -4/3x) is being bisected by OC. To find the angle bisector between these two lines, we can use the formula for the angle bisector between two lines.In general, the angle bisector between two lines can be found by solving:|(a1x + b1y + c1)/sqrt(a1^2 + b1^2)| = |(a2x + b2y + c2)/sqrt(a2^2 + b2^2)|But in this case, the two lines are x = 0 (OA) and y = -4/3x (OB). Since both lines pass through the origin, the angle bisector will also pass through the origin. Therefore, the bisector is a line through the origin.Alternatively, there's a formula for the angle bisector between two lines that pass through the origin. If the lines are given by a1x + b1y = 0 and a2x + b2y = 0, then the angle bisectors can be found by:(a1x + b1y)/sqrt(a1^2 + b1^2) = ±(a2x + b2y)/sqrt(a2^2 + b2^2)In our case, line OA is x = 0, which can be written as 1x + 0y = 0, but since it's x = 0, maybe I need to adjust.Wait, line OA is x = 0, which is the y-axis. Line OB is y = -4/3x, which can be written as 4x + 3y = 0 (by multiplying both sides by 3: 3y = -4x => 4x + 3y = 0). So, equation of OA: 1x + 0y = 0 (since x = 0 is equivalent to 1x + 0y = 0). Wait, actually, x = 0 is just x = 0. So, the general formula for the angle bisector between x = 0 and 4x + 3y = 0.Alternatively, since OA is x=0 (the y-axis) and OB is 4x + 3y =0, which is the line y = -4/3x. Then the angle between them is the angle between the positive y-axis and the line y = -4/3x.The angle bisector would be a line that is equidistant from both OA and OB. But since OA is the y-axis, the distance from a point (x,y) to OA is |x|. The distance from (x,y) to line OB (4x + 3y = 0) is |4x + 3y| / sqrt(4^2 + 3^2) = |4x + 3y| / 5.Setting these distances equal for the bisector: |x| = |4x + 3y| / 5.Since the bisector passes through the origin, and lies in the angle between OA and OB, which is between the positive y-axis and the line OB in the second quadrant, the bisector should lie in the second quadrant. Therefore, x < 0 and y > 0.So, considering the region where x < 0 and y > 0, the equation becomes:|x| = |4x + 3y| / 5Since x is negative, |x| = -x. Also, since in the second quadrant, and the line OB is y = -4/3x, which for x negative gives y positive. The expression 4x + 3y would be 4x + 3y. If we are in the second quadrant, x < 0, y > 0. Let's check the sign of 4x + 3y. Let's take a point on the bisector. Since the bisector is between OA (positive y-axis) and OB (y = -4/3x), for a point on the bisector in the second quadrant, 4x + 3y would be negative? Let's see.Take a point on the bisector. Let's say x = -1, then from the previous calculation, perhaps y would be positive. Let's compute 4*(-1) + 3y. If y is positive, then 4x +3y = -4 + 3y. Depending on y, this could be positive or negative. Hmm, maybe it's not straightforward. Perhaps we need to consider both cases for the absolute value.So, |4x + 3y| /5 = |x|So, 4x + 3y = ±5xBut considering the origin and the region where the bisector lies (second quadrant), which case is appropriate.Case 1: 4x + 3y = 5xThen 3y = 5x - 4x = xTherefore, 3y = x => y = x/3But in the second quadrant, x < 0, y would be negative if y = x/3, but we need y positive. Therefore, this case would give y negative when x is negative, which is not in the angle between OA and OB. Therefore, this case is not the bisector we want.Case 2: 4x + 3y = -5xThen 4x + 3y = -5x => 4x + 5x + 3y = 0 => 9x + 3y = 0 => 3x + y = 0 => y = -3xBut y = -3x. In the second quadrant, x < 0, so y = -3x would be positive. Therefore, this line is y = -3x, passing through the origin, with slope -3, which would go through the second and fourth quadrants. Wait, but the angle between OA (positive y-axis) and OB (y = -4/3x) is in the second quadrant. So, the bisector should be a line in the second quadrant between OA and OB. Let me check if y = -3x is the bisector.Wait, line OA is the positive y-axis (x=0), line OB is y = -4/3x (slope -4/3). The bisector between them should have a slope between 0 (the slope of OA, which is undefined since it's vertical, but direction is upwards) and -4/3. However, the line y = -3x has a slope of -3, which is steeper than OB's slope of -4/3. That can't be right. So, this suggests that perhaps case 2 is not the correct bisector either.Wait, maybe there's a mistake in my approach here. Let's think again. The angle bisector should lie between the two lines OA and OB. Since OA is the positive y-axis (infinite slope) and OB has a slope of -4/3, the bisector should have a slope that is between vertical and -4/3. But if we compute using the distance formula, we got y = -3x, which is steeper than OB. That seems contradictory.Alternatively, maybe the problem is that OA is a vertical line, so the standard angle bisector formula for two lines might need adjustment.Alternatively, let's compute the angle bisector using vector addition. As I did earlier, by adding the unit vectors of OA and OB.OA is (0,1), unit vector is (0,1). OB is (-3,4), unit vector is (-3/5, 4/5). Adding them gives (-3/5, 9/5), which is the direction vector of the bisector. So, the bisector line is parametrized as t*(-3/5, 9/5), t > 0. This line has a slope of (9/5)/(-3/5) = -3. Wait, that's the same as y = -3x. But earlier, that seems too steep. But according to the vector addition method, that's the direction.But wait, let's compute the angle between OA and the bisector, and between OB and the bisector, to check if they are equal.First, the angle between OA (0,1) and the bisector direction (-3/5, 9/5). The cosine of the angle is the dot product divided by the product of magnitudes.Dot product: (0,1) • (-3/5, 9/5) = 0*(-3/5) + 1*(9/5) = 9/5.The magnitude of OA is 1, magnitude of the bisector direction vector is sqrt( (-3/5)^2 + (9/5)^2 ) = sqrt(9/25 + 81/25) = sqrt(90/25) = 3*sqrt(10)/5.Therefore, cos(theta1) = (9/5)/(1 * 3*sqrt(10)/5) = 9/(3*sqrt(10)) = 3/sqrt(10) ≈ 3/3.162 ≈ 0.9486. Therefore, theta1 ≈ arccos(0.9486) ≈ 18.43 degrees.Now, the angle between OB (-3/5, 4/5) and the bisector direction (-3/5, 9/5). Compute the dot product:(-3/5, 4/5) • (-3/5, 9/5) = (-3/5)*(-3/5) + (4/5)*(9/5) = 9/25 + 36/25 = 45/25 = 9/5.The magnitude of OB's unit vector is 1, magnitude of the bisector direction vector is 3*sqrt(10)/5 as before. Therefore, cos(theta2) = (9/5)/(1 * 3*sqrt(10)/5) = 9/(3*sqrt(10)) = 3/sqrt(10), same as theta1. Therefore, theta1 = theta2 ≈ 18.43 degrees. Therefore, the bisector direction is indeed making equal angles with OA and OB. Therefore, even though the slope seems steep, the vector addition method gives the correct bisector.But why does the slope of the bisector come out as -3? Let's see. The original lines OA (vertical) and OB (slope -4/3). The bisector direction is (-3,9) when scaled up, which is a slope of 9/(-3) = -3. So, the bisector line is y = -3x. However, when we look at the angle between OA (vertical) and OB (slope -4/3), the bisector with slope -3 actually makes sense in terms of direction.Wait, but if OA is the positive y-axis, and OB is in the second quadrant with slope -4/3, then the angle between OA and OB is the angle between the positive y-axis and the line going to (-3,4). The bisector should be a line that splits this angle. The direction given by (-3,9) is going towards the second quadrant as well. If we plot the vectors, the unit vector of OA is (0,1), the unit vector of OB is (-3/5,4/5). Adding them gives (-3/5,9/5), which is in the second quadrant. So, the bisector direction is towards the second quadrant, which is correct. Even though the slope is -3, which is steeper than OB's slope of -4/3, the angle calculation shows that it is indeed the bisector.Therefore, perhaps my initial intuition about the steepness was wrong, and the vector addition method is correct. Therefore, the direction of the bisector is (-3/5,9/5), and scaling this to length 2 gives the vector OC as calculated earlier: (-sqrt(10)/5, 3*sqrt(10)/5).But let me verify with another approach.Alternative approach: Let's parametrize the bisector and use the condition that the angle between OC and OA is equal to the angle between OC and OB.Suppose vector OC is (x,y). The angle between OC and OA should be equal to the angle between OC and OB.The angle between two vectors can be found using the dot product formula:cos(theta1) = (OC • OA) / (|OC| |OA|)cos(theta2) = (OC • OB) / (|OC| |OB|)Since theta1 = theta2, we have:(OC • OA)/( |OC| |OA| ) = (OC • OB)/( |OC| |OB| )Simplify, since |OC| cancels out:(OC • OA)/ |OA| = (OC • OB)/ |OB|Given that OA is (0,1), so OC • OA = y. |OA| = 1.OB is (-3,4), so OC • OB = -3x + 4y. |OB| = 5.Therefore, the equation becomes:y / 1 = (-3x + 4y) / 5Multiply both sides by 5:5y = -3x + 4ySubtract 4y:y = -3xSo, the bisector lies on the line y = -3x. Wait, but this is the same result as before from the distance method. But according to the vector addition method, the bisector direction is (-3/5,9/5) which is equivalent to y = -3x, because if x = -3/5, then y = 9/5 = -3*(-3/5) = 9/5. So, indeed, parametrizing t*(-3,9) would be along the line y = -3x.Therefore, the bisector is the line y = -3x, in the second and fourth quadrants. But in our problem, since the original angle is between OA (positive y-axis) and OB (second quadrant), the bisector should be the portion of y = -3x in the second quadrant (where x < 0, y > 0).Therefore, point C is on the line y = -3x, in the second quadrant, and |OC| = 2. So, let's parametrize C as (x, y) = (x, -3x), with x < 0.Then, the magnitude of OC is sqrt(x^2 + y^2) = sqrt(x^2 + 9x^2) = sqrt(10x^2) = |x|*sqrt(10). Since x < 0, |x| = -x. Therefore, sqrt(10x^2) = -x*sqrt(10). We want this to be equal to 2.Therefore, -x*sqrt(10) = 2 => x = -2/sqrt(10) = -sqrt(10)/5.Therefore, x = -sqrt(10)/5, and y = -3x = -3*(-sqrt(10)/5) = 3*sqrt(10)/5.Therefore, point C is (-sqrt(10)/5, 3*sqrt(10)/5), so vector OC is (-sqrt(10)/5, 3*sqrt(10)/5).This matches the result from the vector addition method. Therefore, this confirms that the vector OC is indeed (-sqrt(10)/5, 3*sqrt(10)/5).Wait, but in this alternative approach, the equation derived from equating the cosines of the angles led directly to the line y = -3x, which is the same as the direction obtained by vector addition. So both methods lead to the same result.Therefore, I can be confident that this is the correct vector OC.Just to recap:1. Found unit vectors of OA and OB.2. Added them to get the direction of the angle bisector.3. Scaled the direction vector to have a magnitude of 2.4. Verified using angle equality condition, leading to the same result.Thus, the vector OC is (-√10/5, 3√10/5).Final AnswerThe vector (overrightarrow{OC}) is equal to boxed{left( -dfrac{sqrt{10}}{5}, dfrac{3sqrt{10}}{5} right)}).</think>Given points ( A(0, 1) ) and ( B(-3, 4) ), we need to find the vector (overrightarrow{OC}) where point ( C ) lies on the bisector of (angle AOB) and the magnitude of (overrightarrow{OC}) is 2.1. Unit Vectors of OA and OB: - Vector ( overrightarrow{OA} ) is ((0, 1)) with magnitude 1. The unit vector is ((0, 1)). - Vector ( overrightarrow{OB} ) is ((-3, 4)) with magnitude 5. The unit vector is (left( -frac{3}{5}, frac{4}{5} right)).2. Direction Vector of the Angle Bisector: - Adding the unit vectors of ( overrightarrow{OA} ) and ( overrightarrow{OB} ): [ (0, 1) + left( -frac{3}{5}, frac{4}{5} right) = left( -frac{3}{5}, frac{9}{5} right) ] - This gives the direction vector of the angle bisector.3. Scaling the Direction Vector: - The magnitude of the direction vector (left( -frac{3}{5}, frac{9}{5} right)) is (frac{3sqrt{10}}{5}). - To find the vector (overrightarrow{OC}) with magnitude 2, we scale the direction vector by ( t ) such that: [ t cdot frac{3sqrt{10}}{5} = 2 implies t = frac{2 cdot 5}{3sqrt{10}} = frac{sqrt{10}}{3} ] - Scaling the direction vector by ( t ): [ overrightarrow{OC} = frac{sqrt{10}}{3} left( -frac{3}{5}, frac{9}{5} right) = left( -frac{sqrt{10}}{5}, frac{3sqrt{10}}{5} right) ]4. Verification: - Using the angle equality condition, we confirmed that the line ( y = -3x ) is the angle bisector. - Parametrizing ( C ) on this line and solving for the magnitude 2 confirmed the coordinates of ( C ).Thus, the vector (overrightarrow{OC}) is (boxed{left( -dfrac{sqrt{10}}{5}, dfrac{3sqrt{10}}{5} right)}).]

🔑:Since the magnitudes of vectors overrightarrow {OA} and overrightarrow {OB} are 1 and 5, respectively, let point D(x, y) be the intersection of OC and AB.The ratio AD:BD is 1:5, which means D divides the directed line segment AB in the ratio frac {1}{5}. Hence, we have: begin{cases} x= frac {-3times frac {1}{5}}{1+ frac {1}{5}} y= frac {1+4times frac {1}{5}}{1+ frac {1}{5}}end{cases}Solving these equations yields:begin{cases} x=- frac {1}{2} y= frac {3}{2}end{cases}Therefore, overrightarrow {OD}=(- frac {1}{2}, frac {3}{2}).Since the magnitude of overrightarrow {OC} is 2, we have:overrightarrow {OC} = frac {2 overrightarrow {OD}}{left|overrightarrow {OD}right|}Now, we need to calculate left|overrightarrow {OD}right|, which is:left|overrightarrow {OD}right| = sqrt{left(-frac{1}{2}right)^2 + left(frac{3}{2}right)^2} = sqrt{frac{1}{4} + frac{9}{4}} = sqrt{frac{10}{4}} = frac{sqrt{10}}{2}Thus, we have:overrightarrow {OC} = frac {2 overrightarrow {OD}}{frac{sqrt{10}}{2}} = frac{2}{frac{sqrt{10}}{2}} (-frac{1}{2}, frac{3}{2}) = frac{4}{sqrt{10}}(-frac{1}{2}, frac{3}{2}) = left(-frac{2}{sqrt{10}}, frac{6}{sqrt{10}}right)After simplifying, we get the final vector:overrightarrow {OC} = boxed{left(-frac{sqrt{10}}{5}, frac{3sqrt{10}}{5}right)}

🧠:[<think>Okay, let's see. The problem says that there's this eccentric person who has N unit squares. He managed to form a rectangle with sides differing by 9. Then, using the same N squares, he formed another rectangle with sides differing by 6. We need to find N.So, first, I need to understand what the problem is asking. Essentially, N is a number that can be expressed as the product of two integers that differ by 9, and also as the product of two other integers that differ by 6. Our job is to find such an N.Let me start by breaking this down. Let's denote the sides of the first rectangle as a and b, where |a - b| = 9. Then, the area N is a*b. Similarly, for the second rectangle, let the sides be c and d, where |c - d| = 6, and N = c*d. So, N must have two pairs of factors with differences of 9 and 6 respectively.So, the problem reduces to finding an integer N such that:1. There exist integers a and b where a*b = N and |a - b| = 9.2. There exist integers c and d where c*d = N and |c - d| = 6.Our task is to find the smallest such N, or perhaps there's only one such N?But the problem doesn't specify that N is the smallest; it just says "Find N." So, maybe there's only one possible N, or perhaps we need to find all possible N and the problem expects a specific one. Let's proceed.First, let's model the equations.For the first rectangle:Let’s suppose a > b, so a - b = 9, and a*b = N. So, a = b + 9. Then, substituting into the area equation: (b + 9)*b = N. So, N = b^2 + 9b.For the second rectangle:Similarly, let’s suppose c > d, so c - d = 6, and c*d = N. Then, c = d + 6. Substituting into the area equation: (d + 6)*d = N. So, N = d^2 + 6d.Therefore, we have:b^2 + 9b = d^2 + 6d.So, we need to find integers b and d such that b^2 + 9b = d^2 + 6d.So, this is a Diophantine equation. Let me rearrange it:b^2 + 9b - d^2 - 6d = 0.Maybe we can complete the square for both variables.Let me try completing the square for both b and d.Starting with b:b^2 + 9b = (b + 4.5)^2 - (4.5)^2 = (b + 9/2)^2 - 81/4.Similarly, for d:d^2 + 6d = (d + 3)^2 - 9.Therefore, substituting back into the equation:(b + 9/2)^2 - 81/4 = (d + 3)^2 - 9Let me rearrange terms:(b + 9/2)^2 - (d + 3)^2 = 81/4 - 9Compute the right-hand side:81/4 - 9 = 81/4 - 36/4 = 45/4.So, we have:(b + 9/2)^2 - (d + 3)^2 = 45/4.Multiply both sides by 4 to eliminate denominators:4*(b + 9/2)^2 - 4*(d + 3)^2 = 45.But note that 4*(b + 9/2)^2 is (2b + 9)^2 and 4*(d + 3)^2 is (2d + 6)^2.Therefore:(2b + 9)^2 - (2d + 6)^2 = 45.This is a difference of squares, which factors as:[(2b + 9) - (2d + 6)][(2b + 9) + (2d + 6)] = 45.Simplify the terms inside the brackets:First factor: (2b + 9 - 2d - 6) = (2b - 2d + 3) = 2(b - d) + 3.Second factor: (2b + 9 + 2d + 6) = 2b + 2d + 15 = 2(b + d) + 15.Therefore, the equation becomes:[2(b - d) + 3][2(b + d) + 15] = 45.Let me denote x = b - d and y = b + d. Then, the equation becomes:[2x + 3][2y + 15] = 45.Our goal is to find integer solutions x and y, since b and d are integers.But x = b - d and y = b + d. Since b and d are positive integers (they are sides of a rectangle), y must be positive and greater than x (since y = b + d and x = b - d, assuming b > d). Wait, actually, if b > d, then x is positive, but if d > b, then x is negative. However, in our case, since we assumed a > b (so a = b + 9) and c > d (c = d + 6), but b and d themselves could be any positive integers. Wait, actually, in the first rectangle, sides are a and b, with a > b, so a = b + 9. In the second rectangle, sides are c and d, with c > d, so c = d + 6. Therefore, b and d are positive integers.So, x = b - d can be positive or negative. However, since we need integer solutions, let's consider possible factor pairs of 45.Given that 45 can be factored into positive integers as:1 * 453 * 155 * 9Also, considering negative factors:(-1) * (-45)(-3) * (-15)(-5) * (-9)But since [2x + 3] and [2y + 15] must be integers. Let's list all possible factor pairs (both positive and negative) and solve for x and y.Let’s denote:[2x + 3] = m[2y + 15] = nWith m * n = 45.So, possible (m, n) pairs are:(1, 45), (3, 15), (5, 9), (9, 5), (15, 3), (45, 1),(-1, -45), (-3, -15), (-5, -9), (-9, -5), (-15, -3), (-45, -1).Now, for each pair, solve for x and y.First, let's take positive factors.Case 1: m = 1, n = 45.So,2x + 3 = 1 => 2x = -2 => x = -12y + 15 = 45 => 2y = 30 => y = 15So, x = -1, y = 15.But x = b - d = -1 => d = b + 1y = b + d = 15Substituting d = b + 1 into y = b + d:b + (b + 1) = 15 => 2b + 1 = 15 => 2b = 14 => b = 7Then, d = 7 + 1 = 8Check if these values work.Original equations:From first rectangle: N = b*(b + 9) = 7*16 = 112From second rectangle: N = d*(d + 6) = 8*14 = 112So, N = 112. Hmm, that works. Let me check other cases to see if there are more solutions.Case 2: m = 3, n = 15.So,2x + 3 = 3 => 2x = 0 => x = 02y + 15 = 15 => 2y = 0 => y = 0But y = b + d = 0. Since b and d are positive integers, y cannot be zero. So, invalid.Case 3: m = 5, n = 9.2x + 3 = 5 => 2x = 2 => x = 12y + 15 = 9 => 2y = -6 => y = -3But y = b + d = -3, which is invalid. Discard.Case 4: m = 9, n = 5.2x + 3 = 9 => 2x = 6 => x = 32y + 15 = 5 => 2y = -10 => y = -5Again, y is negative. Invalid.Case 5: m = 15, n = 3.2x + 3 = 15 => 2x = 12 => x = 62y + 15 = 3 => 2y = -12 => y = -6Invalid y.Case 6: m = 45, n = 1.2x + 3 = 45 => 2x = 42 => x = 212y + 15 = 1 => 2y = -14 => y = -7Invalid.Now, negative factors:Case 7: m = -1, n = -45.2x + 3 = -1 => 2x = -4 => x = -22y + 15 = -45 => 2y = -60 => y = -30But y = b + d = -30. Invalid.Case 8: m = -3, n = -15.2x + 3 = -3 => 2x = -6 => x = -32y + 15 = -15 => 2y = -30 => y = -15Invalid.Case 9: m = -5, n = -9.2x + 3 = -5 => 2x = -8 => x = -42y + 15 = -9 => 2y = -24 => y = -12Invalid.Case 10: m = -9, n = -5.2x + 3 = -9 => 2x = -12 => x = -62y + 15 = -5 => 2y = -20 => y = -10Invalid.Case 11: m = -15, n = -3.2x + 3 = -15 => 2x = -18 => x = -92y + 15 = -3 => 2y = -18 => y = -9Invalid.Case 12: m = -45, n = -1.2x + 3 = -45 => 2x = -48 => x = -242y + 15 = -1 => 2y = -16 => y = -8Invalid.So, the only valid solution is from Case 1: x = -1, y = 15, leading to b = 7, d = 8, and N = 112.Wait, but let me check if there are other factor pairs. For example, 45 can also be factored as ( -1, -45 ), but we considered that. Hmm, seems like the only possible solution is N = 112.But let me verify again. Let's check N = 112.First rectangle: sides differing by 9. So, we need two numbers that multiply to 112 and differ by 9.Let's list the factor pairs of 112:1 * 112 (difference 111)2 * 56 (diff 54)4 * 28 (diff 24)7 * 16 (diff 9) → yes, here it is. 16 - 7 = 9.Second rectangle: sides differing by 6. Let's check factors of 112 again:Looking for two numbers with product 112 and difference 6.Check 8 * 14 = 112, 14 - 8 = 6. Bingo.So, yes, N = 112 is a valid solution.But is this the only solution? Let's see.Wait, but in the process above, we only found one solution. Maybe N = 112 is the only possible value. Let me check if there are other N's.Suppose there is another solution. Let's think about possible values of b and d.Alternatively, maybe there's a different approach. Let's list possible rectangles with sides differing by 9 and 6, and see if they can have the same area.For sides differing by 9:Let’s denote the sides as x and x + 9. Then area is x(x + 9).Similarly, for sides differing by 6: y and y + 6. Area is y(y + 6).So, x(x + 9) = y(y + 6).Let’s solve for integers x and y.So:x² + 9x = y² + 6yBring all terms to left side:x² + 9x - y² - 6y = 0This is similar to before. Maybe rearrange terms:x² - y² + 9x - 6y = 0Factor x² - y² as (x - y)(x + y):(x - y)(x + y) + 9x - 6y = 0Not sure if that helps. Alternatively, complete the square for x and y.For x:x² + 9x = (x + 4.5)² - (4.5)²Similarly, for y:y² + 6y = (y + 3)² - 9So:(x + 4.5)² - 20.25 = (y + 3)² - 9Bring constants to the right:(x + 4.5)² - (y + 3)² = 20.25 - 9 = 11.25Multiply both sides by 4 to eliminate decimals:4*(x + 4.5)² - 4*(y + 3)² = 45Which simplifies to:(2x + 9)² - (2y + 6)² = 45Which is the same equation as before. So, same as before.Thus, we end up with the same equation, leading to the same solution. Therefore, N=112 is the only solution.But wait, maybe there's a larger N? Let's test with higher numbers.Suppose we take another factor pair. Wait, but in our factorization of 45, we considered all possible integer pairs. So, unless there are more factorizations, which there aren't, then N=112 is the only solution.Alternatively, maybe I need to consider that the factors m and n could be fractions? But no, because m and n are factors of 45, which is an integer, and m and n must be integers because 2x + 3 and 2y + 15 are integers (since x and y are integers). Therefore, m and n must be integer divisors of 45.Therefore, since we checked all possible integer factor pairs of 45, and only one of them gave a valid solution, N=112 is the only possible answer.Alternatively, let me check if there are other solutions by brute force for small N.Let’s list numbers which can be written as a product of two numbers differing by 9 and also as a product of two numbers differing by 6.Starting from N=1, 2,... up to, say, 200.For N=112, we saw that 7*16 and 8*14 work.Let me check N=144.Factors of 144: 1*144, 2*72, 3*48, 4*36, 6*24, 8*18, 9*16, 12*12. Differences: 143, 70, 45, 32, 18, 10, 7, 0. So, differences of 9? 9*16=144, difference 7. No. Next, 8*18=144, difference 10. 6*24=144, difference 18. 4*36=144, difference 32. So, no factors differing by 9. So, 144 cannot be formed into a rectangle with sides differing by 9. So, N=144 is out.Next, N=180.Factors of 180: 1*180, 2*90, 3*60, 4*45, 5*36, 6*30, 9*20, 10*18, 12*15. Differences: 179, 88, 57, 41, 31, 24, 11, 8, 3. So, sides differing by 9? Let's see: 9*20=180, difference 11. 12*15=180, difference 3. 10*18=180, difference 8. 5*36=180, difference 31. No. So, no factors differing by 9. So, not suitable.N=60.Factors of 60: 1*60, 2*30, 3*20, 4*15, 5*12, 6*10. Differences: 59, 28, 17, 11, 7, 4. No difference of 9 or 6? Wait, for the first rectangle, sides differing by 9. Let's check if 60 has factors differing by 9. 5*12=60, difference 7. 4*15=60, difference 11. 3*20=60, difference 17. So, no. So, not.N=72.Factors: 1*72, 2*36, 3*24, 4*18, 6*12, 8*9. Differences: 71, 34, 21, 14, 6, 1. So, sides differing by 9: 8*9=72, difference 1. 6*12=72, difference 6. 4*18=72, difference 14. So, 72 can be formed as 6*12 (difference 6) but does it have factors differing by 9? 1*72 (71), no. So, no. So, not.N=96.Factors: 1*96, 2*48, 3*32, 4*24, 6*16, 8*12. Differences: 95, 46, 29, 20, 10, 4. Does 96 have factors differing by 9? 3*32=96, difference 29. 4*24=96, difference 20. 6*16=96, difference 10. 8*12=96, difference 4. No. So, no.N=112: as before, 7*16=112 (diff 9) and 8*14=112 (diff 6). So, that's our candidate.Another N=240.Factors of 240: 1*240, 2*120, 3*80, 4*60, 5*48, 6*40, 8*30, 10*24, 12*20, 15*16. Differences: 239, 118, 77, 56, 43, 34, 22, 14, 8, 1. Does 240 have factors differing by 9? Let's see. 15*16=240, diff 1. 12*20=240, diff 8. 10*24=240, diff 14. 8*30=240, diff 22. 6*40=240, diff 34. 5*48=240, diff 43. 4*60=240, diff 56. 3*80=240, diff 77. 2*120=240, diff 118. 1*240=240, diff 239. So, no. So, 240 is out.Alternatively, N=112 is the only one so far. Let's check another.N=40.Factors:1*40, 2*20, 4*10, 5*8. Differences:39, 18, 6, 3. So, 4*10=40, difference 6. So, N=40 can form a rectangle with sides differing by 6. But can it form a rectangle with sides differing by 9? 1*40: difference 39, 2*20: difference 18, 4*10: 6, 5*8:3. No. So, N=40 is invalid.N=84.Factors:1*84, 2*42, 3*28, 4*21, 6*14, 7*12. Differences:83,40,25,17,8,5. Any difference of 9? 7*12=84, difference 5. 6*14=84, difference 8. 4*21=84, difference 17. No. So, no.N=126.Factors:1*126, 2*63, 3*42, 6*21, 7*18, 9*14. Differences:125,61,39,15,11,5. 9*14=126, difference 5. 7*18=126, difference 11. 6*21=126, difference 15. 3*42=126, difference 39. 2*63=126, difference 61. 1*126=126, difference 125. So, no.N=150.Factors:1*150,2*75,3*50,5*30,6*25,10*15. Differences:149,73,47,25,19,5. No.N=90.Factors:1*90,2*45,3*30,5*18,6*15,9*10. Differences:89,43,27,13,9,1. Ah, here we go: 9*10=90, difference 1; 6*15=90, difference 9. So, N=90 can be formed into a rectangle with sides differing by 9 (6 and 15). Then, can it be formed into a rectangle with sides differing by 6? Let's check.Looking for two factors of 90 differing by 6.Factors:1*90 (89), 2*45 (43), 3*30 (27), 5*18 (13), 6*15 (9), 9*10 (1). So, no pair differing by 6. The closest is 5*18=90, difference 13. So, N=90 is invalid.So, N=112 seems unique. Let me confirm once again.Alternatively, let's consider that maybe the equations can have other solutions. For example, if we take the equation (2x + 9)^2 - (2y + 6)^2 = 45, which we converted into [2(b - d) + 3][2(b + d) + 15] = 45, leading to the solution N=112. But perhaps there's another way to factor 45?Wait, 45 can be factored as 15*3, 5*9, etc., but we considered all possible pairs. Since 45 is not a very large number, and factoring into integers is limited, I think N=112 is the only solution.Alternatively, perhaps we can model this problem as two quadratic equations and find their intersection.First equation: N = b² + 9bSecond equation: N = d² + 6dSet equal:b² + 9b = d² + 6dLet’s rearrange:b² - d² + 9b - 6d = 0Factor:(b - d)(b + d) + 3(3b - 2d) = 0Hmm, not sure if helpful. Alternatively, express in terms of variables.Let’s express b in terms of d or vice versa.From the equation:b² + 9b = d² + 6dLet’s move all terms to left:b² - d² + 9b - 6d = 0Factor as (b - d)(b + d) + 3(3b - 2d) = 0. Not obvious.Alternatively, let's treat this as a quadratic in b:b² + 9b - (d² + 6d) = 0So, for each integer d, we can check if the discriminant is a perfect square.Quadratic in b: b² + 9b - (d² + 6d) = 0Discriminant D = 81 + 4(d² + 6d) = 4d² + 24d + 81This needs to be a perfect square.So, 4d² + 24d + 81 = k², for some integer k.Let’s denote k = 2m + p (some integer). Let's see:4d² + 24d + 81 = k²Let me write this as:(2d + 6)^2 + 45 = k²Because (2d + 6)^2 = 4d² + 24d + 36, so 4d² + 24d + 81 = (2d + 6)^2 + 45Therefore:k² - (2d + 6)^2 = 45Which factors as (k - (2d + 6))(k + (2d + 6)) = 45So, similar to before. Let’s set:k - (2d + 6) = mk + (2d + 6) = nWith m*n = 45, and m < n (since k - (2d +6) < k + (2d +6)), and m and n positive integers or both negative. However, since k and d are integers, m and n must be both odd or both even? Let's see.But 45 factors into (1,45), (3,15), (5,9). All odd pairs. So, m and n must be both odd. Therefore, since m and n are both odd, then k = (m + n)/2 and 2d + 6 = (n - m)/2. Since m and n are both odd, (m + n) is even, so k is integer. Similarly, (n - m) is even, so 2d +6 is integer, which is already true.So, possible factor pairs of 45 (positive):(1,45), (3,15), (5,9)For each:Case 1: m=1, n=45Then,k = (1 + 45)/2 = 232d + 6 = (45 - 1)/2 = 22So,2d + 6 = 22 => 2d = 16 => d = 8Then, from N = d² + 6d = 64 + 48 = 112Case 2: m=3, n=15k = (3 +15)/2 = 92d +6 = (15 -3)/2 = 6So,2d +6 =6 => 2d=0 => d=0. Not possible, since d must be positive.Case3: m=5, n=9k=(5+9)/2=72d +6=(9 -5)/2=2So,2d +6=2 => 2d= -4 => d=-2. Not possible.Therefore, only Case1 gives a valid solution: d=8, N=112.Therefore, confirmed again N=112.Thus, the answer is 112.Final Answerboxed{112}</think>Given that the eccentric has ( N ) unit squares which can form a rectangle with sides differing by 9 and another rectangle with sides differing by 6, we need to find ( N ).First, we denote the sides of the first rectangle as ( a ) and ( b ) where ( |a - b| = 9 ) and ( N = a cdot b ). Similarly, for the second rectangle, we denote the sides as ( c ) and ( d ) where ( |c - d| = 6 ) and ( N = c cdot d ).We start by setting up the equations for the areas:1. For the first rectangle: ( N = b(b + 9) )2. For the second rectangle: ( N = d(d + 6) )Equating the two expressions for ( N ):[b^2 + 9b = d^2 + 6d]Rearranging and completing the square:[b^2 + 9b - d^2 - 6d = 0][(b + frac{9}{2})^2 - (d + 3)^2 = frac{45}{4}]Multiplying through by 4 to clear the fraction:[(2b + 9)^2 - (2d + 6)^2 = 45]Factoring as a difference of squares:[[(2b + 9) - (2d + 6)][(2b + 9) + (2d + 6)] = 45]Simplifying the factors:[(2(b - d) + 3)(2(b + d) + 15) = 45]We solve for integer pairs of factors of 45. The valid factor pair (1, 45) leads to:[2(b - d) + 3 = 1 quad text{and} quad 2(b + d) + 15 = 45]Solving these, we find:[2(b - d) + 3 = 1 implies b - d = -1 implies d = b + 1][2(b + d) + 15 = 45 implies 2(b + (b + 1)) = 30 implies 4b + 2 = 30 implies b = 7 implies d = 8]Substituting ( b = 7 ) and ( d = 8 ) into the area equations:[N = 7 cdot (7 + 9) = 7 cdot 16 = 112][N = 8 cdot (8 + 6) = 8 cdot 14 = 112]Thus, the value of ( N ) is confirmed to be ( 112 ).[boxed{112}],<think>Okay, so I need to find the number N of unit squares that can form two different rectangles. The first rectangle has sides differing by 9, and the second one has sides differing by 6. Hmm, let me think about how to approach this.First, since N is the number of unit squares, it must be equal to the product of the lengths of the sides of each rectangle. Let me denote the sides of the first rectangle as a and b, where |a - b| = 9. Similarly, the sides of the second rectangle can be c and d, where |c - d| = 6. Since both rectangles use the same N squares, we have:N = a * b = c * d.So, my goal is to find an integer N that can be expressed both as the product of two integers with a difference of 9 and as the product of two integers with a difference of 6.Let me formalize this. For the first rectangle, suppose the sides are x and x + 9. Then, N = x(x + 9). For the second rectangle, the sides could be y and y + 6, so N = y(y + 6). Therefore, I need to find integers x and y such that:x(x + 9) = y(y + 6).Let me expand both sides:x² + 9x = y² + 6y.Rearranging terms:x² + 9x - y² - 6y = 0.Hmm, this looks like a quadratic in terms of x and y. Maybe I can find a way to express this equation differently. Let's try to complete the squares for both x and y.For the x terms: x² + 9x. To complete the square, take half of 9, which is 4.5, square it to get 20.25. So, x² + 9x = (x + 4.5)² - 20.25.Similarly, for the y terms: y² + 6y. Half of 6 is 3, squared is 9. So, y² + 6y = (y + 3)² - 9.Substituting back into the equation:(x + 4.5)² - 20.25 - (y + 3)² + 9 = 0.Simplify:(x + 4.5)² - (y + 3)² - 11.25 = 0.Hmm, so (x + 4.5)² - (y + 3)² = 11.25.This resembles a difference of squares: (A)² - (B)² = (A - B)(A + B) = 11.25.But 11.25 is a decimal. Maybe I can multiply both sides by 16 to eliminate decimals? Wait, 11.25 is 45/4. Let's see:Original equation after completing squares:(x + 9/2)² - (y + 3)² = 45/4.Multiply both sides by 4 to eliminate denominators:4(x + 9/2)² - 4(y + 3)² = 45.But 4(x + 9/2)² is (2x + 9)², and 4(y + 3)² is (2y + 6)². So:(2x + 9)² - (2y + 6)² = 45.Now, this is a difference of squares:(2x + 9 - 2y - 6)(2x + 9 + 2y + 6) = 45.Simplify the terms inside the parentheses:(2x - 2y + 3)(2x + 2y + 15) = 45.Let me factor this equation. Let me denote A = 2x - 2y + 3 and B = 2x + 2y + 15. So, A * B = 45.Since x and y are positive integers (since they are lengths of sides), let's check possible factor pairs of 45 where both A and B are positive integers.The positive divisors of 45 are 1, 3, 5, 9, 15, 45.So possible pairs (A, B) are:(1, 45), (3, 15), (5, 9), (9, 5), (15, 3), (45, 1).But since A and B must be positive, we can consider all these pairs. Let's check each pair.Case 1: A = 1, B = 45.So,2x - 2y + 3 = 1 --> 2x - 2y = -2 --> x - y = -1.and2x + 2y + 15 = 45 --> 2x + 2y = 30 --> x + y = 15.So, we have:x - y = -1x + y = 15Adding these equations:2x = 14 --> x = 7.Then y = x + 1 = 8.Check if these are positive integers. x=7, y=8. Yes.So let's check if N = x(x + 9) = 7*16 = 112, and N = y(y + 6) = 8*14 = 112. Yes, that works. So 112 is a possible solution.But let's check other cases to see if there are other solutions.Case 2: A = 3, B = 15.Then,2x - 2y + 3 = 3 --> 2x - 2y = 0 --> x = y.and2x + 2y +15 =15 --> 2x + 2y =0 --> x + y =0.But x and y are positive integers, so this case is impossible.Case 3: A =5, B =9.So,2x - 2y + 3 =5 --> 2x - 2y =2 --> x - y =1.and2x + 2y +15=9 -->2x + 2y= -6 -->x + y= -3.Again, x and y are positive, so this is impossible.Case 4: A=9, B=5.2x - 2y +3=9 -->2x -2y=6 -->x - y=3.and2x +2y +15=5 -->2x +2y= -10 -->x + y= -5.Impossible.Case5: A=15, B=3.2x -2y +3=15 -->2x -2y=12 -->x - y=6.and2x +2y +15=3 -->2x +2y= -12 -->x + y= -6.Impossible.Case6: A=45, B=1.2x -2y +3=45 -->2x -2y=42 -->x - y=21.and2x +2y +15=1 -->2x +2y= -14 -->x + y= -7.Impossible.So the only valid case is case1: x=7, y=8, giving N=112.But wait, let me verify if there are other possible factor pairs. Since 45 is positive, both A and B must be positive. So the pairs I considered are all possible.Therefore, N=112 is the only solution.Wait, but let me check again. Because when we multiplied by 4, we assumed x and y are integers. So 2x and 2y must be integers as well. So x and y could be half-integers? Wait, but x and y are the sides of the rectangles, which must be integers. Therefore, x and y must be integers, so 2x and 2y are even integers. Then, in the equations for A and B:A = 2x -2y +3. Since 2x and 2y are even, their difference is even, so A is even +3, which is odd.Similarly, B = 2x +2y +15. 2x +2y is even, so B is even +15, which is odd. Therefore, A and B are both odd. The factors of 45 that are odd are (1,45), (3,15), (5,9), etc. So the cases considered already cover all possible pairs. Therefore, indeed, the only solution is N=112.But let me confirm again by checking the factors of N=112.First rectangle: sides differing by 9. So factors of 112: 1 and 112 (diff 111), 2 and 56 (diff 54), 4 and 28 (diff 24), 7 and 16 (diff 9). Oh, yes, 7 and 16. So 16-7=9. Good.Second rectangle: sides differing by 6. Factors: 8 and 14 (14-8=6). 8*14=112. Correct. So yes, 112 is the number. Therefore, the answer is 112.But just to make sure there are no other Ns. Suppose, for example, if there is another factor pair. But according to the equation above, the only possible solution is N=112. Therefore, the answer should be 112.Final AnswerThe value of ( N ) is boxed{112}.</think>Given that the eccentric has ( N ) unit squares which can form two different rectangles with sides differing by 9 and 6 respectively, we need to find ( N ).First, we denote the sides of the first rectangle as ( x ) and ( x + 9 ), so ( N = x(x + 9) ). For the second rectangle, the sides are ( y ) and ( y + 6 ), so ( N = y(y + 6) ). Therefore, we have the equation:[x(x + 9) = y(y + 6)]Expanding and rearranging terms, we get:[x^2 + 9x = y^2 + 6y]Completing the square for both ( x ) and ( y ):[(x + 4.5)^2 - 20.25 = (y + 3)^2 - 9]Simplifying, we get:[(x + 4.5)^2 - (y + 3)^2 = 11.25]Multiplying through by 4 to clear decimals:[(2x + 9)^2 - (2y + 6)^2 = 45]This can be factored as a difference of squares:[(2x + 9 - 2y - 6)(2x + 9 + 2y + 6) = 45]Simplifying the factors:[(2x - 2y + 3)(2x + 2y + 15) = 45]Considering the factor pairs of 45, we solve for ( x ) and ( y ). The valid pair leads to:[2x - 2y + 3 = 1 quad text{and} quad 2x + 2y + 15 = 45]Solving these, we find:[x - y = -1 quad text{and} quad x + y = 15]Adding these equations, we get ( x = 7 ) and ( y = 8 ). Thus, ( N = 7 times 16 = 112 ) and ( N = 8 times 14 = 112 ).Therefore, the value of ( N ) is (boxed{112}).]

🔑:Solution 1: If we denote the smaller sides of the constructed rectangles by x and y, we get the equation N=x(x+9)=y(y+6). Multiply both sides by 4.begin{gathered}4 x^{2}+36 x=4 y^{2}+24 y (2 x+9)^{2}-81=(2 y+6)^{2}-36 (2 x+2 y+15)(2 x-2 y+3)=45end{gathered}Both brackets on the left side are integers, and the left one is strictly greater than 15. The only divisor of the number 45 that is greater than 15 is the number 45 itself. Therefore, 2 x+2 y+15=45, 2 x-2 y+3=1. Solving the system, we find x=7, y=8. Hence, N=7 cdot 16.Solution 2: Notice that if x geqslant y, then the left side is at least y(y+9)>y(y+6), and if y geqslant x+2, then the right side is at least (x+2)(x+8)=x^{2}+10 x+16>x^{2}+9 x=x(x+9). Therefore, y=x+1, from which it is not difficult to obtain x=7 and y=8.## Criteria:- When checking divisors, only natural divisors are considered, and the positivity of the two factors is not obvious - up to 2 points are deducted;- The answer contains extraneous solutions with non-positive x and y - 2 points are deducted.

🧠:[<think>Okay, so I need to find the value of k such that the points A, B, and C are collinear. The coordinates of these points are given in vector form: OA is (k,2), OB is (1,2k), and OC is (1−k,−1). Since they are distinct points and collinear, the slope between any two pairs of points should be the same. Alternatively, I can use the area of the triangle formed by these points; if the area is zero, they are collinear. Let me think which method would be better here.First, let's recall that three points are collinear if the vectors formed between them are parallel. For example, the vector from A to B and the vector from A to C should be scalar multiples of each other. Or, equivalently, the determinant of the matrix formed by these vectors should be zero. That might be a straightforward approach.Alternatively, using slopes: the slope from A to B should equal the slope from B to C. Let me try both methods and see which one works better.Starting with the slope method. Let's compute the coordinates of points A, B, and C. Since OA is (k,2), point A is (k,2). Similarly, point B is (1,2k), and point C is (1−k,−1).So, the slope between A and B is (2k - 2)/(1 - k). The slope between B and C is (-1 - 2k)/( (1 - k) - 1 ). Let me compute that denominator: (1 - k) - 1 = -k. So slope BC is (-1 - 2k)/(-k) = (1 + 2k)/k.Setting the two slopes equal:(2k - 2)/(1 - k) = (1 + 2k)/kCross-multiplying:k(2k - 2) = (1 - k)(1 + 2k)Let's expand both sides:Left side: 2k^2 - 2kRight side: (1)(1) + 1(2k) - k(1) - k(2k) = 1 + 2k - k - 2k^2 = 1 + k - 2k^2So set left = right:2k^2 - 2k = 1 + k - 2k^2Bring all terms to left side:2k^2 - 2k - 1 - k + 2k^2 = 0Combine like terms:(2k^2 + 2k^2) + (-2k - k) + (-1) = 04k^2 - 3k - 1 = 0So quadratic equation: 4k² -3k -1 =0Let me solve this quadratic. The discriminant is D = 9 + 16 = 25. So sqrt(D)=5.Solutions: k = [3 ±5]/8Thus, k = (3+5)/8 = 8/8 =1, or k=(3-5)/8= -2/8= -1/4But we need to check if these solutions give distinct points. The problem states that the points A, B, C are distinct.First, check k=1:If k=1, then OA=(1,2), OB=(1,2*1)=(1,2), OC=(1-1,-1)=(0,-1). So points A and B would both be (1,2), which are not distinct. Hence, k=1 is invalid.Now check k=-1/4:OA=(-1/4, 2), OB=(1, 2*(-1/4))=(1, -1/2), OC=(1 - (-1/4), -1)=(1 +1/4, -1)=(5/4, -1). All three points are different. So k=-1/4 is valid.Therefore, the answer is k=-1/4.Wait, but let me verify with the determinant method to make sure.The determinant for collinearity of three points (x1,y1), (x2,y2), (x3,y3) is:|x1 y1 1||x2 y2 1||x3 y3 1|If this determinant is zero, then the points are collinear.So let's compute this determinant with points A, B, C.Expressed as a matrix:|k 2 1||1 2k 1||1−k −1 1|Compute the determinant:k*(2k*1 - (-1)*1) - 2*(1*1 - (1−k)*1) + 1*(1*(-1) - (1−k)*2k)Let me compute each term step by step.First term: k*(2k*1 - (-1)*1) = k*(2k +1)Second term: -2*(1*1 - (1−k)*1) = -2*(1 - (1 -k)) = -2*(k)Third term: 1*(1*(-1) - (1 -k)*2k) = 1*(-1 -2k +2k²)So the entire determinant is:k*(2k +1) -2k + (-1 -2k +2k²)Simplify term by term:First term: 2k² +kSecond term: -2kThird term: -1 -2k +2k²Combine all terms:2k² +k -2k -1 -2k +2k²Combine like terms:(2k² +2k²) + (k -2k -2k) + (-1)=4k² -3k -1Set determinant to zero: 4k² -3k -1=0, which is the same equation as before. So determinant method gives same equation. Hence, the solutions k=1 and k=-1/4. As before, k=1 invalid, so k=-1/4 is the answer.So the answer is k=-1/4.Just to make sure, let's check the slopes with k=-1/4.First, compute the coordinates:OA=(-1/4, 2), so A is (-1/4, 2)OB=(1, 2*(-1/4))=(1, -1/2)OC=(1 - (-1/4), -1)=(5/4, -1)Compute slope AB: (yB - yA)/(xB - xA)= (-1/2 - 2)/(1 - (-1/4))= (-5/2)/(5/4)= (-5/2)*(4/5)= -2Slope BC: (yC - yB)/(xC - xB)= (-1 - (-1/2))/(5/4 -1)= (-1/2)/(1/4)= -2Slopes are equal, so they are collinear. Therefore, correct.Another check: compute vectors AB and AC and see if they are scalar multiples.Vector AB: OB - OA = (1 - (-1/4), -1/2 -2) = (5/4, -5/2)Vector AC: OC - OA = (5/4 - (-1/4), -1 -2) = (6/4, -3) = (3/2, -3)Check if AB is a scalar multiple of AC:Let’s see. If AB = m * AC, then:5/4 = m*(3/2) => m= (5/4)/(3/2)= (5/4)*(2/3)=10/12=5/6Then check the y-component: -5/2 = m*(-3)= (5/6)*(-3)= -15/6= -5/2. Which matches. Therefore, vectors AB and AC are scalar multiples (m=5/6), so points are collinear. Correct.So the answer is definitely k=-1/4.Final AnswerThe value of ( k ) is boxed{-dfrac{1}{4}}.</think>Given the vectors (overrightarrow{OA} = (k, 2)), (overrightarrow{OB} = (1, 2k)), and (overrightarrow{OC} = (1 - k, -1)), we need to find the value of (k) such that the points (A), (B), and (C) are collinear.First, we compute the coordinates of points (A), (B), and (C):- Point (A) is ((k, 2))- Point (B) is ((1, 2k))- Point (C) is ((1 - k, -1))To determine collinearity, we use the condition that the slope between any two pairs of points should be the same. We calculate the slopes between (A) and (B), and between (B) and (C):1. Slope between (A) and (B): [ text{slope}_{AB} = frac{2k - 2}{1 - k} ]2. Slope between (B) and (C): [ text{slope}_{BC} = frac{-1 - 2k}{(1 - k) - 1} = frac{-1 - 2k}{-k} = frac{1 + 2k}{k} ]Setting these slopes equal gives:[frac{2k - 2}{1 - k} = frac{1 + 2k}{k}]Cross-multiplying and simplifying:[k(2k - 2) = (1 - k)(1 + 2k)][2k^2 - 2k = 1 + k - 2k^2][4k^2 - 3k - 1 = 0]Solving the quadratic equation (4k^2 - 3k - 1 = 0) using the quadratic formula:[k = frac{3 pm sqrt{25}}{8} = frac{3 pm 5}{8}][k = 1 quad text{or} quad k = -frac{1}{4}]Checking the distinctness of points for (k = 1) and (k = -frac{1}{4}):- For (k = 1), points (A) and (B) coincide, so (k = 1) is invalid.- For (k = -frac{1}{4}), all points (A), (B), and (C) are distinct.Verification using the determinant method confirms the result. The determinant of the matrix formed by coordinates of (A), (B), and (C) is zero for (k = -frac{1}{4}), confirming collinearity.Thus, the value of (k) is (boxed{-dfrac{1}{4}}).,<think>Okay, so I need to find the value of k where the points A, B, and C are collinear. The coordinates of these points are given in vector form: OA is (k, 2), OB is (1, 2k), and OC is (1 - k, -1). Since they are distinct and collinear, they must lie on the same straight line. First, I remember that three points are collinear if the slope between any two pairs of points is the same. So, maybe I can calculate the slopes between A and B, B and C, and A and C, then set them equal to each other. That should give me equations to solve for k.Alternatively, another method I recall is that three points are collinear if the area of the triangle they form is zero. The area can be calculated using the determinant formula: Area = (1/2) |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|If the area is zero, then the points are collinear. Maybe that's a more straightforward way because it's just one equation instead of comparing multiple slopes. Let me try that.So, let's assign the coordinates:Point A: (k, 2)Point B: (1, 2k)Point C: (1 - k, -1)Plugging into the area formula:0 = (1/2) |k*(2k - (-1)) + 1*(-1 - 2) + (1 - k)*(2 - 2k)|Since the area is zero, the expression inside the absolute value must be zero. So:k*(2k + 1) + 1*(-3) + (1 - k)*(2 - 2k) = 0Let me compute each term step by step.First term: k*(2k + 1) = 2k² + kSecond term: 1*(-3) = -3Third term: (1 - k)*(2 - 2k). Let me expand this:Multiply 1 by 2: 2Multiply 1 by (-2k): -2kMultiply (-k) by 2: -2kMultiply (-k) by (-2k): 2k²So altogether: 2 - 2k - 2k + 2k² = 2 - 4k + 2k²So adding all three terms:First term: 2k² + kSecond term: -3Third term: 2k² - 4k + 2Combine all terms:2k² + k - 3 + 2k² - 4k + 2Combine like terms:2k² + 2k² = 4k²k - 4k = -3k-3 + 2 = -1So the equation becomes:4k² - 3k - 1 = 0Now, solve the quadratic equation 4k² - 3k - 1 = 0Quadratic formula: k = [3 ± sqrt(9 + 16)] / 8Because discriminant D = (-3)² - 4*4*(-1) = 9 + 16 = 25So sqrt(D) = 5Thus, k = [3 ± 5]/8So two solutions:k = (3 + 5)/8 = 8/8 = 1k = (3 - 5)/8 = (-2)/8 = -1/4But the problem states that the points A, B, C are distinct. So we need to check if k = 1 or k = -1/4 result in any overlapping points.First, check k = 1:Point A: OA = (1, 2)Point B: OB = (1, 2*1) = (1, 2)Point C: OC = (1 - 1, -1) = (0, -1)Wait, points A and B would both be (1, 2). That means A and B are the same point, which contradicts the condition that they are distinct. Therefore, k = 1 is invalid.Now check k = -1/4:Point A: OA = (-1/4, 2)Point B: OB = (1, 2*(-1/4)) = (1, -1/2)Point C: OC = (1 - (-1/4), -1) = (5/4, -1)So all three points:A: (-1/4, 2)B: (1, -1/2)C: (5/4, -1)Are these distinct? Let's see:Coordinates of A: (-0.25, 2)Coordinates of B: (1, -0.5)Coordinates of C: (1.25, -1)All x-coordinates and y-coordinates are different. None of the points share both x and y. So yes, they are distinct. Therefore, k = -1/4 is the valid solution.Hence, the value of k is -1/4.Wait, just to make sure I didn't make any mistakes in my calculations. Let me verify the collinearity when k = -1/4.Compute the slope between A and B:Slope AB = (y_B - y_A)/(x_B - x_A) = (-1/2 - 2)/(1 - (-1/4)) = (-5/2)/(5/4) = (-5/2) * (4/5) = -2Slope between B and C:Slope BC = (y_C - y_B)/(x_C - x_B) = (-1 - (-1/2))/(5/4 - 1) = (-1/2)/(1/4) = -2Slope between A and C:Slope AC = (y_C - y_A)/(x_C - x_A) = (-1 - 2)/(5/4 - (-1/4)) = (-3)/(6/4) = (-3)*(4/6) = -2All slopes are equal to -2, so they are collinear. That checks out. So k = -1/4 is correct. The other solution k = 1 leads to overlapping points A and B, which is invalid, so we discard it. Therefore, the answer is k = -1/4.Final AnswerThe value of ( k ) is boxed{-dfrac{1}{4}}.</think>Given the vectors (overrightarrow{OA} = (k, 2)), (overrightarrow{OB} = (1, 2k)), and (overrightarrow{OC} = (1 - k, -1)), we need to find the value of (k) such that the distinct points (A), (B), and (C) are collinear.To determine collinearity, we use the area of the triangle formed by the points. The area must be zero for the points to be collinear. The area formula is:[text{Area} = frac{1}{2} left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) right|]Substituting the coordinates of points (A), (B), and (C):[0 = frac{1}{2} left| k(2k - (-1)) + 1(-1 - 2) + (1 - k)(2 - 2k) right|]Simplifying the expression inside the absolute value:[k(2k + 1) + 1(-3) + (1 - k)(2 - 2k) = 0]Expanding and combining like terms:[2k^2 + k - 3 + 2k^2 - 4k + 2 = 0][4k^2 - 3k - 1 = 0]Solving the quadratic equation (4k^2 - 3k - 1 = 0) using the quadratic formula (k = frac{-b pm sqrt{b^2 - 4ac}}{2a}):[k = frac{3 pm sqrt{9 + 16}}{8} = frac{3 pm 5}{8}]This gives two solutions:[k = frac{8}{8} = 1 quad text{and} quad k = frac{-2}{8} = -frac{1}{4}]Checking for distinct points:- For (k = 1), points (A) and (B) coincide, which is invalid.- For (k = -frac{1}{4}), all points (A), (B), and (C) are distinct.Verifying collinearity for (k = -frac{1}{4}) confirms that the slopes between all pairs of points are equal (-2), confirming collinearity.Thus, the value of (k) is (boxed{-dfrac{1}{4}}).]

🔑:Since overrightarrow{OA}=(k,2), overrightarrow{OB}=(1,2k), overrightarrow{OC}=(1-k,-1), and the distinct points A, B, C are collinear, we have:overrightarrow{AB}=(1-k,2k-2),quad overrightarrow{BC}=(-k,-1-2k).Since A, B, and C are collinear, overrightarrow{AB} and overrightarrow{BC} are parallel. Therefore, the determinant of the matrix formed by these vectors should be 0:(1-k)(-1-2k)-(2k-2)(-k)=0.Expanding and simplifying the equation, we obtain:k^2 - frac{5}{2}k + 1 = 0.Solving for k, we have k=1 or k=-frac{1}{4}. However, when k=1, points A and B coincide, which contradicts the condition that A, B, and C are distinct points. Hence, we discard this solution.Thus, the answer is boxed{k=-frac{1}{4}}.

🧠:[<think>Okay, let's see. I need to check which of these four statements are correct. The options are A, B, C, D, indicating how many are correct. Let me go through each statement one by one.First, statement ①: "If a + b = 0, then a and b are opposite numbers." Hmm. Opposite numbers... I think that means additive inverses. So, if a + b = 0, then b = -a, right? So yes, they are opposites. So that should be correct. So ① is correct.Statement ②: "3² = 6." Wait, 3 squared is 9, not 6. So that's definitely incorrect. So ② is wrong.Statement ③: "If a is a rational number, then a must be greater than -a." Let me think. Rational numbers include positive and negative numbers, and zero. So if a is positive, then -a is negative, so a > -a. If a is negative, then -a is positive, so -a > a. If a is zero, then a = -a. So the statement says "must be greater than," but if a is negative or zero, it's not true. For example, take a = -1, which is rational. Then -a = 1, so -1 < 1, so the statement "a must be greater than -a" is false here. Similarly, if a = 0, then 0 is not greater than 0. So statement ③ is incorrect.Statement ④: "If |a| = |b|, then a = b." The absolute values being equal means that a and b are either equal or negatives of each other. For example, if a = 1 and b = -1, then |a| = |b|, but a ≠ b. So the conclusion a = b doesn't necessarily follow. Therefore, statement ④ is false.So summarizing:① Correct② Incorrect③ Incorrect④ IncorrectSo only one correct statement. Therefore, the answer should be A: 1 correct statement.Final Answerboxed{A}</think>Let's analyze each statement one by one:1. Statement ①: "If (a + b = 0), then (a) and (b) are opposite numbers." This is correct because if (a + b = 0), then (b = -a), which means (a) and (b) are additive inverses (opposite numbers).2. Statement ②: "(3^2 = 6)." This is incorrect because (3^2 = 9), not 6.3. Statement ③: "If (a) is a rational number, then (a) must be greater than (-a)." This is incorrect. For example, if (a) is a negative rational number (e.g., (a = -1)), then (-a) is positive, so (a < -a). If (a = 0), then (a = -a), so the statement does not hold.4. Statement ④: "If (|a| = |b|), then (a = b)." This is incorrect because (|a| = |b|) implies (a = b) or (a = -b). For example, (a = 1) and (b = -1) have the same absolute value but are not equal.Summarizing the results:- Correct statement: ①- Incorrect statements: ②, ③, ④Thus, there is only 1 correct statement. The answer is (boxed{A}).,<think>Alright, let's see. I have four statements here, and I need to figure out which ones are correct. Then, based on that, choose the answer from A to D. Let me go through each statement one by one.Starting with statement ①: "If a + b = 0, then a and b are opposite numbers." Hmm. Opposite numbers... I think that means additive inverses, right? So if a + b = 0, then b = -a. So yes, that should mean that a and b are opposites. For example, if a is 5, then b has to be -5. So statement ① seems correct. Let me check again. If two numbers add up to zero, they must be negatives of each other. Yep, that's the definition. So I agree, ① is correct.Moving on to statement ②: "3² = 6." Wait, 3 squared is 9, not 6. That's straightforward. 3 times 3 is 9. So ② is definitely incorrect. So ② is wrong.Statement ③: "If a is a rational number, then a must be greater than -a." Hmm. Rational numbers are numbers that can be expressed as a fraction of integers. But does that imply a > -a? Let's think. If a is positive, then -a is negative, so a > -a. For example, a = 2, then -a = -2, and 2 > -2. But if a is negative, then -a is positive. So, let's take a = -3. Then -a = 3. So, is -3 > 3? No, that's not true. So if a is negative, then a < -a. Therefore, the statement says "if a is rational, then a > -a." But this isn't always true. It depends on whether a is positive or negative. So the statement is false because it doesn't hold for negative rational numbers. So ③ is incorrect.Statement ④: "If |a| = |b|, then a = b." The absolute values being equal means that a and b are either equal or opposites. For example, |5| = |5|, so a = b. But also, |5| = |-5|, but 5 ≠ -5. Therefore, the statement is not necessarily true. So ④ is incorrect.So, summarizing:① Correct② Incorrect③ Incorrect④ IncorrectSo only one correct statement, which is ①. Therefore, the answer should be option A: 1 correct statement.Wait, let me double-check statement ③. The problem says, "If a is a rational number, then a must be greater than -a." So if a is a rational number, does it have to be greater than its negative? As I thought before, if a is positive, yes. If a is zero, then a = -a, so 0 = 0. So 0 is not greater than 0. So even if a is zero, which is rational, the statement "a must be greater than -a" fails. So if a is zero, which is rational, then a is equal to -a. So the statement would be false for a = 0 as well. So that gives another reason why ③ is incorrect. So yes, definitely, ③ is wrong.And ④, the absolute value. If |a| = |b|, then a and b are equal or opposites. So unless both a and b are non-negative, they don't have to be equal. So yeah, ④ is incorrect.Therefore, only statement ① is correct. So the correct answer is A.Final Answerboxed{A}</think>Let's analyze each statement one by one:1. Statement ①: If (a + b = 0), then (a) and (b) are opposite numbers. - This is correct because if (a + b = 0), then (b = -a), which means (a) and (b) are additive inverses (opposite numbers).2. Statement ②: (3^2 = 6). - This is incorrect because (3^2 = 9), not 6.3. Statement ③: If (a) is a rational number, then (a) must be greater than (-a). - This is incorrect. If (a) is a positive rational number, then (a > -a). However, if (a) is negative, then (a < -a). Additionally, if (a = 0), then (a = -a), so (a) is not greater than (-a).4. Statement ④: If (|a| = |b|), then (a = b). - This is incorrect. The absolute values being equal means (a) and (b) can be either equal or opposites (e.g., (a = 5) and (b = -5)).Summarizing the results:- Statement ① is correct.- Statements ②, ③, and ④ are incorrect.Thus, there is only 1 correct statement. The answer is (boxed{A}).]

🔑:Let's analyze each statement step by step:1. Statement ①: If a+b=0, then a and b are opposite numbers. This statement is based on the definition of opposite numbers. If a+b=0, then we can rewrite this as a=-b, which means a and b are indeed opposite numbers. Therefore, statement ① is boxed{text{correct}}.2. Statement ②: 3^{2}=6. Calculating the left side gives us 3^{2} = 3 times 3 = 9, which is not equal to 6. Therefore, statement ② is boxed{text{incorrect}}.3. Statement ③: If a is a rational number, then a must be greater than -a. This statement does not hold true for all rational numbers. For example, let's consider a=-1, which is a rational number. In this case, -a = -(-1) = 1, which is greater than a. Therefore, statement ③ is boxed{text{incorrect}}.4. Statement ④: If |a|=|b|, then a=b. The absolute value of a number represents its distance from 0 on the number line, regardless of direction. Therefore, if |a|=|b|, it means that a and b are the same distance from 0, but they could be on opposite sides of 0. This implies a could be b or -b. Hence, statement ④ is boxed{text{incorrect}}.In conclusion, after analyzing each statement, we find that only 1 statement is correct.Therefore, the answer is: boxed{A}.