🧠:[<think>Okay, so I need to find the number of space diagonals in this convex polyhedron P. Let me recall what a space diagonal is. The problem says it's a line segment connecting two non-adjacent vertices that do not belong to the same face. Hmm. So first, I think I should figure out the total number of diagonals possible in the polyhedron and then subtract the ones that are either face diagonals or edges. Wait, the total number of line segments between any two vertices is given by the combination formula C(n, 2), where n is the number of vertices. Here, there are 26 vertices, so that would be 26*25/2 = 325. But this counts all possible line segments between vertices. From this, we need to subtract the edges (since edges are not diagonals) and the face diagonals (since those are diagonals but not space diagonals). Then what's left should be the space diagonals.Let me write that down:Total space diagonals = Total number of vertex pairs - Number of edges - Number of face diagonals.So, total vertex pairs: 26C2 = 325. Number of edges is given as 60. So subtract 60 from 325, which gives 265. Then subtract the face diagonals. Now, how many face diagonals are there?The polyhedron has two types of faces: triangular and quadrilateral. For each face, we can compute the number of diagonals it has. For a triangle, since it's a 3-sided polygon, there are no diagonals because all vertices are adjacent. For a quadrilateral (4-sided polygon), the number of diagonals in each is 2. Because in a quadrilateral, each vertex connects to two non-adjacent vertices, but each diagonal is counted twice (once from each end), so it's actually 4*1/2 = 2 diagonals per quadrilateral.So, since there are 24 triangular faces and 12 quadrilateral faces. The triangular faces contribute 0 diagonals each, so 24*0=0. The quadrilateral faces contribute 2 diagonals each, so 12*2=24. So total face diagonals are 24.Therefore, the total space diagonals would be 325 - 60 - 24 = 325 - 84 = 241.Wait, but hold on. Is that correct? Let me double-check. The problem states that a space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. So, the face diagonals are diagonals on the same face, hence they are not space diagonals. So yes, we need to subtract those. So 24 face diagonals. So 325 total vertex pairs minus 60 edges minus 24 face diagonals is 241. But I need to confirm if this is correct.Alternatively, another way to compute space diagonals is to consider for each vertex, how many space diagonals come out of it. For a single vertex, the number of space diagonals would be total vertices minus 1 (all other vertices) minus the number of adjacent vertices (connected by edges) minus the number of vertices on the same face but not adjacent (i.e., face diagonals). Then, since each space diagonal is counted twice (once from each end), we can divide by 2.But maybe this approach would help verify. Let me try that.First, let's consider one vertex. How many edges does it have? Well, in a polyhedron, each vertex is where several edges meet. To find the number of edges per vertex, we can use the formula related to the degrees. But maybe Euler's formula first? Wait, Euler's formula says V - E + F = 2. Let's check if the given numbers satisfy Euler's formula. 26 - 60 + 36 = 26 - 60 is -34, plus 36 is 2. Yes, that works. So Euler's formula holds, which is good.Now, to find the number of edges per vertex. The total number of edges is 60. Each edge is connected to two vertices, so the sum of the degrees of all vertices is 2*60 = 120. Therefore, the average degree per vertex is 120/26 ≈ 4.615. Hmm, but individual degrees might vary. However, since the faces are triangles and quadrilaterals, perhaps the vertices are part of different numbers of edges.But maybe instead of getting into that, let's consider a general vertex. Let's assume that each vertex is part of some triangular faces and some quadrilateral faces. Each face that a vertex is part of contributes edges. For a triangular face, each vertex is part of two edges, and for a quadrilateral face, each vertex is part of two edges as well. Wait, no. Wait, in a triangular face, each vertex is connected to two other vertices, forming the triangle. Similarly, in a quadrilateral face, each vertex is connected to two adjacent vertices. So each face that a vertex is part of contributes two edges. However, if a vertex is part of, say, m triangular faces and n quadrilateral faces, then the total number of edges at that vertex is m + n. Wait, no. Wait, actually, each triangular face that a vertex is part of gives two edges, but since each edge is shared by two faces, maybe not. Hmm, perhaps this is getting complicated. Maybe there's a better way.Alternatively, since the polyhedron is convex, each edge is shared by exactly two faces. The total number of face edges can be calculated as follows: each triangular face has 3 edges, each quadrilateral face has 4 edges. So total face edges would be 24*3 + 12*4 = 72 + 48 = 120. But since each edge is shared by two faces, the total number of edges is 120/2 = 60, which matches the given number of edges. That checks out.But how does this help me with the number of edges per vertex? Maybe not directly. Let me return to the original problem. The space diagonals are those connecting two vertices that are neither adjacent nor on the same face. So, for a given vertex, the number of space diagonals from it would be total vertices minus 1 (all other vertices) minus adjacent vertices (edges) minus the number of other vertices on the same face but not adjacent (face diagonals). So, for a vertex, if it is part of, say, t triangular faces and q quadrilateral faces, how many vertices are adjacent to it? Each triangular face contributes two adjacent vertices, and each quadrilateral face contributes two adjacent vertices. Wait, but each edge is shared between two faces. So if a vertex is part of m edges, then it has m adjacent vertices. Wait, the degree of the vertex is equal to the number of edges incident to it, which is the same as the number of adjacent vertices. So, if the average degree is 120/26 ≈ 4.615, but individual vertices can vary. But since all faces are triangles and quadrilaterals, the degrees might be consistent?Wait, in a convex polyhedron where all faces are triangles and quadrilaterals, the degree of each vertex can be calculated based on the number of faces meeting at that vertex. Each triangular face has three vertices, each contributing one 'connection', and each quadrilateral face has four vertices. But how does that translate to the degree? Hmm. Wait, each edge is shared by two faces. So, if a vertex is part of t triangular faces and q quadrilateral faces, then the number of edges at that vertex is equal to the number of edges contributed by the faces. For each triangular face, the vertex is part of two edges (since in a triangle, each vertex is connected to two others), and for each quadrilateral face, the vertex is part of two edges as well. Wait, but if a face is a triangle, then each vertex in the triangle has two edges in that face. Similarly, a quadrilateral face has each vertex connected to two edges. However, if a vertex is part of multiple faces, the edges are shared between the faces. Therefore, the degree of the vertex (number of edges) is equal to the number of faces meeting at that vertex. Wait, no. For example, in a cube, which is a convex polyhedron with quadrilateral faces, each vertex is part of three faces, and indeed each vertex has three edges. So, in general, for a convex polyhedron, the degree of a vertex is equal to the number of faces meeting at that vertex. Therefore, if all the faces are triangles and quadrilaterals, the degree of each vertex is equal to the number of faces meeting at it.But in our case, the polyhedron has 24 triangular faces and 12 quadrilateral faces. The total number of face-vertex incidences is 24*3 + 12*4 = 72 + 48 = 120. Since each vertex is part of several faces, the number of faces per vertex can be calculated by dividing this total by the number of vertices: 120/26 ≈ 4.615. But that would be the average number of faces per vertex. However, since each face is a triangle or quadrilateral, each face contributes 3 or 4 vertices. Hmm. Wait, but each vertex is part of a certain number of faces, so the average degree is 120/26 ≈ 4.615, which matches the average degree we calculated earlier (sum of degrees is 120). So each vertex is part of approximately 4.615 faces, which is the same as the average degree. Therefore, the degree of each vertex is equal to the number of edges incident to it, which is equal to the number of faces meeting at that vertex.But since the polyhedron is convex and all faces are triangles and quadrilaterals, perhaps all vertices have degrees either 4 or 5? Because 4.615 is between 4 and 5. Let me check. If some vertices have degree 4 and others 5, then 4x + 5y = 120, and x + y = 26. So solving these equations: x = 26 - y, so 4(26 - y) + 5y = 120 => 104 - 4y +5y = 120 => y = 16. Then x = 10. So 10 vertices of degree 4 and 16 vertices of degree 5. Hmm, possible. But does this matter for the problem? Maybe not. Since the problem doesn't specify the degrees of the vertices, perhaps we can proceed without that information.But going back to calculating the number of space diagonals per vertex. For each vertex, the number of space diagonals is total vertices minus 1 (all other vertices) minus adjacent vertices (edges) minus vertices on the same face but not adjacent (face diagonals). So, let's denote for a vertex v:space_diagonals(v) = (26 - 1) - degree(v) - face_diagonals(v)But we need to compute face_diagonals(v). For each face that v is part of, how many non-adjacent vertices are there in that face? For a triangular face, there are no non-adjacent vertices. For a quadrilateral face, each vertex is part of two edges, and the other two vertices in the face are non-adjacent (but connected by a face diagonal). Wait, in a quadrilateral, each vertex is connected to two adjacent vertices, and the other two vertices are diagonally opposite. So for each quadrilateral face that a vertex is part of, there is one face diagonal from that vertex. Because in a quadrilateral, each vertex has two adjacent vertices and one non-adjacent vertex (but since diagonals are two, but per vertex, it's one). Wait, no. Wait, in a quadrilateral, each vertex is not adjacent to one other vertex. Wait, no, each vertex in a quadrilateral is connected to two adjacent vertices, and not connected to the opposite vertex. But in a quadrilateral, each vertex has one face diagonal. Because connecting to the vertex not adjacent. Wait, no. For a quadrilateral, the diagonals are two: connecting opposite vertices. So each quadrilateral has two face diagonals, each connecting a pair of opposite vertices. Therefore, for each quadrilateral face that a vertex is part of, that vertex is an endpoint of one face diagonal. Because there are two diagonals in the quadrilateral, each involving two vertices. So each vertex in a quadrilateral is part of one face diagonal.Therefore, if a vertex is part of q quadrilateral faces, then the number of face diagonals from that vertex is q. Because in each quadrilateral face, it has one face diagonal. But also, we need to ensure we are not double-counting. Wait, but if we are calculating per vertex, then for each quadrilateral face that the vertex is part of, it can have a face diagonal to the opposite vertex. So if a vertex is part of, say, two quadrilateral faces, then in each of those two quadrilaterals, it has one face diagonal. Therefore, for a vertex that is part of m quadrilateral faces, it has m face diagonals. However, note that these face diagonals could be duplicates if two quadrilaterals share a face diagonal, but in reality, each face is distinct, so the diagonals are in different faces. So, no, actually, each face is a separate face, so the diagonals are unique to each face.Therefore, the number of face diagonals from a vertex is equal to the number of quadrilateral faces that the vertex is part of. So, if we can find how many quadrilateral faces each vertex is part of, we can compute the face diagonals per vertex. But we don't have that information. However, since there are 12 quadrilateral faces, each has 4 vertices, so total vertex-face incidences for quadrilaterals are 12*4 = 48. Since there are 26 vertices, the average number of quadrilateral faces per vertex is 48/26 ≈ 1.846. Similarly, triangular faces: 24*3 = 72 vertex-face incidences, so 72/26 ≈ 2.769 per vertex. Therefore, the average vertex is part of approximately 2.769 triangular faces and 1.846 quadrilateral faces. But since you can't have fractions of faces, the actual numbers must be integers. So some vertices are part of 1 or 2 quadrilateral faces, others 2 or 3, etc.But maybe instead of worrying about individual vertices, since we know the total number of face diagonals in the polyhedron. Earlier, we calculated that there are 12 quadrilateral faces, each with 2 face diagonals, so total face diagonals is 24. Therefore, regardless of how they're distributed among vertices, the total is 24. So in the first approach, subtracting 24 from the total gives us 241. But is that correct?Wait, but in the first approach, we calculated total vertex pairs (325), subtracted edges (60), subtracted face diagonals (24), to get 241. But does this account for all the non-space diagonals? Let me confirm.Yes, because in a convex polyhedron, any two vertices are either connected by an edge, a face diagonal, or a space diagonal. Since it's convex, there are no intersecting edges or anything; the diagonals are either on the face or through the interior. So yes, the total number of vertex pairs is edges + face diagonals + space diagonals. Therefore, space diagonals = total vertex pairs - edges - face diagonals. So 325 - 60 - 24 = 241. So the answer should be 241. But let me check with another method to confirm.Alternatively, using the formula for space diagonals in a polyhedron: for any convex polyhedron, the number of space diagonals is (V*(V-3)/2) - E - F_d, where V is the number of vertices, E edges, and F_d face diagonals. Wait, but (V*(V - 3)/2) is the total number of diagonals (both face and space) in a complete graph of V vertices, minus the edges. Wait, no. Wait, in a polygon with V vertices, the number of diagonals is V*(V - 3)/2. But in 3D, for a polyhedron, the total number of line segments between vertices is C(V, 2). Then, subtract edges and face diagonals to get space diagonals. So it's the same as the first approach. Therefore, the answer is indeed 241.But let me check with a simple example. Take a cube, which has 8 vertices, 12 edges, 6 faces (all quadrilaterals). Total vertex pairs: 28. Subtract edges (12) and face diagonals. Each face of the cube has 2 face diagonals, so 6*2=12. So total space diagonals would be 28 - 12 -12 = 4. But in a cube, the space diagonals are the ones connecting opposite vertices through the center, and there are 4 of them. So that works. So this formula works. Therefore, applying it here, 325 - 60 -24=241. So the answer is 241. Therefore, I think that's correct. But let me check another example. Take a tetrahedron, which has 4 vertices, 6 edges, 4 triangular faces. Total vertex pairs:6. Subtract edges (6) and face diagonals (0). So space diagonals=0. Which is correct, because in a tetrahedron, any two vertices are connected by an edge; there are no face diagonals or space diagonals. So that works as well. Therefore, the formula seems valid.Therefore, I think the answer is 241. So I should check if my calculation for face diagonals was correct. The problem states that there are 24 triangular faces and 12 quadrilateral faces. Each triangular face has 0 diagonals, each quadrilateral has 2. So 12*2=24. So subtract 24. So 325-60=265, 265-24=241. Yes, that seems right.But wait, just to be thorough, let's consider another way. For each vertex, the number of space diagonals is (26 - 1 - degree(v) - face_diagonals(v)). So if we sum over all vertices, we get sum over v [25 - degree(v) - face_diagonals(v)] = 25*26 - sum(degree(v)) - sum(face_diagonals(v)). Now, sum(degree(v)) is 2*E = 120. Sum(face_diagonals(v)) is 2*F_d, because each face diagonal is counted twice (once for each endpoint). Since there are 24 face diagonals, sum(face_diagonals(v)) is 2*24=48. Therefore, total space diagonals would be (25*26 - 120 - 48)/2 = (650 - 120 -48)/2 = (650 -168)/2=482/2=241. Yes, same answer. So this confirms it.Therefore, the number of space diagonals is 241.Final AnswerThe number of space diagonals in the polyhedron is boxed{241}.</think>To find the number of space diagonals in the convex polyhedron ( P ), we start by calculating the total number of line segments between any two vertices, which is given by the combination formula ( binom{26}{2} ):[binom{26}{2} = frac{26 times 25}{2} = 325]Next, we subtract the number of edges (60) and the number of face diagonals. The polyhedron has 24 triangular faces and 12 quadrilateral faces. Triangular faces have no diagonals, while each quadrilateral face has 2 diagonals. Therefore, the total number of face diagonals is:[12 times 2 = 24]Subtracting the edges and face diagonals from the total number of line segments gives us the number of space diagonals:[325 - 60 - 24 = 241]To verify, we consider the formula for space diagonals using the sum over all vertices. For each vertex, the number of space diagonals is ( 25 - text{degree}(v) - text{face_diagonals}(v) ). Summing over all vertices and dividing by 2 (since each space diagonal is counted twice), we get:[frac{25 times 26 - 120 - 48}{2} = frac{650 - 168}{2} = frac{482}{2} = 241]Thus, the number of space diagonals in the polyhedron is (boxed{241}).,<think>Alright, so I need to find the number of space diagonals in this convex polyhedron P. Let me recall what a space diagonal is. It's a line segment connecting two non-adjacent vertices that don't belong to the same face. Got it. So, first, maybe I should figure out the total number of diagonals possible and then subtract the ones that are on the faces or are edges, right?The problem gives me the number of vertices, edges, and faces. It's a convex polyhedron with 26 vertices, 60 edges, 36 faces (24 triangular and 12 quadrilateral). So, let me start by calculating the total number of line segments between any two vertices. Since each vertex can connect to 25 others, the total number of line segments is C(26, 2). Let me compute that.C(26, 2) = 26*25/2 = 325. So, there are 325 possible line segments connecting pairs of vertices.Now, out of these 325, some are edges of the polyhedron. The problem states there are 60 edges. So, subtract those. Then, the remaining are diagonals. But wait, these diagonals can be of two types: face diagonals and space diagonals. The space diagonals are the ones we need, so we have to subtract both the edges and the face diagonals from the total.So, the formula should be: Space diagonals = Total diagonals - Edges - Face diagonals. Wait, no. Wait, actually, the total line segments are 325. The edges are 60, so the diagonals (both face and space) are 325 - 60 = 265. Then, from these 265 diagonals, subtract the face diagonals to get the space diagonals.Yes, that makes sense. So, Space diagonals = Total line segments - Edges - Face diagonals.Therefore, I need to calculate the number of face diagonals. Each face contributes a certain number of diagonals. Since all faces are either triangular or quadrilateral, I can compute the number of diagonals per face type and sum them up.Triangles have no diagonals. Because in a triangle, any two non-adjacent vertices... but a triangle has three vertices, each connected to the other two, so there are no diagonals. So, each triangular face contributes 0 diagonals.Quadrilateral faces, on the other hand, have two diagonals each. For a quadrilateral (a four-sided figure), the number of diagonals is C(4,2) - 4 edges = 6 - 4 = 2 diagonals per quadrilateral.So, since there are 12 quadrilateral faces, each contributing 2 diagonals, the total number of face diagonals is 12 * 2 = 24.Therefore, the total number of face diagonals is 24. So, putting it all together:Total line segments: 325Minus edges: 60Minus face diagonals: 24Space diagonals = 325 - 60 - 24 = 325 - 84 = 241.Wait, hold on. But let me verify. Is this correct? So, in a convex polyhedron, any space diagonal is a diagonal that's not on a face. So, yes, subtracting edges and face diagonals gives the space diagonals.But let me check again. For each face:Triangular faces (24 in total): Each has 3 edges, but no diagonals. So, 24 * 0 = 0 diagonals.Quadrilateral faces (12 in total): Each has 4 edges and 2 diagonals. So, 12 * 2 = 24 face diagonals.So total face diagonals are 24. So, that seems right.Therefore, total number of space diagonals is 325 (total line segments) - 60 (edges) - 24 (face diagonals) = 241. Hmm. But I have a feeling that maybe this is not the answer they are expecting. Wait, let me check again.Alternatively, another way to compute space diagonals is to use the formula:Total number of space diagonals = (Number of vertices choose 2) - Number of edges - Number of face diagonals.Which is exactly what I did. So, 325 - 60 - 24 = 241. Hmm. So, 241.But wait, another thought: in a convex polyhedron, two vertices not on the same face cannot see each other through the interior, so a space diagonal is a line segment connecting two vertices not on the same face and not connected by an edge. But in a convex polyhedron, all space diagonals lie entirely inside the polyhedron. But is that affecting the count? Probably not. The count should still be as calculated.Alternatively, maybe there's a different formula. Let me recall that in a polyhedron, the number of space diagonals can also be calculated using the formula:Space diagonals = [V(V - 3)/2] - FdWhere V is the number of vertices, and Fd is the number of face diagonals. Let's see. Wait, V(V - 3)/2 is the total number of diagonals (both face and space) in a convex polyhedron. Because for each vertex, there are V - 1 connections, subtract 3: the vertex itself (but we're connecting to others, so maybe subtract the adjacent edges? Wait, no. Wait, perhaps in a polygon, the number of diagonals per vertex is (n - 3). So, for a polyhedron, each vertex is part of multiple faces, but maybe the formula is similar.Wait, actually, in general, for any convex polyhedron, the total number of diagonals (both face and space) is C(V, 2) - E. So that's 325 - 60 = 265. Then, subtract face diagonals (24) to get space diagonals: 265 - 24 = 241. So, same result. So, seems like 241 is the answer.But wait, let me check with an example. Let's take a cube. A cube has 8 vertices, 12 edges, 6 faces (all quadrilateral). So, total line segments: C(8,2)=28. Edges:12. So diagonals: 28 -12=16. Face diagonals: each face has 2 diagonals, 6 faces: 12. So, space diagonals: 16 -12=4. Which is correct, a cube has 4 space diagonals (connecting opposite vertices through the center). So, the formula works here. So, that seems to confirm the method.So, applying the same logic to the given polyhedron: 26 vertices. So, total line segments: 325. Edges:60. Total diagonals (face + space):325 -60=265. Face diagonals:24. So, space diagonals:265 -24=241. Therefore, the answer should be 241.Wait, but let me think again. The definition says a space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the same face. So, in other words, it's a diagonal that is not on any face and is not an edge. So, yes, so the total number is computed as:Total number of possible line segments: C(26,2)=325.Subtract edges:60. Subtract face diagonals:24.So, 325 -60 -24=241. So, yes, 241.Therefore, I think that's the answer.But just to be thorough, let me check if there is another way to compute this. For example, using Euler's formula. Wait, Euler's formula says V - E + F = 2. Let's check if that's satisfied here.Given V=26, E=60, F=36.26 -60 +36=2. 26+36=62, 62 -60=2. Yes, that's correct. So, the given polyhedron satisfies Euler's formula. So, that's a good check.Also, in terms of the faces, 24 triangular and 12 quadrilateral. Let's check the total number of edges contributed by the faces.Each triangular face has 3 edges, each quadrilateral has 4 edges. So total edges counted by faces:24*3 +12*4=72 +48=120. However, each edge is shared by two faces, so the actual number of edges should be 120/2=60, which matches the given. So, that's another check. So, the given data is consistent.Therefore, the previous calculation should hold. So, the number of space diagonals is 241.Wait, but let me check once more.Total number of line segments between vertices:26*25/2=325. Edges:60. So, non-edges:325 -60=265. These are all diagonals (face or space). Now, of these 265 diagonals, how many lie on the faces? Each triangular face has 0 diagonals. Each quadrilateral face has 2 diagonals. There are 12 quadrilateral faces, so 12*2=24 face diagonals. Therefore, the remaining diagonals are space diagonals:265 -24=241. So, yes, 241.Therefore, the answer is 241. So, boxed{241}Final Answerboxed{241}</think>To find the number of space diagonals in the convex polyhedron ( P ), we start by calculating the total number of line segments between any two vertices. This is given by the combination formula ( binom{26}{2} ):[binom{26}{2} = frac{26 times 25}{2} = 325]Next, we subtract the number of edges (60) from this total, as edges are not diagonals:[325 - 60 = 265]These 265 diagonals include both face diagonals and space diagonals. We need to subtract the face diagonals. For the given polyhedron, there are 24 triangular faces and 12 quadrilateral faces. Triangular faces have no diagonals, while each quadrilateral face has 2 diagonals. Therefore, the total number of face diagonals is:[12 times 2 = 24]Subtracting these face diagonals from the 265 diagonals gives the number of space diagonals:[265 - 24 = 241]Thus, the number of space diagonals in the polyhedron ( P ) is (boxed{241}).]

🔑:1. Calculate the total number of line segments connecting any two vertices: The total number of line segments connecting any two vertices in a polyhedron with (26) vertices is given by the binomial coefficient: [ binom{26}{2} = frac{26 times 25}{2} = 325 ]2. Subtract the number of edges: We are given that the polyhedron has (60) edges. These edges are line segments connecting adjacent vertices, so we need to subtract these from the total number of line segments: [ 325 - 60 = 265 ]3. Subtract the diagonals of the quadrilateral faces: Each quadrilateral face has (4) vertices and (2) diagonals. Since there are (12) quadrilateral faces, the total number of diagonals in these faces is: [ 2 times 12 = 24 ] We need to subtract these (24) diagonals because they connect vertices that belong to the same face.4. Calculate the number of space diagonals: Subtract the number of diagonals of the quadrilateral faces from the previous result: [ 265 - 24 = 241 ]Thus, the number of space diagonals in the polyhedron is:[boxed{241}]

🧠:[<think>Okay, so I have this problem here: It's given that sin α + sin β equals 2 sin(α + β), and α + β is not equal to 2πn for any integer n. I need to find the product of tan(α/2) and tan(β/2). Hmm, let's see how I can approach this.First, I remember that there are some trigonometric identities for the sum of sines. Maybe I can use one of those to rewrite the left side of the equation. The formula for sin α + sin β is 2 sin[(α + β)/2] cos[(α - β)/2]. Let me write that down:sin α + sin β = 2 sin[(α + β)/2] cos[(α - β)/2]According to the problem, this is equal to 2 sin(α + β). So, setting them equal:2 sin[(α + β)/2] cos[(α - β)/2] = 2 sin(α + β)I can divide both sides by 2 to simplify:sin[(α + β)/2] cos[(α - β)/2] = sin(α + β)Now, maybe I can use another identity for sin(α + β). Let's recall that sin(2θ) = 2 sinθ cosθ. If I let θ = (α + β)/2, then sin(α + β) = sin[2θ] = 2 sinθ cosθ = 2 sin[(α + β)/2] cos[(α + β)/2]. So substituting back into the right side:sin[(α + β)/2] cos[(α - β)/2] = 2 sin[(α + β)/2] cos[(α + β)/2]Hmm, both sides have sin[(α + β)/2], and since α + β is not equal to 2πn, sin[(α + β)/2] isn't zero. So I can divide both sides by sin[(α + β)/2], giving:cos[(α - β)/2] = 2 cos[(α + β)/2]Alright, so now I have:cos[(α - β)/2] = 2 cos[(α + β)/2]Let me write that as:cos[(α - β)/2] - 2 cos[(α + β)/2] = 0Hmm, maybe I can expand these cosines using sum and difference formulas. Let's recall that cos(A ± B) = cos A cos B ∓ sin A sin B. Let me apply that to both terms.First, cos[(α - β)/2] = cos(α/2 - β/2) = cos(α/2)cos(β/2) + sin(α/2)sin(β/2)Similarly, cos[(α + β)/2] = cos(α/2 + β/2) = cos(α/2)cos(β/2) - sin(α/2)sin(β/2)So substituting back into the equation:[cos(α/2)cos(β/2) + sin(α/2)sin(β/2)] - 2[cos(α/2)cos(β/2) - sin(α/2)sin(β/2)] = 0Let me distribute the -2 into the second bracket:cos(α/2)cos(β/2) + sin(α/2)sin(β/2) - 2 cos(α/2)cos(β/2) + 2 sin(α/2)sin(β/2) = 0Now, combine like terms:[cos(α/2)cos(β/2) - 2 cos(α/2)cos(β/2)] + [sin(α/2)sin(β/2) + 2 sin(α/2)sin(β/2)] = 0Which simplifies to:- cos(α/2)cos(β/2) + 3 sin(α/2)sin(β/2) = 0Let me rewrite that:3 sin(α/2)sin(β/2) = cos(α/2)cos(β/2)Now, divide both sides by cos(α/2)cos(β/2) to get:3 tan(α/2) tan(β/2) = 1Therefore, tan(α/2) tan(β/2) = 1/3So the product is 1/3. Let me check my steps to make sure I didn't make a mistake.Starting with sin α + sin β = 2 sin(α + β), converted to sum formula, set equal, divided by sin[(α + β)/2], ended up with an equation involving cos terms. Expanded those using sum and difference identities, combined terms, and finally got the ratio of tangents. Seems okay.Wait, let me verify with a quick example. Suppose tan(α/2) tan(β/2) = 1/3. Let's pick α and β such that this holds. Let’s suppose α = β. Then tan(α/2)^2 = 1/3, so tan(α/2) = 1/√3, which means α/2 = π/6, so α = π/3. Then β = π/3 as well. Then sin α + sin β = sin(π/3) + sin(π/3) = √3/2 + √3/2 = √3. On the other hand, 2 sin(α + β) = 2 sin(2π/3) = 2*(√3/2) = √3. So that works. And tan(α/2) tan(β/2) = (1/√3)*(1/√3) = 1/3. Perfect. So at least in this case, it works. So maybe the answer is indeed 1/3.Alternatively, let me check another case. Suppose α = 2 arctan(1) = π/2, so tan(α/2) = 1. Then β should be such that tan(β/2) = 1/3. So β = 2 arctan(1/3). Let's compute sin α + sin β. sin(π/2) = 1, sin β = sin(2 arctan(1/3)) = 2*(1/3)/(1 + (1/3)^2) = (2/3)/(10/9) = (2/3)*(9/10) = 3/5. So total sin α + sin β = 1 + 3/5 = 8/5. Now 2 sin(α + β). α + β = π/2 + 2 arctan(1/3). Let me compute sin(α + β). Using sin(A + B) formula: sin π/2 cos(2 arctan(1/3)) + cos π/2 sin(2 arctan(1/3)) = 1 * cos(2 arctan(1/3)) + 0. cos(2 arctan(1/3)) = (1 - (1/3)^2)/(1 + (1/3)^2) = (8/9)/(10/9) = 8/10 = 4/5. So 2 sin(α + β) = 2*(4/5) = 8/5. Which matches. So this example also works. Therefore, the answer is 1/3.Final Answerboxed{dfrac{1}{3}}</think>Given the problem where (sin alpha + sin beta = 2 sin (alpha + beta)) and (alpha + beta neq 2 pi n) for any integer (n), we need to find (operatorname{tg} frac{alpha}{2} operatorname{tg} frac{beta}{2}).First, we use the trigonometric identity for the sum of sines:[sin alpha + sin beta = 2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right)]Given that this equals (2 sin (alpha + beta)), we set up the equation:[2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right) = 2 sin (alpha + beta)]Dividing both sides by 2, we get:[sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right) = sin (alpha + beta)]Using the double-angle identity (sin (2theta) = 2 sin theta cos theta) on the right-hand side, we substitute (theta = frac{alpha + beta}{2}):[sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right) = 2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha + beta}{2} right)]Since (sin left( frac{alpha + beta}{2} right) neq 0), we can divide both sides by it:[cos left( frac{alpha - beta}{2} right) = 2 cos left( frac{alpha + beta}{2} right)]Expanding both sides using sum and difference identities:[cos left( frac{alpha}{2} - frac{beta}{2} right) = 2 cos left( frac{alpha}{2} + frac{beta}{2} right)][cos left( frac{alpha}{2} right) cos left( frac{beta}{2} right) + sin left( frac{alpha}{2} right) sin left( frac{beta}{2} right) = 2 left( cos left( frac{alpha}{2} right) cos left( frac{beta}{2} right) - sin left( frac{alpha}{2} right) sin left( frac{beta}{2} right) right)]Simplifying the right-hand side:[cos left( frac{alpha}{2} right) cos left( frac{beta}{2} right) + sin left( frac{alpha}{2} right) sin left( frac{beta}{2} right) = 2 cos left( frac{alpha}{2} right) cos left( frac{beta}{2} right) - 2 sin left( frac{alpha}{2} right) sin left( frac{beta}{2} right)]Combining like terms:[- cos left( frac{alpha}{2} right) cos left( frac{beta}{2} right) + 3 sin left( frac{alpha}{2} right) sin left( frac{beta}{2} right) = 0]Dividing by (cos left( frac{alpha}{2} right) cos left( frac{beta}{2} right)):[3 tan left( frac{alpha}{2} right) tan left( frac{beta}{2} right) = 1]Thus, the product is:[tan left( frac{alpha}{2} right) tan left( frac{beta}{2} right) = frac{1}{3}][boxed{dfrac{1}{3}}],<think>Okay, so I need to solve this problem where it's given that sin α + sin β equals 2 sin(α + β), and α + β is not equal to 2πn for any integer n. I have to find the product of tan(α/2) and tan(β/2). Hmm, let me start by recalling some trigonometric identities that might help here.First, the sum of sines formula: sin α + sin β = 2 sin[(α + β)/2] cos[(α - β)/2]. The problem states that this is equal to 2 sin(α + β). So, if I set up the equation:2 sin[(α + β)/2] cos[(α - β)/2] = 2 sin(α + β)I can divide both sides by 2 to simplify:sin[(α + β)/2] cos[(α - β)/2] = sin(α + β)Now, maybe I can express sin(α + β) using a double-angle identity. Remember that sin(2θ) = 2 sinθ cosθ. Let me let θ = (α + β)/2, so then sin(α + β) = sin(2θ) = 2 sinθ cosθ. Substituting back, that would be 2 sin[(α + β)/2] cos[(α + β)/2]. So replacing sin(α + β) in the equation:sin[(α + β)/2] cos[(α - β)/2] = 2 sin[(α + β)/2] cos[(α + β)/2]Hmm, both sides have sin[(α + β)/2], and since α + β ≠ 2πn, sin[(α + β)/2] isn't zero. So I can divide both sides by sin[(α + β)/2], which gives:cos[(α - β)/2] = 2 cos[(α + β)/2]Alright, so now I have this equation involving cosines of half the sum and half the difference. Let me write that again:cos[(α - β)/2] = 2 cos[(α + β)/2]I need to find tan(α/2) tan(β/2). Let me recall that tan(α/2) tan(β/2) can be expressed in terms of sum and difference formulas. Maybe there's a way to express this product using the sum and difference angles.Alternatively, perhaps I can express cos[(α - β)/2] and cos[(α + β)/2] in terms of tan(α/2) and tan(β/2). Let me think.Another approach: Let me denote x = α/2 and y = β/2. Then, α = 2x and β = 2y. So, the original equation becomes sin(2x) + sin(2y) = 2 sin(2x + 2y). Let's compute both sides.Left-hand side (LHS): sin(2x) + sin(2y) = 2 sin(x + y) cos(x - y). Wait, that's similar to the sum formula I used earlier.Right-hand side (RHS): 2 sin(2x + 2y) = 2 sin[2(x + y)] = 4 sin(x + y) cos(x + y). Because sin(2θ) = 2 sinθ cosθ, so sin[2(x+y)] = 2 sin(x+y) cos(x+y). Multiply by 2: 4 sin(x + y) cos(x + y).So, setting LHS = RHS:2 sin(x + y) cos(x - y) = 4 sin(x + y) cos(x + y)Assuming sin(x + y) ≠ 0 (since α + β ≠ 2πn, so x + y ≠ πn, which would make sin(x + y) = 0 if n is integer). Therefore, we can divide both sides by 2 sin(x + y):cos(x - y) = 2 cos(x + y)So we have cos(x - y) = 2 cos(x + y). Let's expand both sides using the cosine addition formulas.cos(x - y) = cosx cosy + sinx sinycos(x + y) = cosx cosy - sinx sinySubstituting into the equation:cosx cosy + sinx siny = 2 (cosx cosy - sinx siny)Let me distribute the 2 on the right:cosx cosy + sinx siny = 2 cosx cosy - 2 sinx sinyNow, bring all terms to the left side:cosx cosy + sinx siny - 2 cosx cosy + 2 sinx siny = 0Combine like terms:(cosx cosy - 2 cosx cosy) + (sinx siny + 2 sinx siny) = 0Which simplifies to:(- cosx cosy) + (3 sinx siny) = 0So:- cosx cosy + 3 sinx siny = 0Let me rearrange:3 sinx siny = cosx cosyDivide both sides by cosx cosy (assuming cosx ≠ 0 and cosy ≠ 0, which should be valid since if, say, cosx = 0, then x = π/2 + kπ, which would make α = π + 2kπ, but combined with β, maybe it's possible, but given that α + β ≠ 2πn, maybe we can assume cosx and cosy are non-zero here).So, 3 tanx tany = 1Hence, tanx tany = 1/3But x = α/2 and y = β/2, so tan(α/2) tan(β/2) = 1/3Therefore, the answer is 1/3. Wait, that seems straightforward. Let me verify.Starting from the original equation:sin α + sin β = 2 sin(α + β)Using sum to product:2 sin[(α + β)/2] cos[(α - β)/2] = 2 sin(α + β)Divide both sides by 2:sin[(α + β)/2] cos[(α - β)/2] = sin(α + β)Express sin(α + β) as 2 sin[(α + β)/2] cos[(α + β)/2]Then:sin[(α + β)/2] cos[(α - β)/2] = 2 sin[(α + β)/2] cos[(α + β)/2]Cancel sin[(α + β)/2]:cos[(α - β)/2] = 2 cos[(α + β)/2]Then, as before, set x = α/2, y = β/2:cos(x - y) = 2 cos(x + y)Expanding both sides:cosx cosy + sinx siny = 2(cosx cosy - sinx siny)Which leads to:cosx cosy + sinx siny = 2 cosx cosy - 2 sinx sinyMove terms:0 = cosx cosy - 3 sinx sinySo cosx cosy = 3 sinx sinyDivide both sides by cosx cosy:1 = 3 tanx tany => tanx tany = 1/3Hence, tan(α/2) tan(β/2) = 1/3So the answer is 1/3, which in boxed form is boxed{dfrac{1}{3}}.I think that's correct. Let me check with a simple example. Suppose α = β, then the equation would be 2 sin α = 2 sin(2α). Then, sin α = sin(2α) => sin α = 2 sin α cos α. If sin α ≠ 0, then 1 = 2 cos α => cos α = 1/2 => α = π/3 + 2πn or 5π/3 + 2πn. Let's take α = β = π/3. Then tan(α/2) tan(β/2) = tan(π/6) tan(π/6) = (1/√3)(1/√3) = 1/3. Which matches the answer. So that seems to check out. Another example: if α = π/3 and β = π/3, as above. Alternatively, suppose α = 5π/3 and β = 5π/3, same thing. If α and β are different, does this still hold? Let's pick α and β such that their sum is some angle where the equation holds. Let me see.Suppose α + β = θ, so sin α + sin β = 2 sin θ. Let me pick θ = π/2. Then 2 sin θ = 2. So sin α + sin β = 2. The maximum of sin α + sin β is 2, achieved when both sines are 1. So α = β = π/2. Then tan(α/2) tan(β/2) = tan(π/4) tan(π/4) = 1*1 = 1. Wait, but according to our answer, it should be 1/3. But here, if α = β = π/2, then α + β = π, which is allowed (since 2πn would require θ = 0, 2π, etc.). Wait, but in this case, if α + β = π, then according to our equation, sin α + sin β = 2 sin(α + β) = 2 sin π = 0. But in reality, sin α + sin β = sin π/2 + sin π/2 = 1 + 1 = 2. But 2 sin(α + β) = 0. So this is a contradiction. Therefore, θ = π is not allowed? Wait, but the problem states that α + β ≠ 2πn. π is not 2πn, unless n is 1/2, which is not an integer. So θ = π is allowed. But in that case, the equation sin α + sin β = 2 sin(α + β) becomes 2 = 0, which is impossible. Therefore, such α and β cannot exist when θ = π. So my initial test case where α = β = π/2 is invalid because even though their sum is π, which is allowed, the equation cannot hold. Therefore, that case is impossible, which is why when I tried to test with θ = π/2, but the equation would require sin α + sin β = 2 sin θ = 2*1 = 2, which is only possible if α and β are π/2, but then θ would be π, leading to a contradiction. Therefore, such α and β cannot exist. So maybe the only possible solutions are those where the equation holds. Let's see.Alternatively, take another example where α + β = something else. Suppose α = π/6 and β = π/6. Then α + β = π/3. Then sin α + sin β = 1/2 + 1/2 = 1. 2 sin(α + β) = 2 sin(π/3) = 2*(√3/2) = √3 ≈ 1.732. So 1 ≠ 1.732, so the equation doesn't hold here. Therefore, such α and β would not satisfy the equation. So in order for sin α + sin β = 2 sin(α + β) to hold, there must be some specific relationship between α and β.But according to the problem, such α and β exist (given the problem is posed), so through our algebraic manipulation, we arrived at tan(α/2) tan(β/2) = 1/3. So even though certain test cases may not hold because the equation imposes specific constraints on α and β, the algebra leads us to the answer. And the example where α = β = π/3 works. Let's check:If α = β = π/3, then sin α + sin β = 2*(√3/2) = √3. 2 sin(α + β) = 2 sin(2π/3) = 2*(√3/2) = √3. So √3 = √3, which holds. Then tan(α/2) tan(β/2) = tan(π/6) tan(π/6) = (1/√3)*(1/√3) = 1/3. Perfect, that works. Another example: Let me choose α = π/6 and β such that the equation holds. Let's see.Suppose α = π/6. Then sin(π/6) + sin β = 2 sin(π/6 + β). Let’s denote θ = π/6 + β. So the equation is 1/2 + sin β = 2 sin θ. But θ = π/6 + β => β = θ - π/6. So substituting:1/2 + sin(θ - π/6) = 2 sin θ.Compute sin(θ - π/6) = sin θ cos(π/6) - cos θ sin(π/6) = sin θ*(√3/2) - cos θ*(1/2)So the equation becomes:1/2 + [√3/2 sin θ - 1/2 cos θ] = 2 sin θMultiply out:1/2 + (√3/2) sin θ - (1/2) cos θ = 2 sin θBring all terms to left:1/2 + (√3/2) sin θ - (1/2) cos θ - 2 sin θ = 0Combine like terms:1/2 + [ (√3/2 - 2) sin θ ] - (1/2) cos θ = 0Simplify coefficients:1/2 + [ ( (√3 - 4)/2 ) sin θ ] - (1/2) cos θ = 0Multiply both sides by 2 to eliminate denominators:1 + (√3 - 4) sin θ - cos θ = 0So:(√3 - 4) sin θ - cos θ = -1This is a linear combination of sin θ and cos θ. Let's write it as:A sin θ + B cos θ = CWhere A = √3 - 4, B = -1, C = -1We can solve this equation for θ. The general solution is:θ = arcsin(C / sqrt(A² + B²)) - arctan(B/A)But this might get complicated. Let me compute sqrt(A² + B²):sqrt( (√3 - 4)^2 + (-1)^2 ) = sqrt( (3 - 8√3 + 16) + 1 ) = sqrt(20 - 8√3)Hmm, approximately sqrt(20 - 13.856) = sqrt(6.144) ≈ 2.478Then C / sqrt(A² + B²) = -1 / 2.478 ≈ -0.403So θ = arcsin(-0.403) - arctan( (-1)/(√3 - 4) )Compute arcsin(-0.403) ≈ -0.414 radians (about -23.7 degrees)Compute arctan( (-1)/(√3 - 4) )First, √3 ≈ 1.732, so √3 - 4 ≈ -2.268So (-1)/(-2.268) ≈ 0.4409arctan(0.4409) ≈ 0.415 radians (≈ 23.8 degrees)Therefore, θ ≈ -0.414 - 0.415 ≈ -0.829 radians. Alternatively, considering periodicity, θ ≈ π - (-0.829) ≈ 3.97 radians? Hmm, maybe not. This is getting messy, but the point is, there exists a θ that satisfies the equation, hence a β exists for α = π/6. Then, once we find θ, we can find β = θ - π/6, and then compute tan(α/2) tan(β/2). According to our solution, it should be 1/3 regardless of α and β as long as they satisfy the original equation. So even though calculating specific angles is complicated, the algebraic derivation shows that the product must be 1/3. Therefore, the answer is consistent.Hence, after careful step-by-step reasoning and verification with a valid example, I conclude that the value of tan(α/2) tan(β/2) is 1/3.Final Answerboxed{dfrac{1}{3}}</think>Given the problem where (sin alpha + sin beta = 2 sin (alpha + beta)) and (alpha + beta neq 2 pi n) for any integer (n), we need to find (operatorname{tg} frac{alpha}{2} operatorname{tg} frac{beta}{2}).First, we use the sum-to-product identity for sines:[sin alpha + sin beta = 2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right)]Given the equation:[2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right) = 2 sin (alpha + beta)]Dividing both sides by 2:[sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right) = sin (alpha + beta)]Using the double-angle identity for sine, (sin (alpha + beta) = 2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha + beta}{2} right)), we substitute:[sin left( frac{alpha + beta}{2} right) cos left( frac{alpha - beta}{2} right) = 2 sin left( frac{alpha + beta}{2} right) cos left( frac{alpha + beta}{2} right)]Since (sin left( frac{alpha + beta}{2} right) neq 0), we divide both sides by (sin left( frac{alpha + beta}{2} right)):[cos left( frac{alpha - beta}{2} right) = 2 cos left( frac{alpha + beta}{2} right)]Let (x = frac{alpha}{2}) and (y = frac{beta}{2}). Then, (alpha = 2x) and (beta = 2y), so the equation becomes:[cos (x - y) = 2 cos (x + y)]Expanding using cosine addition formulas:[cos x cos y + sin x sin y = 2 (cos x cos y - sin x sin y)]Simplifying:[cos x cos y + sin x sin y = 2 cos x cos y - 2 sin x sin y]Rearranging terms:[cos x cos y + sin x sin y - 2 cos x cos y + 2 sin x sin y = 0]Combining like terms:[- cos x cos y + 3 sin x sin y = 0]Dividing by (cos x cos y):[3 tan x tan y = 1]Thus:[tan x tan y = frac{1}{3}]Since (x = frac{alpha}{2}) and (y = frac{beta}{2}), we have:[tan frac{alpha}{2} tan frac{beta}{2} = frac{1}{3}]The final answer is (boxed{dfrac{1}{3}}).]

🔑:4.89 We have2 sin frac{alpha+beta}{2} cos frac{alpha-beta}{2}=4 sin frac{alpha+beta}{2} cos frac{alpha+beta}{2}from whichfrac{cos frac{alpha-beta}{2}}{cos frac{alpha+beta}{2}}=2.Therefore,operatorname{tg} frac{alpha}{2} operatorname{tg} frac{beta}{2}=frac{2 sin frac{alpha}{2} sin frac{beta}{2}}{2 cos frac{alpha}{2} cos frac{beta}{2}}=frac{cos frac{alpha-beta}{2}-cos frac{alpha+beta}{2}}{cos frac{alpha-beta}{2}+cos frac{alpha+beta}{2}}=frac{1}{3}

🧠:[<think>Okay, let's see. We have two gears. When connected, the transmission ratio is either 7:9 or 9:7. Then, by replacing one gear with one that has 3 fewer teeth and the other with a gear that has 3 more teeth, the ratio becomes 1:3 or 3:1. We need to figure out which gear was replaced with the one with fewer teeth and find out the number of teeth on all four gears (original two and the two replacements). First, transmission ratio is the ratio of rotational speeds, which is inversely proportional to the number of teeth. So, if gear A has N teeth and gear B has M teeth, then the ratio of their rotational speeds is M:N (assuming gear A is driving). So if the ratio is 7:9, that would mean that the driven gear has 9 teeth for every 7 teeth on the driving gear. Wait, no, actually, the ratio of rotational speeds is driving:driven, which is inverse of the teeth ratio. So if the driving gear has N teeth and the driven has M teeth, then the ratio is N:M for teeth, but the speed ratio is M:N. Hmm, maybe I need to clarify.Transmission ratio (velocity ratio) is driven speed / driving speed. So if driving gear has N teeth and driven has M teeth, then the speed ratio is N/M. Because the number of teeth is inversely proportional to the speed. So when they say the transmission ratio is 7:9, that means driving speed : driven speed = 7:9. Which would be equivalent to driven speed / driving speed = 9/7. Therefore, the teeth ratio driving:driven = 9:7. Wait, that's confusing. Let me think again.If two gears are meshed, the speed ratio is inversely proportional to the number of teeth. So if gear 1 has T1 teeth and gear 2 has T2 teeth, then speed1/speed2 = T2/T1. So if the transmission ratio (speed1/speed2) is 7:9, that would mean T2/T1 = 7/9, so T1/T2 = 9/7. Therefore, the teeth ratio is 9:7. Similarly, if the transmission ratio is 9:7, then T1/T2 = 7/9.Wait, maybe the problem states the transmission ratio as driving:driven. So if driving gear is rotating at speed1 and driven at speed2, then speed1:speed2 is the transmission ratio. So speed1/speed2 = T2/T1. Therefore, if the ratio is 7:9, then T2/T1 = 7/9, so T1:T2 = 9:7. Similarly, if the ratio is 9:7, then T1:T2 = 7:9. So the problem says the ratio is either 7:9 or 9:7. So the original gears could have teeth in the ratio 9:7 or 7:9. So there are two possibilities here.But the problem is, we need to find out the actual numbers of teeth. Then after replacing one gear with 3 fewer teeth and the other with 3 more teeth, the ratio becomes 1:3 or 3:1. So again, similar logic. So first, let's denote the original gears. Let's say the original gears have T1 and T2 teeth. Then, the transmission ratio (speed1/speed2) is either T2/T1 = 7/9 or T2/T1 = 9/7. So either T1/T2 = 9/7 or 7/9.But the problem says "the ratio of the rotational speeds... is 7:9 or 9:7". So depending on which is driving. So maybe the gears can be connected either way. So regardless, the two gears have teeth numbers such that T1/T2 is either 7/9 or 9/7. So T1 and T2 are in the ratio 7:9 or 9:7.But since gears must have integer numbers of teeth, T1 and T2 must be multiples of 7 and 9 (if ratio is 7:9) or 9 and 7. But maybe they have some common divisor. So T1 = 7k, T2 = 9k, or T1 = 9k, T2 = 7k, for some integer k.Then, after replacing one gear with 3 fewer teeth and the other with 3 more teeth, the transmission ratio becomes 1:3 or 3:1. So the new teeth numbers are T1 ±3 and T2 ∓3? Wait, replacing one with 3 fewer and the other with 3 more. So if original gears are T1 and T2, then replacing one with T1 - 3 and the other with T2 + 3, or replacing one with T1 + 3 and the other with T2 - 3? Wait, the problem says "replacing one of the gears with a gear that has 3 fewer teeth and the other with a gear that has 3 more teeth". So one is replaced with 3 fewer, the other with 3 more. So total teeth change is 0. So original total teeth T1 + T2, new total is (T1 - 3) + (T2 + 3) = T1 + T2. So same total. So the total number of teeth remains the same. So that's a key point.Therefore, the sum of the teeth of the two gears remains the same after replacement. So original gears T1 and T2, sum S = T1 + T2. After replacement, gears are T1 - 3 and T2 + 3, sum S. Or T1 + 3 and T2 - 3. But the problem says replacing one with 3 fewer and the other with 3 more. So either (T1 -3, T2 +3) or (T1 +3, T2 -3). So the two possibilities. Then, the new transmission ratio is either 1:3 or 3:1. So again, the ratio could be speed1/speed2 = 1/3 or 3/1. Which would correspond to teeth ratios 3:1 or 1:3. So (T_new1 / T_new2) = 3 or 1/3, depending on which gear is driving.But let's formalize this.Original gears: T1 and T2. Their ratio is either 7:9 or 9:7, so T1/T2 = 7/9 or 9/7. So T1 = (7/9) T2 or T1 = (9/7) T2. Then, after replacement, gears become either (T1 -3, T2 +3) or (T1 +3, T2 -3). The transmission ratio is either 1:3 or 3:1. So the new ratio (T_new1 / T_new2) is either 1/3 or 3/1, depending on which gear is driving. But since the problem says the transmission ratio becomes 1:3 or 3:1, similar to before. So the new ratio could be either T_new1/T_new2 = 1/3 or 3/1, but we have to consider which one is driving.But perhaps it's better to write the speed ratio. So if we denote the original gears, the speed ratio (driving:driven) is 7:9 or 9:7. Then after replacement, it's 1:3 or 3:1. So we need to find which replacement leads to such ratios.Let me denote variables. Let’s assume the original gears have T1 and T2 teeth. Then, the transmission ratio (driving:driven) is either T2/T1 or T1/T2, depending on which is connected as driving. But the problem says the ratio is 7:9 or 9:7. So that implies that T1/T2 is either 7/9 or 9/7. Therefore, the two gears must be in a 7:9 ratio. So either T1 = 7k and T2 = 9k, or T1 = 9k and T2 = 7k for some integer k.Then, when we replace one gear with 3 fewer teeth and the other with 3 more, the new gears are (T1 -3, T2 +3) or (T1 +3, T2 -3). Then, the new transmission ratio is either 1:3 or 3:1, which would correspond to the teeth ratio either 1/3 or 3/1. So the ratio of the new gears (either (T1 -3)/(T2 +3) or (T1 +3)/(T2 -3)) is either 1/3 or 3/1.So let's set up equations. Let's first suppose that the original gears have T1 and T2 teeth with T1/T2 = 7/9. So T1 = (7/9) T2. Let’s denote T1 = 7k and T2 = 9k. Then, total teeth S = T1 + T2 = 16k.After replacement, if we replace T1 with T1 -3 and T2 with T2 +3, then new teeth are T1' = 7k -3 and T2' = 9k +3. The new ratio would be T1'/T2' = (7k -3)/(9k +3). This should be equal to either 1/3 or 3. Alternatively, if we replace T1 with T1 +3 and T2 with T2 -3, then T1' = 7k +3 and T2' = 9k -3. The ratio would be (7k +3)/(9k -3). This should equal 1/3 or 3.Alternatively, if the original ratio was 9:7, then T1 =9k and T2=7k. Then similar equations.So maybe we need to check both cases.Case 1: Original ratio 7:9, so T1=7k, T2=9k.Subcase 1a: Replace T1 with T1 -3 and T2 with T2 +3. New ratio (7k -3)/(9k +3). This equals either 1/3 or 3.Set (7k -3)/(9k +3) = 1/3.Cross multiply: 3(7k -3) = 9k +321k -9 =9k +321k -9k = 3 +912k =12 => k=1.Check if k=1 is valid. Then original gears T1=7, T2=9. Replace T1 with 7-3=4, T2 with 9+3=12. Then ratio is 4/12=1/3. So the transmission ratio is 1:3. Which is allowed. So this works. So k=1.Alternatively, set (7k -3)/(9k +3)=3. Then:7k -3 = 3(9k +3)7k -3=27k +97k -27k=9 +3-20k=12 =>k= -12/20= -3/5. Not possible. So invalid.Subcase 1b: Replace T1 with T1 +3 and T2 with T2 -3. Then ratio (7k +3)/(9k -3). Set equal to 1/3 or 3.First set to 1/3:(7k +3)/(9k -3)=1/3Cross multiply: 3(7k +3)=9k -321k +9=9k -321k -9k= -3 -912k= -12 =>k= -1. Invalid.Set equal to 3:(7k +3)/(9k -3)=37k +3=27k -97k -27k= -9 -3-20k= -12 =>k= 12/20=3/5. Not integer. So invalid.Thus, in Case 1, only possible solution is k=1, original gears 7 and 9. Replacing driving gear (7 teeth) with 4 and driven (9) with 12, ratio 4:12=1:3. So this works. So if original gears were 7 and 9, replacing the 7 teeth gear with 4 (3 fewer) and the 9 teeth with 12 (3 more) gives ratio 1:3. So which gear was replaced with fewer teeth? The original 7 teeth gear was replaced with 4, which is 3 fewer. So answer: the gear with 7 teeth was replaced with the one with fewer teeth. Then, the four gears would be original:7,9; new:4,12. So that's possible.But let's check the other case.Case 2: Original ratio 9:7, so T1=9k, T2=7k.Subcase 2a: Replace T1 with T1 -3 and T2 with T2 +3. New ratio (9k -3)/(7k +3). Set equal to 1/3 or 3.First set to 1/3:(9k -3)/(7k +3)=1/327k -9=7k +320k=12 =>k=12/20=3/5. Not integer.Set to 3:(9k -3)/(7k +3)=39k -3=21k +9-12k=12 =>k= -1. Invalid.Subcase 2b: Replace T1 with T1 +3 and T2 with T2 -3. New ratio (9k +3)/(7k -3). Set equal to 1/3 or 3.First set to 1/3:(9k +3)/(7k -3)=1/327k +9=7k -320k= -12 =>k= -12/20= -3/5. Invalid.Set to 3:(9k +3)/(7k -3)=39k +3=21k -9-12k= -12 =>k=1.So k=1. Original gears T1=9*1=9, T2=7*1=7. Replace T1 with 9+3=12, T2 with 7-3=4. Then new ratio is 12/4=3/1. So transmission ratio 3:1. Which is allowed. So this works. So here, original gears were 9 and 7. Replacing T1 (9 teeth) with 12 (3 more) and T2 (7) with 4 (3 fewer). So in this case, the gear that was replaced with fewer teeth is the original 7 teeth gear, which is replaced with 4. Wait, no: original gears are 9 and 7. They replaced one with 3 fewer and the other with 3 more. So in this case, T1 (9) was replaced with 12 (3 more) and T2 (7) was replaced with 4 (3 fewer). So the gear with 7 teeth was replaced with one that has 3 fewer teeth: 7 -3=4. So same as previous case.Wait, in Case 1, original gears 7 and 9: replacing 7 with 4 (fewer) and 9 with 12 (more). In Case 2, original gears 9 and 7: replacing 9 with 12 (more) and 7 with 4 (fewer). So in both cases, the 7 teeth gear is the one being replaced with fewer teeth. But wait, in Case 2, original gears are 9 and 7. If we replace the 9 with 12 and 7 with 4, then the 7 teeth gear is replaced with 4, which is 3 fewer. So regardless of the original arrangement, the gear with 7 teeth is the one being replaced with fewer teeth. However, in the problem statement, the original gears could be in either order. Wait, but in the first case, the gears are 7 and 9, and in the second case, 9 and 7. So depending on which one is driving, the ratio is different. But the problem says "by connecting them, the ratio... is 7:9 or 9:7". So the two gears can be connected either way, so the ratio depends on which is the driver. So the problem doesn't specify which one is driving initially, just that the ratio can be 7:9 or 9:7. Therefore, the gears have 7k and 9k teeth. Then replacing one with 3 fewer and the other with 3 more, leading to 1:3 or 3:1 ratio. So in both cases, the gear with 7k teeth is replaced with 3 fewer (assuming k=1). So original gears 7 and 9. So the answer is the gear with 7 teeth was replaced with one with fewer teeth (4 teeth), and the other gear (9 teeth) was replaced with 12 teeth. So the four gears are original 7 and 9, new 4 and 12. But wait, the problem says "replacing one of the gears with a gear that has 3 fewer teeth and the other with a gear that has 3 more teeth". So only two gears are involved: the original two, and then replacing each with a new one. So the four gears are original two and the two replacements. So the answer is that the gear which had 7 teeth was replaced with 4 (3 fewer), and the gear which had 9 teeth was replaced with 12 (3 more). So the four gears are 7, 9, 4, 12. But the problem asks "how many teeth does each of the 4 gears have?" So original gears and the replacements. So original gears were 7 and 9, replacements are 4 and 12.But wait, in Case 2, where original gears are 9 and 7, replacing 9 with 12 and 7 with 4. So the four gears are 9,7,12,4. But same numbers. So regardless of the original order, the gears are 7,9,4,12. So the answer would be that the gear with 7 teeth was replaced with 4 teeth (3 fewer), and the other gear (9) was replaced with 12 (3 more). So which gear was replaced with fewer teeth? The one that originally had 7 teeth.But let's verify the equations again. Let me check with k=1.Original gears: 7 and 9. Sum 16.After replacement: 4 and 12. Sum 16. So same total.Transmission ratio original: 7:9 or 9:7.After replacement: 4:12=1:3 or 12:4=3:1. So correct.Similarly, original gears 9 and 7. Replace 9 with 12 and 7 with 4. Same result.But in both cases, the gear that had 7 teeth was replaced with 4. So regardless of which was driving, the 7 teeth gear is the one being replaced with fewer teeth.Therefore, the answer is the gear with 7 teeth was replaced with one that has 3 fewer teeth. The four gears are 7,9,4,12.But let's check if there are other possible solutions. For example, could k be higher? Suppose k=2.Case1: T1=14, T2=18. Then replacing T1 with 11 and T2 with 21. Then ratio 11/21≈0.5238. Not 1/3 or 3. Or replacing T1 with 17 and T2 with 15. Ratio 17/15≈1.133. Not 1/3 or 3. Similarly, for k=2, subcase 1a and 1b don't give the required ratios.Similarly, if original gears were 9k and 7k, with k=2, T1=18, T2=14. Replacing 18-3=15, 14+3=17. Ratio 15/17≈0.882, not 1/3 or 3. Replace 18+3=21, 14-3=11. Ratio 21/11≈1.909, not 1/3 or 3. So k=1 is the only solution.Therefore, the only possible solution is original gears 7 and 9, replaced with 4 and 12. So the answer is the 7-teeth gear was replaced with the one with fewer teeth (4 teeth), and the four gears are 7,9,4,12.Final AnswerThe gear with boxed{7} teeth was replaced with one that has fewer teeth. The four gears have boxed{7}, boxed{9}, boxed{4}, and boxed{12} teeth respectively.</think>Given two gears, the transmission ratio (rotational speed ratio) is either 7:9 or 9:7. When one gear is replaced with a gear that has 3 fewer teeth and the other with a gear that has 3 more teeth, the transmission ratio becomes 1:3 or 3:1. We need to determine which gear was replaced with fewer teeth and the number of teeth on each of the four gears.1. Original Gears Analysis: - The transmission ratio of 7:9 or 9:7 implies the gears have teeth in the ratio 7:9. Let the original gears have ( T_1 ) and ( T_2 ) teeth such that ( T_1 = 7k ) and ( T_2 = 9k ) for some integer ( k ).2. Replacement Analysis: - After replacing one gear with 3 fewer teeth and the other with 3 more teeth, the new transmission ratio is 1:3 or 3:1. This implies the new teeth numbers must be in the ratio 1:3 or 3:1.3. Equations and Solutions: - Case 1: Original gears ( T_1 = 7k ) and ( T_2 = 9k ) - Replacing ( T_1 ) with ( T_1 - 3 ) and ( T_2 ) with ( T_2 + 3 ): - New ratio: ( frac{7k - 3}{9k + 3} = frac{1}{3} ) - Solving gives ( k = 1 ), so original gears are 7 and 9. - New gears: 4 (7 - 3) and 12 (9 + 3), giving a ratio of 1:3. - Replacing ( T_1 ) with ( T_1 + 3 ) and ( T_2 ) with ( T_2 - 3 ) results in invalid solutions. - Case 2: Original gears ( T_1 = 9k ) and ( T_2 = 7k ) - Replacing ( T_1 ) with ( T_1 + 3 ) and ( T_2 ) with ( T_2 - 3 ): - New ratio: ( frac{9k + 3}{7k - 3} = 3 ) - Solving gives ( k = 1 ), so original gears are 9 and 7. - New gears: 12 (9 + 3) and 4 (7 - 3), giving a ratio of 3:1.4. Conclusion: - The gear with 7 teeth was replaced with 4 teeth (3 fewer), and the gear with 9 teeth was replaced with 12 teeth (3 more). - The four gears are 7, 9, 4, and 12. Final AnswerThe gear with boxed{7} teeth was replaced with one that has fewer teeth. The four gears have boxed{7}, boxed{9}, boxed{4}, and boxed{12} teeth respectively.,<think>Okay, let's see. So there are two gears. When connected, their transmission ratio is either 7:9 or 9:7. Then, when one gear is replaced with one that has 3 fewer teeth and the other is replaced with a gear that has 3 more teeth, the ratio becomes 1:3 or 3:1. The question is, which gear was replaced with the one that has fewer teeth, and how many teeth do each of the four gears have?Hmm. Let's break this down. First, transmission ratio is the ratio of rotational speeds. But in gears, the transmission ratio is inversely proportional to the number of teeth. So if gear A is driving gear B, the ratio of their rotational speeds (ω_A/ω_B) is equal to the ratio of the number of teeth on gear B to gear A (N_B/N_A). So if the ratio is 7:9, that could mean that N_driven/N_driver = 7/9 or 9/7, depending on which is driving. Wait, the problem says "the ratio of the rotational speeds of the driving and driven gears (the transmission ratio) is 7:9 or 9:7". So if driving is first, then ω_driving/ω_driven = 7:9 or 9:7. Since ω is inversely proportional to the number of teeth, ω_driving/ω_driven = N_driven/N_driver. So, if the transmission ratio is 7:9, then N_driven/N_driver = 7/9, so N_driver/N_driven = 9/7. Hmm, maybe I need to be careful here.Let me clarify. The transmission ratio is defined as (rotational speed of driving gear)/(rotational speed of driven gear) = (number of teeth on driven gear)/(number of teeth on driving gear). So, if the ratio is 7:9, that would mean that the driving gear has 9 teeth and the driven has 7, or vice versa? Wait, no. Let me think again.Suppose the driving gear has N1 teeth and the driven gear has N2 teeth. The transmission ratio is ω1/ω2 = N2/N1. So if the transmission ratio is 7:9, that is ω1/ω2 = 7/9, which would mean N2/N1 = 7/9, so N1/N2 = 9/7. Therefore, the driving gear has 9 teeth and the driven has 7? Wait, but if the driving gear has more teeth, then the driven gear would spin faster. But if the ratio is 7:9, then the driven gear spins faster. So maybe the driving gear has 9 teeth, driven has 7 teeth, but since the ratio is 7:9, ω1/ω2 = 7/9, which would mean that the driving gear is spinning slower? Wait, no. If the driving gear has more teeth, the driven gear has fewer teeth, so the driven gear spins faster. So if the driving gear has 9 teeth, driven has 7, then ω1/ω2 = 7/9. So ω1 is 7 parts, ω2 is 9 parts. So driving gear is spinning slower than driven gear. That's correct. So the transmission ratio (driving/driven) is 7:9, so driving speed is 7, driven speed is 9, which is achieved by having the driving gear have 9 teeth and driven have 7.Alternatively, if the ratio is 9:7, then driving speed is 9, driven is 7, which would require driving gear to have 7 teeth and driven to have 9. Because then ω1/ω2 = N2/N1 = 9/7. So if driving gear has 7 teeth (N1=7), driven has 9 (N2=9), then ω1/ω2 = 9/7. So depending on which gear is driving, the ratio can be either 7:9 or 9:7. So the original gears can be paired either way. So maybe the original gears have 7 and 9 teeth? Wait, but gears usually have more teeth than that. Maybe the numbers are in ratio. So maybe the original gears have numbers of teeth in the ratio 7:9. Let me denote the original gears as having N and M teeth. Then, the transmission ratio can be either N/M or M/N, depending on which is driving. So the possible ratios are 7:9 or 9:7. Therefore, either N/M = 7/9 or M/N = 7/9. So either way, N and M are in the ratio 7:9. So N = 7k, M = 9k for some integer k.Then, when we replace one gear with one that has 3 fewer teeth and the other with 3 more teeth, the new transmission ratio is 1:3 or 3:1. So similarly, the new gears would have either (N-3) and (M+3) or (N+3) and (M-3). Then, the transmission ratio would be either (N-3)/(M+3) or (M+3)/(N-3), or (N+3)/(M-3) or (M-3)/(N+3), depending on which is driving. But the problem states that after replacement, the ratio becomes 1:3 or 3:1. So, similarly, the ratio of the number of teeth would need to be either 1:3 or 3:1, depending on which is driving. Therefore, either (new driven teeth)/(new driving teeth) = 1/3 or 3/1.So let's set up equations. Let me define the original gears as N and M teeth. So, original ratio possibilities:Case 1: If N is driving, M is driven, then transmission ratio is N/M = 7/9 or 9/7. Wait, no. Wait, ω_driving / ω_driven = (teeth on driven) / (teeth on driving). So if N is driving, M is driven, ratio is M/N. So for the original ratio to be 7:9 or 9:7, we have either M/N = 7/9 or M/N = 9/7. So either way, M/N is 7/9 or 9/7. Therefore, the original gears have teeth in the ratio 7:9 or 9:7.Therefore, N and M satisfy either N/M = 9/7 or 7/9. Let's take N and M such that N = 7k and M = 9k, or N = 9k and M = 7k, for some integer k.Then, after replacing one gear with 3 fewer teeth and the other with 3 more teeth, the new gears would be either:If we replace N with N-3 and M with M+3, then the new gears are N-3 and M+3.Or replace N with N+3 and M with M-3.Depending on which was replaced. Then, the new transmission ratio is either 1:3 or 3:1. So, similarly, (new driven teeth)/(new driving teeth) = 1/3 or 3/1.So let's consider both possibilities.First, suppose the original gears are N = 7k and M = 9k. Then, if we replace N with N-3 and M with M+3, the new gears are 7k - 3 and 9k + 3. Then the ratio would be either (9k + 3)/(7k - 3) or (7k - 3)/(9k + 3). The problem states that after replacement, the ratio is 1:3 or 3:1. So either (9k + 3)/(7k - 3) = 3/1 or 1/3, or similarly for the inverse.Alternatively, if we replace N with N+3 and M with M-3, the new gears are 7k + 3 and 9k - 3. Then the ratios would be (9k - 3)/(7k + 3) or (7k + 3)/(9k - 3), which again should equal 3 or 1/3.Alternatively, if the original gears were N = 9k and M = 7k, similar reasoning.Let me try with N = 7k and M = 9k first.Case 1: Original gears 7k and 9k.Subcase 1a: Replaced N (7k) with N-3 (7k - 3) and M (9k) with M+3 (9k + 3).Then, possible new ratios:If driving gear is N-3, driven is M+3: ratio is (M+3)/(N-3) = (9k + 3)/(7k - 3). This should be equal to 3 or 1/3.So set (9k + 3)/(7k - 3) = 3.Multiply both sides by (7k - 3):9k + 3 = 3*(7k - 3) => 9k + 3 = 21k - 9 => 9k + 3 + 9 = 21k => 9k + 12 = 21k => 12 = 12k => k = 1.Alternatively, set (9k + 3)/(7k - 3) = 1/3.Multiply both sides:9k + 3 = (1/3)(7k - 3) => Multiply both sides by 3: 27k + 9 = 7k - 3 => 20k = -12 => k = -12/20 = -3/5. Not possible, since teeth can't be negative. So only k=1 is possible here.So if k=1, original gears are N=7*1=7 and M=9*1=9. Then replacing N with 7-3=4 and M with 9+3=12. Then the ratio is 12/4=3, so 3:1. Which is one of the possible new ratios. So that works. Then the question is, which gear was replaced with fewer teeth? Originally, N was 7, replaced with 4 (3 fewer), and M was 9, replaced with 12 (3 more). So the driving gear was replaced with fewer teeth? Wait, but which gear is driving? Wait, in this case, if the original gears were 7 and 9, depending on which is driving, the ratio could be 7:9 or 9:7. But in the problem statement, it's just given that the ratio is 7:9 or 9:7. So they could be used either way. But when we replaced N (which was originally 7) with 4 and M (originally 9) with 12. So gear N was replaced with fewer teeth, and gear M with more. Then the new gears are 4 and 12. If 4 is driving and 12 is driven, ratio is 12/4=3, so 3:1. If 12 is driving and 4 is driven, ratio is 4/12=1/3, which is 1:3. So both possible. So that's valid. So in this case, the gear that was replaced with fewer teeth was the original 7-tooth gear.But gears with 7 and 4 teeth? That seems very small. Maybe k is larger? Wait, but in the problem statement, they didn't specify that the gears must have a minimum number of teeth. But maybe there's another solution. Let's check the other subcase.Subcase 1b: Replace N with N+3 and M with M-3. So new gears are 7k +3 and 9k -3.Then, the ratio (9k -3)/(7k +3) or (7k +3)/(9k -3). Let's set (9k -3)/(7k +3) =3. Solve for k:9k -3 = 3*(7k +3) => 9k -3 =21k +9 => -12k =12 => k= -1. Not possible.Set (9k -3)/(7k +3) =1/3:9k -3 = (1/3)(7k +3) => 27k -9 =7k +3 =>20k=12 =>k=12/20=3/5. Not integer. So no solution here.So in Case 1, only k=1 works, leading to original gears 7 and 9, replaced to 4 and 12. Then, which gear was replaced with fewer teeth? The original 7-tooth gear was replaced with 4 (3 fewer), and 9 was replaced with 12 (3 more). So the answer would be that the gear with 7 teeth was replaced with the one with fewer teeth.But let's check Case 2 where original gears are N=9k and M=7k.Case 2: Original gears N=9k, M=7k.Subcase 2a: Replace N with N-3 (9k -3) and M with M+3 (7k +3).New ratios: (7k +3)/(9k -3) or (9k -3)/(7k +3).Set (7k +3)/(9k -3) =3:7k +3 =3*(9k -3) =>7k +3=27k -9 => -20k= -12 =>k=12/20=3/5. Not integer.Set (7k +3)/(9k -3)=1/3:7k +3=(1/3)(9k -3) =>21k +9=9k -3 =>12k= -12 =>k=-1. Not possible.Subcase 2b: Replace N with N+3 (9k +3) and M with M-3 (7k -3).New gears: 9k +3 and 7k -3.Ratios: (7k -3)/(9k +3) or (9k +3)/(7k -3).Set (7k -3)/(9k +3)=3:7k -3=3*(9k +3) =>7k -3=27k +9 =>-20k=12 =>k= -12/20= -3/5. Not possible.Set (7k -3)/(9k +3)=1/3:7k -3=(1/3)(9k +3) =>21k -9=9k +3 =>12k=12 =>k=1.So k=1. Then original gears N=9*1=9, M=7*1=7. Then replacing N with 9+3=12 and M with 7-3=4. New gears 12 and 4. Then the ratio is either 4/12=1/3 or 12/4=3. So 1:3 or 3:1. Which fits. So in this case, original gears were 9 and 7. Then, replaced N (9) with 12 and M (7) with 4. Wait, but the problem states that we replace one gear with 3 fewer and the other with 3 more. But here, replacing N (9) with 12 is adding 3, and M (7) with 4 is subtracting 3. So that's correct. So in this case, the gear with 7 teeth was replaced with 4 (3 fewer), and the gear with 9 teeth was replaced with 12 (3 more). So which gear was replaced with fewer teeth? The original 7-tooth gear.Wait, but in Case 1, original gears were 7 and 9, replacing 7 with 4 and 9 with 12. In Case 2, original gears were 9 and 7, replacing 9 with 12 and 7 with 4. So regardless of labeling, the 7-tooth gear is the one being replaced with 4 (3 fewer). So in both cases, it's the gear that originally had 7 teeth that's replaced with fewer teeth. But gears with 7, 9, 4, 12 teeth? That seems possible. But maybe there are other solutions where k is larger?Wait, the problem says "replacing one of the gears with a gear that has 3 fewer teeth and the other with a gear that has 3 more teeth". So, in the solutions we found, replacing the 7 with 4 and 9 with 12, or vice versa. But is there a solution where the original gears are multiples of 7 and 9, like 14 and 18, then replacing them with 11 and 21?Let me check. Suppose k=2, so original gears N=14 and M=18 (Case 1: 7k and 9k). Then replacing N with 11 and M with 21. The new ratio would be 21/11 or 11/21. 21/11 is approximately 1.9, which is not 3 or 1/3. Similarly, 11/21≈0.52, not 1/3. So that doesn't work. If we tried k=3, original gears 21 and 27. Replace with 18 and 30. Ratios 30/18=5/3≈1.666 or 18/30=0.6. Not 3 or 1/3. So no. Similarly for k=4, 28 and 36. Replace with 25 and 39. Ratios 39/25≈1.56, 25/39≈0.64. Not 3 or 1/3. So seems like only k=1 works. Similarly in Case 2, if k=2, original gears 18 and 14. Replace with 21 and 11. Same issue. So only k=1 gives integer solutions where the new ratio is exactly 3:1 or 1:3.Therefore, the original gears must be 7 and 9 teeth. Then, after replacement, they are 4 and 12. So the gear with 7 teeth was replaced with 4 (3 fewer), and the gear with 9 teeth was replaced with 12 (3 more). Therefore, the answer is the gear that originally had 7 teeth was replaced with the one with fewer teeth. The four gears are the original two: 7 and 9, and the replacements: 4 and 12.Wait, but the problem says "replacing one of the gears with a gear that has 3 fewer teeth and the other with a gear that has 3 more teeth". So they replaced both gears? So we have four gears in total: original two (7 and 9), and the new two (4 and 12). But the question is asking "how many teeth does each of the 4 gears have?" So the original two and the replacement two. So the answer would be 7, 9, 4, 12. And the gear replaced with fewer teeth is the original 7-tooth gear.Alternatively, if the original gears were 9 and 7, then replacing the 9 with 12 and the 7 with 4. So same result. Either way, the gear that had 7 teeth was replaced with 4. So the answer is that the 7-tooth gear was replaced with the one with fewer teeth, and the four gears have 7, 9, 4, and 12 teeth.But let's verify once more. Original gears: 7 and 9. Transmission ratio is either 7:9 or 9:7. If 7 is driving, ratio is 9/7, so ω1/ω2 =9/7. If 9 is driving, ratio is 7/9. Then after replacing 7 with 4 and 9 with 12, the new gears are 4 and 12. If 4 is driving, ratio is 12/4=3, so 3:1. If 12 is driving, ratio is 4/12=1/3, so 1:3. Perfect, that's exactly what the problem states. Therefore, the solution is correct.Final AnswerThe gear originally with boxed{7} teeth was replaced with one that has fewer teeth. The four gears have boxed{7}, boxed{9}, boxed{4}, and boxed{12} teeth.</think>Given two gears with a transmission ratio of (7:9) or (9:7), we need to determine which gear was replaced with one that has 3 fewer teeth and the other with 3 more teeth, resulting in a new transmission ratio of (1:3) or (3:1). 1. Original Gears Analysis: - Let the original gears have (N) and (M) teeth. - The transmission ratio (rotational speed of driving gear to driven gear) is given by (frac{N_{text{driven}}}{N_{text{driving}}}). - The possible ratios are (7:9) or (9:7), implying (N) and (M) are in the ratio (7:9). Therefore, (N = 7k) and (M = 9k) for some integer (k).2. Replacement Analysis: - After replacement, the new gears have (N-3) and (M+3) or (N+3) and (M-3) teeth. - The new transmission ratio is (1:3) or (3:1), implying the new gears must be in the ratio (1:3) or (3:1).3. Solving for (k): - Assume original gears are (7k) and (9k). Replacing (7k) with (7k - 3) and (9k) with (9k + 3): - The new ratio (frac{9k + 3}{7k - 3} = 3) or (frac{1}{3}). - Solving (frac{9k + 3}{7k - 3} = 3) gives (k = 1). - For (k = 1), original gears are (7) and (9). Replacing (7) with (4) (3 fewer) and (9) with (12) (3 more) results in new gears (4) and (12). - The new transmission ratio is (frac{12}{4} = 3) (or (frac{4}{12} = frac{1}{3})), which fits the required ratios.4. Verification: - Original gears (7) and (9) can achieve transmission ratios (7:9) or (9:7). - New gears (4) and (12) achieve transmission ratios (1:3) or (3:1).Thus, the gear originally with (7) teeth was replaced with one that has fewer teeth. The four gears have (7), (9), (4), and (12) teeth.[boxed{7}, boxed{9}, boxed{4}, boxed{12}]The gear originally with (boxed{7}) teeth was replaced with one that has fewer teeth.]

🔑:The ratio of the number of revolutions of the two wheels (over the same time period) being 7:9 means that there is a positive integer k such that the number of teeth on the first wheel is 9k, and on the second wheel, it is 7k. Indeed, with 63k teeth passing, the first wheel makes 7 complete revolutions, and the second wheel makes 9 complete revolutions. Therefore, the number of teeth on the wheels after the replacement is either 9k+3 and 7k-3, or 9k-3 and 7k+3.In the first case, it is certain that 9k+3 > 7k-3, so only(9k+3):(7k-3) = 3:1 text{ is possible, from which } k=1In the second case, we must consider both the subcases 9k-3 > 7k+3 and 9k-3 < 7k+3, but neither yields a positive integer k:begin{aligned}& (9k-3):(7k+3) = 3:1 text{ yields } k=-1 & (9k-3):(7k+3) = 1:3 text{ yields } k=3/5end{aligned}Therefore, the number of teeth is k=1, so 9 and 7, and the 9-toothed wheel was replaced with a 12-toothed wheel, and the 7-toothed wheel with a 4-toothed wheel.Kócza Éva (Fót-Gyermekváros, specialized secondary school I. o. t.)Note. Several people naturally assumed that "since the ratio after the replacement, 3:1 or 1:3, deviates more strongly from the value 1 than the original ratios 9:7 or 7:9, the only possible solution is to replace the wheels with fewer teeth with even fewer teeth." This needed to be proven. The above solution shows that indeed only this case is possible. If the increase or decrease in the number of teeth had not been the same, the mentioned "solvers" would have likely shown more caution.

🧠:[<think>Alright, let's see. I need to compare the values of a, b, and c where a is log base 3 of π, b is the natural logarithm of 2, and c is the logarithm base 10 of 0.8. The options are given as A, B, C, D with different orderings. Hmm. Let me break this down step by step.First, let me recall the values of some common logarithms to get a sense of scale. For example, log base 3 of 3 is 1, and π is approximately 3.14, which is a bit more than 3. So log base 3 of π should be a little more than 1. But wait, let me check that. If 3^1 = 3 and 3^1.1 is approximately 3^(1/10) more. Let me compute 3^0.1. Since 3^0.1 is the same as e^(0.1 ln3) ≈ e^(0.1*1.0986) ≈ e^0.10986 ≈ 1.116. So 3^1.1 ≈ 3*1.116 ≈ 3.348, which is bigger than π (3.1416). Therefore, log base 3 of π is less than 1.1. Wait, but π is 3.1416, so 3^1 = 3, 3^1.05? Let's calculate 3^0.05. That's e^(0.05*ln3) ≈ e^(0.05*1.0986) ≈ e^0.0549 ≈ 1.056. So 3^1.05 ≈ 3*1.056 ≈ 3.168, which is a bit more than π. So log base 3 of π is a bit less than 1.05. Let me approximate. Let's set 3^x = π. Take natural log: x ln3 = lnπ. So x = lnπ / ln3 ≈ (1.1447) / (1.0986) ≈ 1.042. So a ≈ 1.042. Okay, so a is approximately 1.04.Next, b is ln2, which is approximately 0.6931. That's a standard value. So b ≈ 0.693.Then c is log base 10 of 0.8. Since 0.8 is less than 1, the logarithm will be negative. Let's calculate that. log10(0.8) = log10(4/5) = log10(4) - log10(5) ≈ 0.60206 - 0.69897 ≈ -0.09691. So c ≈ -0.0969.Now, comparing the three: a ≈ 1.04, b ≈ 0.693, c ≈ -0.0969. So ordering from largest to smallest is a > b > c. Wait, that's option A. But let me double-check because the options given are A: a > b > c, B: b > a > c, C: c > a > b, D: b > c > a.But according to my calculations, a is around 1.04, which is bigger than b (~0.693), and c is negative, so definitely the order should be a > b > c, which is option A. Hmm, but let me check if I messed up any calculations.First, verifying a: log base 3 of π. As 3^1 = 3, 3^1.1 ≈ 3.348, which is more than π. So log3(π) is between 1 and 1.1. The exact value using natural logs: ln(π)/ln(3) ≈ 1.1447 / 1.0986 ≈ 1.042. That seems right.Then b is ln2 ≈ 0.693. Correct.c is log10(0.8). Since 10^(-0.0969) ≈ 10^(-0.1) ≈ 0.794, which is close to 0.8, so that's correct. Actually, 10^(-0.09691) ≈ 0.8, so the calculation is accurate.Therefore, the order is a (1.04) > b (0.693) > c (-0.0969). So the answer should be A. But wait, let me check again. Maybe there's a mistake in the problem statement? It says "lg0.8". In some contexts, "lg" is log base 2, but usually in math problems, "lg" is log base 10. But the problem states "lg0.8". Let's confirm. In Chinese contexts, "lg" often refers to log base 10, while "ln" is natural log, and "log" without base might be base 10 or base e depending on context, but here since a is log base 3, they specified the base. So "lg" is likely log base 10, so c is log10(0.8). So my calculation of c as approximately -0.0969 is correct.Alternatively, if "lg" was log base 2, then log2(0.8) would be ln(0.8)/ln(2) ≈ (-0.2231)/0.6931 ≈ -0.322, which is more negative. But since the answer options have c as the smallest in A and B, which is negative, so even if c was log2(0.8) ≈ -0.322, the order would still be a > b > c because -0.322 is less than 0.693. But since the problem states "lg0.8", and given the answer options, I think it's safe to assume "lg" is log base 10 here.Therefore, my conclusion is that the correct answer is A: a > b > c.Wait, but looking back, maybe I made a mistake with log base 3 of π. Let's cross-validate. Since 3^1 = 3, 3^1.04 ≈ 3^(1 + 0.04) = 3 * 3^0.04. 3^0.04 ≈ e^(0.04 ln3) ≈ e^(0.04 * 1.0986) ≈ e^0.0439 ≈ 1.0449. So 3^1.04 ≈ 3 * 1.0449 ≈ 3.1347, which is less than π (3.1416). Then 3^1.05 ≈ 3.168 as before, which is higher. So log3(π) is between 1.04 and 1.05. So 1.04 is a good approximation. So a ≈ 1.04, which is indeed greater than b ≈ 0.693, and c ≈ -0.0969. Therefore, the order is a > b > c, which is option A.But wait, the answer given in some references might be different? Let me check once again. Alternatively, maybe there's a trick here. For example, using logarithm properties or comparing without calculating exact values.Alternatively, let's compare a and b without calculating their exact decimal values. Let's see:a = log3(π). Let's convert this to natural logs: a = lnπ / ln3 ≈ 1.1447 / 1.0986 ≈ 1.042.b = ln2 ≈ 0.693.So a is indeed greater than b.Alternatively, if we didn't use calculator approximations, can we compare a and b in another way?Let me see: a = log3(π) and b = ln2. To compare a and b, maybe convert a to natural logs: a = lnπ / ln3, so a ≈ 1.1447 / 1.0986 ≈ 1.042. But if we didn't know the exact values, how else could we compare?Alternatively, since 3^a = π, and e^b = 2. Let's compute 3^a ≈ π ≈ 3.1416, and e^b = 2. So 3^a ≈ 3.14, which is greater than 3^1 = 3, so a > 1. Similarly, e^b = 2, so b ≈ 0.693. Since a is greater than 1, which is already more than b (0.693), so a > b. Therefore, a > b.Then c is log10(0.8) which is negative, so c is the smallest. So the order is a > b > c, which is option A. Therefore, the answer should be A.Wait, but sometimes multiple choice questions can be tricky. Let me check if any of the answer options could be misrepresented. Let me confirm once again:a = log3(π) ≈ 1.04b = ln2 ≈ 0.693c = log10(0.8) ≈ -0.0969Thus, a > b > c. So the correct answer is A. Hmm. But maybe there is a mistake in the problem? For example, if c was log base 2 of 0.8, then c ≈ -0.3219, which is more negative, but still c would be less than b. So regardless, a > b > c.Therefore, unless there's an error in the problem statement, the answer should be A.Wait, but maybe the problem is using different notations? For example, sometimes in some contexts, "lg" is log base 2, but that's more common in computer science. In mathematics, "lg" typically is log base 10, and "ln" is natural log. The problem defines a as log base 3, so it's specified with subscript, then b is "ln" which is natural log, and c is "lg" which is likely log base 10. So my previous reasoning holds.So, conclusion: A is the correct answer.Final Answerboxed{A}</think>Given ( a = log_{3}pi ), ( b = ln 2 ), and ( c = lg 0.8 ), we need to compare their values.1. Calculating ( a ): - ( a = log_{3}pi ) - Convert to natural logarithm: ( a = frac{ln pi}{ln 3} ) - Using approximate values: ( ln pi approx 1.1447 ) and ( ln 3 approx 1.0986 ) - Therefore, ( a approx frac{1.1447}{1.0986} approx 1.042 )2. Calculating ( b ): - ( b = ln 2 ) - Approximate value: ( ln 2 approx 0.6931 )3. Calculating ( c ): - ( c = lg 0.8 ) - Convert to natural logarithm: ( c = frac{ln 0.8}{ln 10} ) - Using approximate values: ( ln 0.8 approx -0.2231 ) and ( ln 10 approx 2.3026 ) - Therefore, ( c approx frac{-0.2231}{2.3026} approx -0.0969 )Comparing the values:- ( a approx 1.042 )- ( b approx 0.6931 )- ( c approx -0.0969 )The order from largest to smallest is ( a > b > c ).Thus, the correct answer is (boxed{A}).,<think>Alright, so I need to figure out the order of a, b, and c where a is log base 3 of pi, b is natural log of 2, and c is log base 10 of 0.8. The options are A: a > b > c, B: b > a > c, C: c > a > b, D: b > c > a. Hmm, let's start by recalling some logarithm properties and maybe approximate their values.First, let's tackle each one individually.Starting with a = log₃π. Since π is approximately 3.1416, which is a bit more than 3. The logarithm base 3 of 3 is 1, so log₃π should be slightly more than 1. Maybe around 1.04 or something? Let me check. If 3^1 = 3, 3^1.1 is 3*3^(0.1). 3^0.1 is approximately 1.116, so 3^1.1 ≈ 3.348. That's higher than π (3.1416). So actually, log₃π is less than 1.1. Maybe around 1.05? Let's see: 3^1.05 = 3^(1 + 0.05) = 3 * 3^0.05. 3^0.05 is e^(0.05 ln3) ≈ 1 + 0.05*1.0986 ≈ 1 + 0.0549 ≈ 1.0549. So 3^1.05 ≈ 3 * 1.0549 ≈ 3.1647. That's still a bit higher than π (3.1416). So maybe 1.04? Let's compute 3^1.04. Similarly, 3^0.04 ≈ e^(0.04*ln3) ≈ 1 + 0.04*1.0986 ≈ 1 + 0.0439 ≈ 1.0439. Then 3^1.04 ≈ 3 * 1.0439 ≈ 3.1317. Hmm, π is 3.1416, so between 1.04 and 1.05. Let's try 1.045: 3^0.045 ≈ e^(0.045*1.0986) ≈ e^0.0494 ≈ 1 + 0.0494 + (0.0494)^2/2 ≈ 1 + 0.0494 + 0.0012 ≈ 1.0506. Then 3^1.045 ≈ 3 * 1.0506 ≈ 3.1518. Still higher than π. So maybe around 1.042? Let's approximate. Let me use linear approximation. Between 1.04 (3.1317) and 1.045 (3.1518). The difference between 3.1416 and 3.1317 is 0.0099, and between 3.1518 and 3.1317 is 0.0201. So fraction is 0.0099 / 0.0201 ≈ 0.4925. So a ≈ 1.04 + 0.4925*(0.005) ≈ 1.04 + 0.00246 ≈ 1.0425. So approximately 1.0425. So a ≈ 1.0425.Next, b = ln2. I remember that ln2 is approximately 0.6931. So that's about 0.693.Then c = lg0.8. lg is log base 10. So log₁₀0.8. Since 0.8 is 8/10 = 10^(-1) * 8, so log₁₀0.8 = log₁₀(10^(-1)*8) = -1 + log₁₀8. log₁₀8 is log₁₀2^3 = 3 log₁₀2. Since log₁₀2 ≈ 0.3010, so 3*0.3010 ≈ 0.9030. Therefore, log₁₀0.8 = -1 + 0.9030 ≈ -0.0970. Alternatively, maybe directly compute log₁₀0.8. Since 10^0 = 1, 10^-0.1 ≈ 0.794, which is close to 0.8. So log₁₀0.8 is approximately -0.0969, which matches the previous calculation. So c ≈ -0.097.So putting them together:a ≈ 1.0425b ≈ 0.6931c ≈ -0.097Therefore, the order from largest to smallest is a > b > c. Which is option A. Wait, but let me check again. Wait, a is approximately 1.04, b is approximately 0.69, so indeed a is bigger than b, which is bigger than c. So A: a > b > c.But let me confirm if my approximation of a is correct. Maybe there's a smarter way to compare a and b without calculating their exact decimal values.Alternatively, since a = log₃π and b = ln2. Let's try to compare them using change of base formula.log₃π = lnπ / ln3 ≈ (1.1447) / (1.0986) ≈ 1.042. So that's the same as before. So a ≈ 1.042.b = ln2 ≈ 0.6931. So indeed, a is greater than b. And c is negative, so c is the smallest.Therefore, the correct order is a > b > c, which is option A.Wait, but hold on. The answer choices are A: a > b > c, B: b > a > c, C: c > a > b, D: b > c > a. So unless I made a mistake in the calculation, A should be the answer.But maybe I messed up with the a value. Let me cross-verify. If log base 3 of pi is approximately 1.04, that seems right because 3^1 = 3, 3^1.1 ≈ 3.348, so pi is 3.1416, which is between 3^1 and 3^1.1, so log base3 pi is between 1 and 1.1, which we approximated as ~1.04. Seems okay.Alternatively, maybe convert a and b to natural logs to compare.a = lnπ / ln3 ≈ 1.1447 / 1.0986 ≈ 1.042.b = ln2 ≈ 0.6931.So 1.042 > 0.6931, so a > b.c is log10(0.8) ≈ -0.0969, which is negative. So c is the smallest.Therefore, the order is a > b > c, so option A. Therefore, the answer should be A.But wait, let me check if any of the answer options could be confused. Maybe the options are in another order. Wait, the options given are:A: a > b > cB: b > a > cC: c > a > bD: b > c > aSo according to our calculation, A is the correct one. Hmm. Wait, but maybe I made a mistake in the sign for c. Let me check c again.c = log10(0.8). Since 0.8 is less than 1, log10(0.8) is indeed negative. 0.8 is 8*10^-1, so log10(8) + log10(10^-1) = log10(8) -1. log10(8) is positive, around 0.903, so 0.903 -1 = -0.097. So yes, c is negative. So c is less than both a and b. Therefore, a and b are positive, with a bigger than b. So A: a > b > c.But wait, just to confirm, maybe the problem is in Chinese and "lg" is log base 2? Wait, the problem says "lg0.8". Usually, in Chinese contexts, "lg" is log base 10, same as "log" without a base. "ln" is natural log, and "log" with subscript is the specified base. So here, a is log base3 π, b is ln2, c is lg0.8 (so log base10 0.8). So my previous calculations hold. So c is indeed approximately -0.097.Therefore, the order is a (≈1.04) > b (≈0.69) > c (≈-0.097), so option A.But wait, just to make sure, maybe there's a trick here. For example, maybe the problem is written in a different notation. But according to standard notation, "lg" is log base10, "ln" is natural log, and "log" with a subscript is the given base. So I think the original reasoning is correct.Therefore, the answer is A.Final Answerboxed{A}</think>To determine the order of (a = log_{3}pi), (b = ln 2), and (c = lg 0.8):1. Calculating (a = log_{3}pi): - (pi approx 3.1416) - Using the change of base formula: (log_{3}pi = frac{ln pi}{ln 3}) - (ln pi approx 1.1447) and (ln 3 approx 1.0986) - (log_{3}pi approx frac{1.1447}{1.0986} approx 1.042)2. Calculating (b = ln 2): - (ln 2 approx 0.6931)3. Calculating (c = lg 0.8): - (lg 0.8 = log_{10} 0.8) - (0.8 = 8 times 10^{-1}) - (log_{10} 0.8 = log_{10} 8 - 1 = 3 log_{10} 2 - 1) - (log_{10} 2 approx 0.3010) - (log_{10} 0.8 approx 3 times 0.3010 - 1 = 0.9030 - 1 = -0.097)Comparing the values:- (a approx 1.042)- (b approx 0.6931)- (c approx -0.097)Thus, the order from largest to smallest is (a > b > c), which corresponds to option A.(boxed{A})]

🔑:Since a=log_{3}pi>log_{3}3=1; b=ln2ln0; c=lg0.8b>c, thus the correct choice is boxed{A}.

🧠:[<think>Okay, let me try to figure out this problem. So, we have a point P on the line l: y = x + 9. We need to construct an ellipse with the shortest possible major axis, and the foci are given as F1(-3, 0) and F2(3, 0). Then, we have to find the equation of that ellipse.First, I remember that an ellipse is defined as the set of points where the sum of the distances from any point on the ellipse to the two foci is constant, which is equal to the length of the major axis. So, the major axis length is 2a, and the constant sum is 2a. Therefore, to minimize the major axis, we need to minimize this sum. Since point P lies on the line y = x + 9 and is also on the ellipse, the sum of the distances from P to F1 and F2 should be equal to 2a. So, to find the shortest possible major axis, we need to find the point P on the line y = x + 9 such that this sum is minimized. Once we find that minimal sum, 2a will be equal to that sum, and then we can write the equation of the ellipse.So, the problem reduces to finding the minimum value of PF1 + PF2 where P is on the line y = x + 9. Then, the minimal sum will be 2a, and we can find the equation of the ellipse.To find the minimal sum, I recall that in optimization problems involving distances, reflection properties might be useful. For example, in problems where you want the shortest path that reflects off a line, you reflect a point across the line and find the straight line distance. Maybe something similar applies here.Alternatively, since we need to minimize PF1 + PF2 for P on a line, maybe we can use calculus. Let me parametrize point P on the line y = x + 9. Let’s let x = t, then y = t + 9. So, P can be written as (t, t + 9). Then, PF1 is the distance from (t, t + 9) to (-3, 0), and PF2 is the distance from (t, t + 9) to (3, 0). Then, the sum PF1 + PF2 is sqrt[(t + 3)^2 + (t + 9)^2] + sqrt[(t - 3)^2 + (t + 9)^2]. We need to minimize this expression with respect to t.Alternatively, perhaps there's a geometric interpretation. Since the sum of the distances is minimized when the ellipse is tangent to the line y = x + 9. Because the ellipse with the smallest major axis that still touches the line would be tangent to it. So, maybe we can use the condition for tangency between the line and the ellipse.But first, let me try the calculus approach. Let me set up the function S(t) = sqrt[(t + 3)^2 + (t + 9)^2] + sqrt[(t - 3)^2 + (t + 9)^2]. We need to find the minimum of S(t).To find the minimum, take the derivative of S(t) with respect to t, set it to zero, and solve for t.Let me compute the derivative S'(t). Let’s denote:A = sqrt[(t + 3)^2 + (t + 9)^2]B = sqrt[(t - 3)^2 + (t + 9)^2]So, S(t) = A + BThen, the derivative S’(t) = (dA/dt) + (dB/dt)Compute dA/dt:dA/dt = [2(t + 3) + 2(t + 9)] / (2A) = [2t + 6 + 2t + 18]/(2A) = [4t + 24]/(2A) = (2t + 12)/ASimilarly, dB/dt:dB/dt = [2(t - 3) + 2(t + 9)] / (2B) = [2t - 6 + 2t + 18]/(2B) = [4t + 12]/(2B) = (2t + 6)/BSo, S’(t) = (2t + 12)/A + (2t + 6)/BSet S’(t) = 0:(2t + 12)/A + (2t + 6)/B = 0Multiply both sides by AB to eliminate denominators:(2t + 12)B + (2t + 6)A = 0So, (2t + 12)B = - (2t + 6)ASquare both sides to eliminate A and B. Wait, but squaring may introduce extraneous solutions. Alternatively, perhaps we can find a relation between A and B.Alternatively, rearrange the terms:[(2t + 12)/A] = - [(2t + 6)/B]So, cross-multiplying:(2t + 12)B = - (2t + 6)ABut this seems complicated. Maybe plug in A and B expressions in terms of t.Alternatively, perhaps we can set u = t + 3 and v = t - 3, but not sure.Alternatively, note that point P lies on y = x + 9, so P is (t, t + 9). The sum of distances from P to F1 and F2. To minimize this sum.Alternatively, using reflection property. For the minimal sum, the ellipse would be tangent to the line y = x + 9. So, the tangent at point P on the ellipse satisfies the reflection property: the angle between the tangent and the line to F1 is equal to the angle between the tangent and the line to F2.But maybe we can use the method of reflecting one focus across the tangent line, and then the line joining the reflected focus to the other focus passes through the point of tangency.Wait, I recall that in ellipse tangents, the reflection property states that the tangent at any point P on the ellipse makes equal angles with the lines from P to each focus. Alternatively, there's a method where if you reflect one focus over the tangent line, the reflected point lies on the line joining the other focus to the point of tangency.But since we don't know the tangent line, but we know that the tangent line is y = x + 9. So, if we reflect one focus over the tangent line, then the line joining the other focus to the reflected focus should pass through the point P.Let me try this approach. Let's reflect one of the foci across the line y = x + 9. Let's choose, say, F2 = (3, 0). Let's find the reflection of F2 over the line l: y = x + 9. The reflection point, let's call it F2', will lie such that the line l is the perpendicular bisector of the segment joining F2 and F2'.To find the reflection of a point over a line, here's the formula. Given a line ax + by + c = 0 and a point (x0, y0), the reflection of the point is given by:(x', y') = (x0 - 2a(ax0 + by0 + c)/(a² + b²), y0 - 2b(ax0 + by0 + c)/(a² + b²))First, write the line y = x + 9 in standard form: x - y + 9 = 0. So, a = 1, b = -1, c = 9.So, reflecting F2 (3, 0):Compute ax0 + by0 + c = 1*3 + (-1)*0 + 9 = 12.Then,x' = 3 - 2*1*(12)/(1 + 1) = 3 - 24/2 = 3 - 12 = -9y' = 0 - 2*(-1)*(12)/2 = 0 + 24/2 = 12So, the reflection of F2 over line l is F2'(-9, 12)Now, according to the reflection property, the point P lies on the line l, and also lies on the line joining F1 to F2'. Because the reflection over the tangent line causes the path from F1 to P to F2 to be equivalent to the straight line from F1 to F2' via P.Therefore, the point P is the intersection point of the line joining F1(-3, 0) and F2'(-9, 12), and the line l: y = x + 9.So, let's find the equation of the line joining F1(-3, 0) and F2'(-9, 12).First, compute the slope:m = (12 - 0)/(-9 - (-3)) = 12 / (-6) = -2So, the line equation is y - 0 = -2(x + 3), which simplifies to y = -2x -6Now, find intersection point P between y = -2x -6 and y = x + 9.Set x + 9 = -2x -6x + 2x = -6 -93x = -15x = -5Then, y = (-5) + 9 = 4So, point P is (-5, 4)Now, verify that this point is on both lines:For line l: y = x + 9 → 4 = -5 + 9 = 4 ✔️For the line joining F1 and F2': y = -2x -6 → 4 = -2*(-5) -6 → 4 = 10 -6 = 4 ✔️So, point P is (-5, 4). Now, compute the sum of distances PF1 + PF2, which will be 2a.Compute PF1:Distance from (-5,4) to (-3,0):sqrt[(-5 + 3)^2 + (4 - 0)^2] = sqrt[(-2)^2 + 4^2] = sqrt[4 + 16] = sqrt[20] = 2*sqrt(5)Compute PF2:Distance from (-5,4) to (3,0):sqrt[(-5 -3)^2 + (4 -0)^2] = sqrt[(-8)^2 + 4^2] = sqrt[64 + 16] = sqrt[80] = 4*sqrt(5)Sum: 2*sqrt(5) + 4*sqrt(5) = 6*sqrt(5). So, 2a = 6*sqrt(5) → a = 3*sqrt(5)Now, the distance between the two foci is 2c. The foci are at (-3,0) and (3,0), so distance between them is 6 units, so 2c = 6 → c = 3.In an ellipse, we have the relationship a^2 = b^2 + c^2. So, we can compute b:b = sqrt(a^2 - c^2) = sqrt[(3*sqrt(5))^2 - 3^2] = sqrt[45 - 9] = sqrt[36] = 6Therefore, the standard equation of the ellipse is (x^2)/a^2 + (y^2)/b^2 = 1. Wait, but we need to confirm the major axis. Since a > b? Wait, a is 3*sqrt(5) ≈ 6.708, and b is 6, so yes, a > b, so the major axis is along the x-axis? Wait, but in standard notation, if the major axis is along the x-axis, then the equation is (x^2/a^2) + (y^2/b^2) = 1. But wait, if the major axis is along the x-axis, then a is the semi-major axis, and b is the semi-minor. However, in our case, since the foci are on the x-axis, the major axis is along the x-axis.Wait, but in our case, the sum of distances is 6*sqrt(5), so 2a = 6*sqrt(5) → a = 3*sqrt(5). The distance between the foci is 6, so 2c = 6 → c = 3. Then, b^2 = a^2 - c^2 = (45) - 9 = 36 → b = 6.Therefore, the equation is (x^2)/( (3*sqrt(5))^2 ) + (y^2)/(6^2) = 1 → (x^2)/45 + (y^2)/36 = 1.Wait, but let me check if the center is at the origin. Since the foci are at (-3,0) and (3,0), the center is the midpoint of the foci, which is (0,0). So yes, the center is at the origin, so the standard equation is (x^2)/a^2 + (y^2)/b^2 = 1 with a = 3*sqrt(5) and b = 6.So, the equation is x²/45 + y²/36 = 1.Therefore, the equation of the ellipse is x²/45 + y²/36 = 1.But let me verify once again. The sum of distances from P(-5,4) is 6*sqrt(5), which is 2a. So a = 3*sqrt(5). Then, since the foci are at (±3,0), c = 3. Then, b² = a² - c² = 45 - 9 = 36, so b = 6. Therefore, the equation is indeed x²/45 + y²/36 = 1. That seems correct.Alternatively, perhaps I made a mistake in assuming that the reflection method works here. Let me cross-validate with calculus.Earlier, we parametrized P as (t, t + 9), and found that the minimal sum occurs at t = -5, leading to P(-5,4). Let's check the derivative.Compute S’(t) at t = -5:First, compute A = sqrt[(-5 + 3)^2 + (-5 + 9)^2] = sqrt[(-2)^2 + (4)^2] = sqrt[4 + 16] = sqrt(20) = 2*sqrt(5)B = sqrt[(-5 -3)^2 + (-5 + 9)^2] = sqrt[(-8)^2 + 4^2] = sqrt[64 + 16] = sqrt(80) = 4*sqrt(5)Then, compute S’(-5) = (2*(-5) + 12)/A + (2*(-5) + 6)/B= (-10 + 12)/ (2*sqrt(5)) + (-10 + 6)/(4*sqrt(5))= (2)/(2*sqrt(5)) + (-4)/(4*sqrt(5))= (1/sqrt(5)) - (1/sqrt(5)) = 0So, indeed, the derivative is zero at t = -5, confirming that this is a critical point. Since this is the only critical point and the function S(t) tends to infinity as t approaches ±∞, this must be the minimum. Therefore, the calculus approach confirms the reflection method.Thus, the ellipse with the shortest major axis through point P(-5,4) has a = 3*sqrt(5), b = 6, centered at the origin, with equation x²/45 + y²/36 = 1.</think>You will be given a problem. Please reason step by step, and put your final answer within boxed{}:7. Through a point P on the line l: y=x+9, construct an ellipse with the shortest major axis, whose foci are F_{1}(-3,0), F_{2}(3,0). Then the equation of the ellipse is qquadThrough a point ( P ) on the line ( l: y = x + 9 ), construct an ellipse with the shortest major axis, whose foci are ( F_1(-3,0) ) and ( F_2(3,0) ). Then the equation of the ellipse is (boxed{dfrac{x^2}{45} + dfrac{y^2}{36} = 1}).,<think>Okay, let's try to figure out this problem step by step. So, we need to construct an ellipse with the shortest possible major axis, given that the foci are at F1(-3,0) and F2(3,0), and the ellipse passes through a point P on the line l: y = x + 9. The goal is to find the equation of this ellipse.First, I remember that the definition of an ellipse is the set of all points where the sum of the distances from the two foci is constant. That constant is equal to the length of the major axis, which is 2a, where a is the semi-major axis. So, for any point on the ellipse, the sum of the distances to F1 and F2 is 2a. Since we want the shortest major axis, we need to minimize 2a, which means we need to find the smallest possible a such that the ellipse still passes through some point P on the line y = x + 9.So, the key here is to find the point P on the line y = x + 9 that minimizes the sum of the distances from P to F1 and F2. Once we find that point, the sum of those distances will be the minimal 2a, and then we can write the equation of the ellipse.Let me denote the point P as (t, t + 9) since it lies on the line y = x + 9. So, P = (t, t + 9). Then, the distance from P to F1(-3, 0) is sqrt[(t + 3)^2 + (t + 9 - 0)^2], and the distance from P to F2(3, 0) is sqrt[(t - 3)^2 + (t + 9 - 0)^2]. The sum of these two distances should be minimized.Therefore, we need to minimize the function f(t) = sqrt[(t + 3)^2 + (t + 9)^2] + sqrt[(t - 3)^2 + (t + 9)^2].This looks like a calculus optimization problem. To find the minimum, I can take the derivative of f(t) with respect to t, set it to zero, and solve for t. However, taking the derivative of such a function might be a bit complicated. Let's see.First, let's simplify the expressions inside the square roots. Let's compute (t + 3)^2 + (t + 9)^2. Expanding that:(t^2 + 6t + 9) + (t^2 + 18t + 81) = 2t^2 + 24t + 90.Similarly, (t - 3)^2 + (t + 9)^2:(t^2 - 6t + 9) + (t^2 + 18t + 81) = 2t^2 + 12t + 90.So, f(t) = sqrt(2t^2 + 24t + 90) + sqrt(2t^2 + 12t + 90).Hmm, maybe factor out a 2? Let's see:sqrt(2(t^2 + 12t + 45)) + sqrt(2(t^2 + 6t + 45)).Which is sqrt(2)*sqrt(t^2 + 12t + 45) + sqrt(2)*sqrt(t^2 + 6t + 45).But that might not help much. Alternatively, perhaps we can write f(t) as sqrt(2t^2 + 24t + 90) + sqrt(2t^2 + 12t + 90).Alternatively, perhaps complete the square for each quadratic inside the square roots.First, for 2t^2 + 24t + 90:Factor out 2: 2(t^2 + 12t + 45). Then complete the square inside the parentheses:t^2 + 12t + 36 + 9 = (t + 6)^2 + 9. So, sqrt(2[(t + 6)^2 + 9]).Similarly, for 2t^2 + 12t + 90:Factor out 2: 2(t^2 + 6t + 45). Completing the square:t^2 + 6t + 9 + 36 = (t + 3)^2 + 36. So, sqrt(2[(t + 3)^2 + 36]).Therefore, f(t) can be written as sqrt(2) [sqrt((t + 6)^2 + 9) + sqrt((t + 3)^2 + 36)].Hmm, but I don't know if that helps. Let's see, maybe consider substituting variables. Let me set u = t + 6. Then, the first term becomes sqrt(2) sqrt(u^2 + 9) and the second term becomes sqrt(2) sqrt((u - 3)^2 + 36). So, f(t) in terms of u would be sqrt(2)[sqrt(u^2 + 9) + sqrt((u - 3)^2 + 36)]. Maybe not helpful.Alternatively, perhaps consider reflecting one of the foci. There's a method in geometry where to find the minimal path that reflects off a line, which involves reflecting a point across the line. Maybe similar idea here. Since the ellipse must pass through a point on line l, which is y = x + 9, perhaps reflecting one of the foci over this line and then the minimal distance would be the straight line between the reflected point and the other focus. The intersection of that line with the original line l would give the point P. Let me think if that applies here.Wait, in the reflection method for ellipses: since the ellipse is the set of points with constant sum of distances to two foci, if we have a constraint that the ellipse passes through a point on a line, then the minimal major axis would correspond to the minimal sum of distances from a point on the line to the two foci. So, geometrically, this is equivalent to finding the point P on line l such that PF1 + PF2 is minimized.To minimize PF1 + PF2, one approach is to use reflection. For example, if you reflect one focus across the line l, then the minimal path from F2 to the reflection F1' would intersect the line l at point P, which would be the point that minimizes PF1 + PF2, since PF1 + PF2 = PF1' + PF2, which is minimized when P is on the straight line between F1' and F2.Therefore, perhaps reflecting F1 over line l: y = x + 9, then the minimal sum would be the distance between F1' and F2, and the point P is the intersection point of line F1'F2 with line l.Let me try this approach.First, find the reflection of F1(-3, 0) across the line l: y = x + 9.To reflect a point over a line, we can use the formula. The formula for reflecting a point (x0, y0) over the line ax + by + c = 0 is:x' = x0 - 2a(ax0 + by0 + c)/(a² + b²)y' = y0 - 2b(ax0 + by0 + c)/(a² + b²)First, write line l in standard form. The line is y = x + 9, which can be rewritten as -x + y - 9 = 0. So, a = -1, b = 1, c = -9.So, reflecting point F1(-3, 0):Compute numerator terms:ax0 + by0 + c = (-1)(-3) + 1*(0) - 9 = 3 + 0 - 9 = -6.Then,x' = -3 - 2*(-1)*(-6)/((-1)^2 + 1^2) = Wait, no. Wait, the formula is:x' = x0 - 2a(ax0 + by0 + c)/(a² + b²)Similarly for y'.So,Compute denominator: a² + b² = (-1)^2 + 1^2 = 1 + 1 = 2.Compute numerator for x':-2a(ax0 + by0 + c) = -2*(-1)*(-6) = -2*1*6 = -12.Wait, wait:Wait, the formula is:x' = x0 - 2a(ax0 + by0 + c)/(a² + b²)Similarly,y' = y0 - 2b(ax0 + by0 + c)/(a² + b²)So, let's compute:ax0 + by0 + c = (-1)(-3) + 1*(0) -9 = 3 -9 = -6.So, x' = -3 - 2*(-1)*(-6)/2First, compute -2a(ax0 + by0 + c)/(a² + b²):-2*(-1)*(-6)/2 = (2* -6)/2 = (-12)/2 = -6So, x' = -3 + (-6) = -9Similarly, y':y' = 0 - 2*1*(-6)/2Compute:-2b(ax0 + by0 + c)/(a² + b²) = -2*1*(-6)/2 = (12)/2 = 6So, y' = 0 + 6 = 6Therefore, the reflection of F1(-3,0) over line l: y = x +9 is F1'(-9,6).Wait, let me check this calculation again because it's easy to make a mistake here.Original point F1(-3,0). Line l: y = x +9. Let's convert line l to standard form: x - y +9 =0. Wait, no: y = x +9 can be written as -x + y -9 =0, which is the same as x - y +9 =0 multiplied by -1. So, the standard form is x - y +9 =0. But different sources might define a, b, c with different signs. Let me confirm.The standard form is ax + by + c =0. Given y = x +9, subtract x and subtract 9: -x + y -9 =0. So, coefficients are a=-1, b=1, c=-9. Alternatively, multiplying both sides by -1: x - y +9 =0. But perhaps depending on the formula, reflection might be the same. Let's check both.Let me use the formula correctly. Suppose we use line l: -x + y -9 =0. Then, a=-1, b=1, c=-9.Original point (x0, y0) = (-3,0).Compute ax0 + by0 + c = (-1)(-3) + (1)(0) -9 = 3 +0 -9 = -6.Then, x' = x0 - 2a(ax0 + by0 + c)/(a² + b²)a² + b² = (-1)^2 +1^2 =2.So, x' = -3 -2*(-1)*(-6)/2 = -3 - (12)/2 = -3 -6 = -9Similarly, y' = y0 -2b(ax0 + by0 + c)/(a² + b²) = 0 -2*(1)*(-6)/2 = 0 - (-12)/2 = 0 +6 =6.So, reflection point is (-9,6). Okay, that seems correct.Alternatively, if we had written the line as x - y +9=0, so a=1, b=-1, c=9.Then ax0 + by0 + c =1*(-3) + (-1)(0) +9= -3 +0 +9=6.Then x' = x0 -2a(ax0 + by0 + c)/(a² + b²) = -3 -2*1*6/2= -3 -6= -9y'= y0 -2b(ax0 + by0 + c)/(a² + b²)=0 -2*(-1)*6/2=0 +6=6Same result. So, reflection point is (-9,6).Therefore, the reflection of F1 over line l is F1'(-9,6).Now, the minimal sum PF1 + PF2 is equal to the distance from F1' to F2, because the minimal path from F1' to F2 passing through line l is a straight line, and the intersection point P is the point where this line meets l.Thus, the minimal sum of distances PF1 + PF2 is equal to the distance between F1'(-9,6) and F2(3,0).Compute this distance:sqrt[(3 - (-9))^2 + (0 -6)^2] = sqrt[(12)^2 + (-6)^2] = sqrt[144 +36] = sqrt[180] = sqrt[36*5] = 6*sqrt(5).Therefore, the minimal major axis length is 6*sqrt(5), so semi-major axis a is 3*sqrt(5).Now, to find the equation of the ellipse, we need to know a, the semi-major axis, and the distance between the foci, which is 2c. The distance between F1 and F2 is sqrt[(3 - (-3))^2 + (0 -0)^2] = sqrt[6^2] =6. So, c=3.Then, for an ellipse, we have the relationship a^2 = b^2 + c^2, where b is the semi-minor axis. So, b = sqrt(a^2 - c^2) = sqrt[(3√5)^2 -3^2] = sqrt[45 -9] = sqrt[36] =6.Since the major axis is along the line joining the two foci, which are on the x-axis (since F1 and F2 are at (-3,0) and (3,0)), so the major axis is horizontal. Therefore, the standard equation of the ellipse is (x^2)/a^2 + (y^2)/b^2 =1.Plugging in a =3√5 and b=6, we get:x²/( (3√5)^2 ) + y²/36 =1Simplify (3√5)^2: 9*5=45.Thus, the equation is x²/45 + y²/36 =1.But wait, let's confirm that the center of the ellipse is at the midpoint of F1 and F2. The midpoint between (-3,0) and (3,0) is (0,0). So, the center is (0,0), which matches the standard equation we wrote.However, we need to make sure that the point P is indeed on the ellipse. Wait, but we constructed the ellipse such that it passes through point P on line l, which is the intersection point of line F1'F2 and line l. So, we need to find the coordinates of P to confirm, but maybe it's not necessary for the equation. Since the ellipse is centered at the origin, with major axis length 2a=6√5 and minor axis 2b=12, equation x²/45 + y²/36=1.But let's verify if this ellipse indeed passes through the point P. Let's find the coordinates of point P first.Point P is the intersection of line F1'F2 and line l:y=x+9.F1' is (-9,6), F2 is (3,0). The line connecting (-9,6) and (3,0).Compute the equation of the line through (-9,6) and (3,0).First, compute the slope: (0 -6)/(3 - (-9))= (-6)/12 = -0.5.So, the slope is -1/2. Using point-slope form with point (3,0):y -0 = -1/2(x -3)y= -1/2 x + 3/2.Find intersection with line l:y =x +9.Set -1/2 x + 3/2 = x +9Multiply both sides by 2 to eliminate denominators:- x +3 = 2x +18- x -2x =18 -3-3x =15x= -5Then, y= x +9= -5 +9=4Therefore, point P is (-5,4).Now, let's verify if (-5,4) is on the ellipse x²/45 + y²/36 =1.Compute (-5)^2 /45 + (4)^2 /36 =25/45 +16/36 =5/9 +4/9=9/9=1. Yes, it satisfies the equation.Therefore, the equation of the ellipse is indeed x²/45 + y²/36 =1.So, the answer should be boxed{dfrac{x^2}{45} + dfrac{y^2}{36} = 1}</think>Through a point ( P ) on the line ( l: y = x + 9 ), construct an ellipse with the shortest major axis, whose foci are ( F_1(-3,0) ) and ( F_2(3,0) ). Then the equation of the ellipse is (boxed{dfrac{x^2}{45} + dfrac{y^2}{36} = 1}).Reasoning:1. Definition of an Ellipse: An ellipse is the set of all points where the sum of the distances from two fixed points (foci) is constant. This constant is the length of the major axis (2a).2. Objective: Minimize the major axis length (2a) for an ellipse passing through a point (P) on the line (l: y = x + 9).3. Reflection Method: Reflect one focus across the line (l). The minimal sum of distances from (P) to the foci (F_1) and (F_2) is the distance between the reflection of (F_1) (denoted (F_1')) and (F_2).4. Reflection Calculation: - Reflect (F_1(-3,0)) over (l: y = x + 9). Using the reflection formula, the coordinates of (F_1') are found to be ((-9, 6)). - Calculate the distance between (F_1'(-9, 6)) and (F_2(3, 0)): [ text{Distance} = sqrt{(3 - (-9))^2 + (0 - 6)^2} = sqrt{12^2 + (-6)^2} = sqrt{180} = 6sqrt{5} ] - The minimal major axis length (2a = 6sqrt{5}), so (a = 3sqrt{5}).5. Semi-minor Axis: - Distance between foci (2c = 6), so (c = 3). - Relationship (a^2 = b^2 + c^2) gives: [ b = sqrt{a^2 - c^2} = sqrt{(3sqrt{5})^2 - 3^2} = sqrt{45 - 9} = sqrt{36} = 6 ]6. Equation of the Ellipse: - Center at the midpoint of (F_1) and (F_2), which is ((0,0)). - Standard form: (dfrac{x^2}{a^2} + dfrac{y^2}{b^2} = 1). - Substituting (a = 3sqrt{5}) and (b = 6): [ dfrac{x^2}{45} + dfrac{y^2}{36} = 1 ]7. Verification: - Point (P(-5, 4)) (intersection of line (l) and the line through (F_1') and (F_2)) satisfies the ellipse equation.]

🔑:7. frac{x^{2}}{45}+frac{y^{2}}{36}=1.As shown in Figure 2, let the point on line l be P(t, t+9), and take the symmetric point of F_{1}(-3,0) about line l as Q(-9,6). According to the definition of an ellipse, we have2 a=P F_{1}+P F_{2}=P Q+P F_{2} geqslant Q F_{2}=sqrt{12^{2}+6^{2}}=6 sqrt{5},with equality holding if and only if Q, P, F_{2} are collinear, i.e., k_{P F_{2}}=k_{Q F_{2}}, which means frac{t+9}{t-3}=frac{6}{-12}. In this case, t=-5, P(-5,4).Given c=3, a=3 sqrt{5}, we get a^{2}=45, b^{2}=36. Therefore, the equation of the ellipse is frac{x^{2}}{45}+frac{y^{2}}{36}=1.

🧠:[<think>Okay, let's tackle these two problems one by one. First, the system of equations. The equations are:1) 2x - y = -4 2) 4x - 5y = -23Hmm, systems of equations can be solved either by substitution or elimination. Let me see which method would be better here. If I can solve one of the equations for one variable and substitute into the other, that might work. Let's try solving the first equation for y. From the first equation: 2x - y = -4. If I add y to both sides and add 4 to both sides, I get 2x + 4 = y. So, y = 2x + 4. Now that I have y expressed in terms of x, I can substitute this into the second equation. The second equation is 4x - 5y = -23. Let me replace y with 2x + 4 here. So, substituting: 4x - 5*(2x + 4) = -23. Let's compute that. First, distribute the -5 across the terms inside the parentheses: 4x -10x -20 = -23. Combine like terms: 4x -10x is -6x, so -6x -20 = -23. Now, add 20 to both sides: -6x = -23 + 20, which is -3. Then, divide both sides by -6: x = (-3)/(-6) = 0.5. So, x = 1/2. Now, substitute x back into the expression we had for y: y = 2x + 4. Since x = 1/2, that becomes y = 2*(1/2) + 4 = 1 + 4 = 5. Wait, let me check that again. 2*(1/2) is 1, yes, then 1 + 4 is 5. So y = 5. Let me verify these solutions in both original equations to make sure. First equation: 2x - y = 2*(1/2) - 5 = 1 - 5 = -4. That matches the first equation's right-hand side. Good. Second equation: 4x -5y = 4*(1/2) -5*5 = 2 -25 = -23. Which is the same as the right-hand side. Perfect. So x = 1/2 and y = 5 is the solution. Alright, part (1) is done. Now moving on to part (2). The problem is to simplify the expression (x - 3y)^2 - (2x + y)(y - 2x), and then evaluate it when x = 2 and y = -1. First, let's try to simplify the expression before plugging in the values. Let me write it down again:(x - 3y)^2 - (2x + y)(y - 2x)Let me look at each part. The first term is a square, so we can expand that. The second term is a product of two binomials. Notice that (y - 2x) is the same as -(2x - y). Let me check that. Yes, because y - 2x = -(2x - y). So, the second term is (2x + y)*( - (2x - y) ) = - (2x + y)(2x - y). That might be useful because (2x + y)(2x - y) is a difference of squares, which is (2x)^2 - y^2. Let's see if that works.But let's proceed step by step. First, expand (x - 3y)^2.(x - 3y)^2 = x^2 - 2*x*(3y) + (3y)^2 = x^2 -6xy +9y^2.Now, the second term: (2x + y)(y - 2x). Let's expand this as is. Multiply each term:First, 2x*y = 2xy Then, 2x*(-2x) = -4x^2 Then, y*y = y^2 Then, y*(-2x) = -2xy So combining these terms: 2xy -4x^2 + y^2 -2xy. Combine like terms: 2xy -2xy cancels out, leaving -4x^2 + y^2. So the second term simplifies to -4x^2 + y^2. Therefore, the entire expression becomes:(x - 3y)^2 - (2x + y)(y - 2x) = [x^2 -6xy +9y^2] - [ -4x^2 + y^2 ]Subtracting the second term is the same as adding the opposite:= x^2 -6xy +9y^2 +4x^2 - y^2Now, combine like terms:x^2 +4x^2 = 5x^2-6xy remains as is9y^2 - y^2 = 8y^2So the simplified expression is 5x^2 -6xy +8y^2. Alternatively, if I had noticed earlier that (2x + y)(y - 2x) is equal to -(2x + y)(2x - y) = -[(2x)^2 - y^2] = -[4x^2 - y^2] = -4x^2 + y^2, which is the same result. So that's another way to see it. Either way, the simplified expression is 5x^2 -6xy +8y^2.Now, substituting x = 2 and y = -1 into the simplified expression:5*(2)^2 -6*(2)*(-1) +8*(-1)^2Compute each term:5*(4) = 20-6*(2)*(-1) = -6*(-2) = 128*(1) =8Add them up: 20 +12 +8 =40.Alternatively, if I substitute into the original expression without simplifying first, just to check:Original expression: (x -3y)^2 - (2x + y)(y -2x)Plug in x=2, y=-1:(2 -3*(-1))^2 - (2*2 + (-1))*(-1 -2*2)Calculate each part:First term: (2 +3)^2 =5^2=25Second term: (4 -1)*(-1 -4)=3*(-5)= -15So the entire expression is 25 - (-15) =25 +15=40. Same result. Good, that checks out. Therefore, the simplified expression is 5x² -6xy +8y², and when evaluated at x=2, y=-1, the value is 40.Final Answer(1) The solution is boxed{x = dfrac{1}{2}} and boxed{y = 5}.(2) The simplified value is boxed{40}.</think>(1) To solve the system of equations:[left{begin{array}{l}2x - y = -4 4x - 5y = -23end{array}right.]First, solve the first equation for ( y ):[2x - y = -4 implies y = 2x + 4]Substitute ( y = 2x + 4 ) into the second equation:[4x - 5(2x + 4) = -23 4x - 10x - 20 = -23 -6x - 20 = -23 -6x = -3 x = frac{1}{2}]Substitute ( x = frac{1}{2} ) back into ( y = 2x + 4 ):[y = 2 left( frac{1}{2} right) + 4 = 1 + 4 = 5]Verification in both original equations confirms the solution. Thus, the solution is:[boxed{x = dfrac{1}{2}} text{ and } boxed{y = 5}](2) Simplify the expression ((x - 3y)^2 - (2x + y)(y - 2x)) and evaluate it at ( x = 2 ) and ( y = -1 ):First, expand and simplify the expression:[(x - 3y)^2 - (2x + y)(y - 2x)]Expand ((x - 3y)^2):[x^2 - 6xy + 9y^2]Expand ((2x + y)(y - 2x)):[2xy - 4x^2 + y^2 - 2xy = -4x^2 + y^2]Combine the results:[x^2 - 6xy + 9y^2 - (-4x^2 + y^2) = x^2 - 6xy + 9y^2 + 4x^2 - y^2 = 5x^2 - 6xy + 8y^2]Substitute ( x = 2 ) and ( y = -1 ):[5(2)^2 - 6(2)(-1) + 8(-1)^2 = 5(4) + 12 + 8(1) = 20 + 12 + 8 = 40]Thus, the simplified value is:[boxed{40}],<think>Okay, let me try to solve this problem step by step. There are two parts here. First, solving a system of equations, and second, simplifying an expression and then evaluating it with given values. Let me start with the first part.Part 1: Solving the system of equationsThe system given is:[begin{cases}2x - y = -4 4x - 5y = -23end{cases}]Hmm, I need to find the values of x and y that satisfy both equations. Let me think about how to approach this. There are a couple of methods: substitution or elimination. Since both equations are in standard form, maybe elimination would work here. Let me check.Looking at the coefficients of x in both equations. The first equation has 2x, and the second has 4x. If I multiply the first equation by 2, the coefficient of x would become 4, which matches the second equation. Then I can subtract one equation from the other to eliminate x. Let's try that.Multiply the first equation by 2:[2*(2x - y) = 2*(-4) 4x - 2y = -8]So now the modified first equation is 4x - 2y = -8. The second original equation is 4x - 5y = -23. Now, subtract the modified first equation from the second equation to eliminate x.Subtracting:[(4x - 5y) - (4x - 2y) = -23 - (-8) 4x -5y -4x +2y = -23 +8 (-5y +2y) = -15 -3y = -15]Okay, so -3y = -15. Solving for y, divide both sides by -3:[y = (-15)/(-3) = 5]So y is 5. Now, substitute this value back into one of the original equations to find x. Let me use the first equation because the numbers might be smaller.Original first equation: 2x - y = -4. Substitute y=5:[2x -5 = -4 2x = -4 +5 2x = 1 x = 1/2]Wait, x is 1/2? Let me check if that's correct by plugging both values into the second equation to verify.Second equation: 4x -5y = -23. Substitute x=1/2 and y=5:[4*(1/2) -5*5 = 2 -25 = -23]Yes, that's correct. So x=1/2 and y=5 are the solutions. Let me just recap to make sure I didn't make a mistake in the elimination step. Multiplying the first equation by 2, then subtracting to eliminate x. Then solving for y. Then back-substituting. Seems okay. So part 1 answer is x=1/2, y=5.Part 2: Simplify the expression, then evaluate with x=2 and y=-1The expression is: ((x - 3y)^2 - (2x + y)(y - 2x)). Need to simplify this first before plugging in the values. Let me work on expanding and simplifying.First, let's take each part separately. The first term is ((x - 3y)^2). Let's expand that.[(x - 3y)^2 = x^2 - 2*x*3y + (3y)^2 = x^2 -6xy +9y^2]Okay, so that's the first part. The second term is ((2x + y)(y - 2x)). Let's expand this product. Notice that (y - 2x) is the same as -(2x - y). Alternatively, multiply term by term.Let me compute it:[(2x + y)(y - 2x) = 2x*y + 2x*(-2x) + y*y + y*(-2x)]Calculating each term:- First term: 2x*y = 2xy- Second term: 2x*(-2x) = -4x²- Third term: y*y = y²- Fourth term: y*(-2x) = -2xyCombine all these terms:2xy -4x² + y² -2xyCombine like terms:2xy -2xy cancels out, leaving -4x² + y²So the second term simplifies to -4x² + y²Now, the entire expression is:First term - second term:[(x^2 -6xy +9y^2) - (-4x² + y²)]Distribute the negative sign:x² -6xy +9y² +4x² - y²Combine like terms:x² +4x² =5x²-6xy remains as is9y² - y² =8y²So the entire expression simplifies to:5x² -6xy +8y²Alright, so that's the simplified form. Now, substitute x=2 and y=-1 into this simplified expression.Compute each term:First term: 5x² =5*(2)^2=5*4=20Second term: -6xy = -6*(2)*(-1) = -6*(-2)=12Third term:8y²=8*(-1)^2=8*1=8Now, add them together:20 +12 +8=40Wait, that's 20 +12 is 32, plus 8 is 40? Let me verify.Yes, 20 (from 5x²) +12 (from -6xy) +8 (from 8y²) =40.But let me check the substitution step again to be thorough. x=2, y=-1.Original simplified expression:5x² -6xy +8y²Compute each term:5x²:5*(2)^2=5*4=20-6xy: -6*(2)*(-1)= -6*(-2)=128y²:8*(-1)^2=8*1=8Sum:20+12+8=40. Yes, correct.Alternatively, if I substitute into the original expression without simplifying first, just to check.Original expression: (x -3y)^2 - (2x + y)(y -2x)Substitute x=2, y=-1.Compute each part:First part: (2 -3*(-1))^2 = (2 +3)^2 =5^2=25Second part: (2*2 + (-1))*( (-1) -2*2 )=(4 -1)*(-1 -4)=3*(-5)=-15So the expression is 25 - (-15)=25 +15=40. Same result. So that's a good check. So the value is 40.Summary:Part1: x=1/2, y=5Part2: After simplifying, substituting gives 40.Final Answer(1) The solution is boxed{x=dfrac{1}{2}} and boxed{y=5}.(2) The value is boxed{40}.</think>Part 1: Solving the system of equationsThe given system of equations is:[begin{cases}2x - y = -4 4x - 5y = -23end{cases}]To solve this system, we use the elimination method. First, multiply the first equation by 2:[4x - 2y = -8]Subtract this modified equation from the second equation:[(4x - 5y) - (4x - 2y) = -23 - (-8) -3y = -15 y = 5]Substitute (y = 5) back into the first original equation:[2x - 5 = -4 2x = 1 x = frac{1}{2}]Verification by substituting (x = frac{1}{2}) and (y = 5) into the second equation confirms the solution is correct.Part 2: Simplifying and evaluating the expressionThe given expression is:[(x - 3y)^2 - (2x + y)(y - 2x)]First, expand and simplify the expression:[(x - 3y)^2 = x^2 - 6xy + 9y^2][(2x + y)(y - 2x) = -4x^2 + y^2][(x - 3y)^2 - (2x + y)(y - 2x) = (x^2 - 6xy + 9y^2) - (-4x^2 + y^2) = 5x^2 - 6xy + 8y^2]Substitute (x = 2) and (y = -1) into the simplified expression:[5(2)^2 - 6(2)(-1) + 8(-1)^2 = 20 + 12 + 8 = 40] Final Answer(1) The solution is (boxed{x=dfrac{1}{2}}) and (boxed{y=5}).(2) The value is (boxed{40}).]

🔑: Problem 1: Solve the system of equationsGiven the system of equations:left{begin{array}{l}2x-y=-4 quad (1) 4x-5y=-23 quad (2)end{array}right.Step 1: Multiply equation (1) by 2 to eliminate y when subtracted from equation (2).- 2(2x - y) = 2(-4)- 4x - 2y = -8Step 2: Subtract the modified equation (1) from equation (2).- (4x - 5y) - (4x - 2y) = -23 - (-8)- -3y = -15Step 3: Solve for y.- y = frac{-15}{-3} = 5Step 4: Substitute y = 5 into equation (1) to solve for x.- 2x - 5 = -4- 2x = 1- x = frac{1}{2}Therefore, the solution to the system of equations is boxed{left{begin{array}{l}{x=frac{1}{2}}{y=5}end{array}right.}. Problem 2: Simplify the expression and find its value given x=2 and y=-1Given the expression: left(x-3yright)^{2}-left(2x+yright)left(y-2xright)Step 1: Expand and simplify the expression.- left(x-3yright)^{2} = x^2 - 6xy + 9y^2- left(2x+yright)left(y-2xright) = 2xy + y^2 - 4x^2 - 2xy = y^2 - 4x^2- Combining both parts: x^2 - 6xy + 9y^2 - (y^2 - 4x^2) = x^2 - 6xy + 9y^2 - y^2 + 4x^2 = 5x^2 - 6xy + 8y^2Step 2: Substitute x=2 and y=-1 into the simplified expression.- 5(2)^2 - 6(2)(-1) + 8(-1)^2- = 5 cdot 4 + 12 + 8 cdot 1- = 20 + 12 + 8- = 40Therefore, the value of the expression when x=2 and y=-1 is boxed{40}.