💡Okay, so I have this cube problem where each face is painted with a distinct color: blue (B), orange (O), yellow (Y), violet (V), teal (T), and pink (P). There are three different views of the cube, and I need to figure out which color is opposite the teal face.First, let me try to visualize the cube. A cube has six faces: top, bottom, front, back, left, and right. Each of these is painted with a different color. The problem gives me three views of the cube, and in each view, the top face is violet. That means violet is consistently on the top, so I can note that down: top face = violet (V).Next, looking at the front faces in each view: the first view shows blue (B), the second view shows yellow (Y), and the third view shows pink (P). So, the front face changes in each view. That suggests that the cube is being rotated between these views, with different faces becoming the front each time.Also, in all three views, the left face is orange (O). So, regardless of how the cube is rotated, the left face remains orange. That tells me that orange is fixed on the left side.So far, I have:- Top face: violet (V)- Left face: orange (O)- Front faces in different views: blue (B), yellow (Y), pink (P)Since the front face changes, these must be different faces adjacent to the top and left faces. So, blue, yellow, and pink are on the front, right, and maybe back faces? Wait, but the front face is changing, so perhaps each time the cube is rotated, a different face becomes the front.Let me try to map this out. Let's consider the cube with top face violet (V) and left face orange (O). The front face in the first view is blue (B). So, blue is on the front face. Then, in the second view, the front face is yellow (Y). So, the cube must have been rotated such that the face that was previously on the right or back is now the front. Similarly, in the third view, the front face is pink (P), so another rotation.Given that, the colors blue, yellow, and pink are on adjacent faces to the top and left. So, blue is front, yellow could be right, and pink could be back? Or maybe another arrangement.Wait, but the left face is always orange, so the right face must be another color. Since the front face changes, the right face could be one of the colors that isn't fixed.Let me list the colors we have:- Top: V- Left: O- Front: B, Y, P in different views- Remaining colors: T (teal) and one more.Wait, the colors are B, O, Y, V, T, P. So, the remaining color not mentioned yet is T (teal). So, teal must be on either the bottom or the back face.But in the three views, the front face changes, so the back face is not visible. The bottom face is also not visible because the top is always violet. So, the bottom face could be teal or another color, but since all other colors are accounted for except teal, perhaps teal is on the bottom.Wait, but let me think again. The front face in the first view is blue, so blue is front. The left face is orange. The top is violet. So, the right face could be yellow or pink, and the back face could be the other one. Then, the bottom face would be teal.Alternatively, the bottom face could be another color, but since teal is the only color not mentioned in the front, left, or top, it must be on the bottom or back.But since the front face changes in each view, the back face is never shown, so it's possible that teal is on the back. But wait, in the third view, the front face is pink, so if pink is front, then the back face would be opposite to it. But if the back face is teal, then the front face would be pink, and the back is teal.But let me try to figure out the opposite faces.We know that the top is violet, so the bottom must be opposite to violet. But the problem is asking for the face opposite to teal. So, if I can figure out where teal is, I can find its opposite.Given that, let's see. The colors we have are:- Top: V- Left: O- Front: B, Y, P in different views- Remaining: TSo, the remaining faces are right, back, and bottom. Since in each view, the front face is different, the right and back faces must be changing as well.Wait, maybe I should think about the cube's net. Let me try to imagine unfolding the cube.Top face: VLeft face: OFront face: B in the first view.So, if I have top as V, front as B, left as O, then the right face would be adjacent to both top and front, so it could be Y or P. Similarly, the back face would be opposite to front, so if front is B, back is something else.But in the second view, front is Y. So, if front is Y, then the back would be opposite to Y, which would be something else.Wait, maybe I'm overcomplicating. Let's try to assign the colors step by step.Top: VLeft: OFront: B, Y, P in different views.So, in the first view, front is B. So, B is front, O is left, V is top. Therefore, the right face must be adjacent to both top and front, so it could be Y or P.In the second view, front is Y. So, Y is front, O is left, V is top. Therefore, the right face must be adjacent to top and front, which could be B or P.In the third view, front is P. So, P is front, O is left, V is top. Therefore, the right face must be adjacent to top and front, which could be B or Y.Wait, so in each view, the front face is different, but the left face is always O, and the top is always V.So, the right face must be changing as well. So, the right face in the first view is either Y or P, in the second view is either B or P, and in the third view is either B or Y.But since the colors are distinct, each face must have a unique color.So, let's try to assign:In the first view: front=B, left=O, top=V. So, the right face must be either Y or P.In the second view: front=Y, left=O, top=V. So, the right face must be either B or P.In the third view: front=P, left=O, top=V. So, the right face must be either B or Y.Since all colors must be unique, let's see:If in the first view, right face is Y, then in the second view, right face can't be Y, so it must be P. But in the third view, front is P, so right face can't be P, so it must be Y. But Y is already used in the second view as front. Wait, no, Y is front in the second view, but right face is different.Wait, maybe it's better to think that the right face in each view is different.So, in the first view, right face is Y or P.In the second view, right face is B or P.In the third view, right face is B or Y.But since each color is unique, let's try to assign:Suppose in the first view, right face is Y. Then, in the second view, front is Y, so right face can't be Y, so it must be P. Then, in the third view, front is P, so right face can't be P, so it must be B.But then, in the first view, right=Y, second view right=P, third view right=B.But then, the back face would be opposite to front. So, in the first view, front=B, so back face is opposite=B, which would be the color opposite to B. Similarly, in the second view, front=Y, so back face is opposite=Y, and in the third view, front=P, so back face is opposite=P.But since the cube has fixed colors, the back face can't be changing. So, this approach might not work.Alternatively, maybe the back face is fixed, and the front face is rotating around.Wait, perhaps the cube is being rotated around the vertical axis (left-right), so the front face changes, but the top and left remain the same.In that case, the right face would be adjacent to top and front, so it would change as the front changes.But then, the back face would be opposite to the front, so it would also change, but since the back face is not visible, it's hard to tell.Wait, maybe I should consider that the cube is being rotated such that the front face changes, but the top and left remain fixed. So, the right face is adjacent to top and front, so it would change as the front changes.But then, the back face is opposite to the front, so it would also change.But since the back face is not visible, we can't see it, but the colors must be consistent.Wait, perhaps the back face is always the same, but the front face is rotating around.Wait, I'm getting confused. Maybe I should try to assign the colors step by step.We have:Top: VLeft: OFront in first view: BFront in second view: YFront in third view: PSo, front face changes, so the cube is being rotated.Therefore, the right face in each view is adjacent to top and front, so it must be a different color each time.So, in the first view, front=B, top=V, left=O, so right face must be either Y or P.In the second view, front=Y, top=V, left=O, so right face must be either B or P.In the third view, front=P, top=V, left=O, so right face must be either B or Y.Since all colors are unique, let's try to assign:First view: right=YSecond view: right=PThird view: right=BBut then, in the first view, right=Y, so back face would be opposite to front=B, so back face=?Wait, no, the back face is opposite to front, so if front=B, back face is opposite=B, which would be a different color.But we don't know which color is opposite to B yet.Wait, maybe I should think about the cube's structure.In a cube, each face has four adjacent faces and one opposite face.So, top is V, left is O, front is B in the first view.So, B is front, adjacent to V (top), O (left), and two other faces: right and bottom.Similarly, in the second view, front=Y, so Y is front, adjacent to V (top), O (left), and two other faces: right and bottom.In the third view, front=P, so P is front, adjacent to V (top), O (left), and two other faces: right and bottom.Wait, so in each view, the front face is different, but the top and left remain the same.Therefore, the right face and the bottom face must be changing as the front face changes.But the colors are fixed, so the right and bottom faces must be assigned colors such that they are consistent across all views.Wait, but in each view, the front face is different, so the right face must be different as well.But since the colors are fixed, the right face must be a different color in each view, but the colors are fixed on the cube.Wait, that doesn't make sense. The cube's colors are fixed, so the right face must be a fixed color, but in different views, it's showing different colors because the cube is rotated.Wait, no, the right face is a fixed face on the cube, but when the cube is rotated, the right face can become front, back, left, or right in different orientations.Wait, maybe I'm overcomplicating.Let me try to think of the cube with top=V, left=O, and front=B in the first view.So, the four adjacent faces to top=V are front=B, right=?, back=?, and left=O.Similarly, the front face=B is adjacent to top=V, left=O, right=?, and bottom=?.In the second view, front=Y, so Y is front, adjacent to top=V, left=O, right=?, and bottom=?.In the third view, front=P, so P is front, adjacent to top=V, left=O, right=?, and bottom=?.So, the right face in each view is different, but the right face is a fixed face on the cube.Wait, no, the right face is a relative term based on the current orientation.So, when the cube is rotated, the right face changes.Therefore, the right face in the first view is a different face than the right face in the second view.Wait, that can't be, because the cube's faces are fixed.I think I need to approach this differently.Let me consider that the cube has fixed colors on each face, and the views are just different orientations of the same cube.So, in each view, the top is V, left is O, and front is B, Y, or P.Therefore, the front face is changing, but the top and left remain the same.So, the front face must be rotating around the vertical axis (left-right), keeping top and left fixed.Therefore, the right face in each view is adjacent to top and front, so it must be a different color each time.But since the cube's colors are fixed, the right face must be a fixed color, but in different views, it's showing different colors because the cube is rotated.Wait, no, the right face is a fixed face on the cube, but when the cube is rotated, the right face can become front, back, left, or right in different orientations.Wait, I'm getting stuck here.Maybe I should try to assign the colors to the cube's faces based on the given information.We have:Top: VLeft: OFront: B, Y, P in different views.So, front face is changing, so the cube is being rotated such that different faces become front.Therefore, the front face is B, Y, or P, and the back face is the opposite of front.So, if front=B, back=?If front=Y, back=?If front=P, back=?But since the cube has fixed colors, the back face must be a fixed color opposite to the front.But since the front is changing, the back must also change, but that's not possible because the cube's colors are fixed.Wait, that suggests that the back face is fixed, and the front face is rotating around.Wait, maybe the back face is fixed, and the front face is rotating around it.But I'm not sure.Alternatively, perhaps the back face is the same as the bottom face.Wait, no, the bottom face is opposite to the top face, which is V.So, bottom face is opposite to V, so it must be a different color.Given that, the bottom face is either B, Y, P, O, or T.But O is on the left, so bottom can't be O.V is on top, so bottom is opposite to V.So, bottom face is one of B, Y, P, T.But in the views, the front face is B, Y, P, so the bottom face must be T.Wait, that makes sense.Because in the three views, the front face is B, Y, P, which are all different, so the bottom face must be the remaining color, which is T.Therefore, bottom face=T.So, the face opposite to T is the top face=V.Therefore, the face opposite to teal is violet.Wait, but let me confirm.If bottom face=T, then the top face=V is opposite to T.But the question is asking for the face opposite to T, which would be V.But let me make sure.Wait, if the bottom face is T, then the top face=V is opposite to T.Yes, that seems correct.But let me double-check.We have:Top=VLeft=OFront=B, Y, P in different views.Therefore, the front face is rotating, so the right face is changing as well.But the bottom face must be the remaining color, which is T.Therefore, bottom=T, so opposite to T is V.Therefore, the answer is violet.But let me make sure I didn't miss anything.Wait, the colors are B, O, Y, V, T, P.Top=VLeft=OFront=B, Y, P in different views.Therefore, the remaining color is T, which must be on the bottom.Therefore, bottom=T, so opposite to T is V.Yes, that seems correct.So, the face opposite to teal is violet.

💡First, I need to understand the relationship between the cost price and the selling price of the articles. The problem states that the cost price of some number of articles is equal to the selling price of 35 articles. This means that if I denote the cost price of one article as ( C ) and the selling price of one article as ( S ), then the equation can be written as:[X times C = 35 times S]Next, the problem mentions that the gain is 42.857142857142854 percent. Gain is calculated based on the cost price, so I can express the selling price in terms of the cost price and the gain percentage:[text{Gain%} = left( frac{S - C}{C} right) times 100]Given that the gain percentage is 42.857142857142854, I can set up the equation:[42.857142857142854 = left( frac{S - C}{C} right) times 100]To simplify the calculation, I'll convert the percentage to a decimal by dividing by 100:[0.42857142857142854 = frac{S - C}{C}]Now, I'll solve for ( S ) in terms of ( C ):[S = 0.42857142857142854 times C + C][S = 1.4285714285714286 times C]With this expression for ( S ), I'll substitute it back into the first equation:[X times C = 35 times (1.4285714285714286 times C)]Finally, I'll divide both sides by ( C ) to solve for ( X ):[X = 35 times 1.4285714285714286][X = 50]Therefore, the number of articles we are talking about is 50.

💡Okay, so I have this triangle ABC, and the sides opposite angles A, B, and C are a, b, and c respectively. The problem gives me this equation: cos B over cos C equals negative b divided by (2a + c). I need to find the measure of angle B first, and then, given that b is the square root of 13 and a + c equals 4, find the value of a.Alright, starting with part (1). I remember that in triangles, the Law of Sines and the Law of Cosines are super useful. The Law of Sines says that a over sin A equals b over sin B equals c over sin C. Maybe I can use that here.Given the equation cos B / cos C = -b / (2a + c). Hmm, so maybe I can express cos B and cos C in terms of the sides using the Law of Cosines. The Law of Cosines says that for any triangle, c² = a² + b² - 2ab cos C, and similarly for the other angles.But wait, before I dive into that, maybe I can express cos B and cos C in terms of the sides. Let me write down the Law of Cosines for angles B and C.For angle B: cos B = (a² + c² - b²) / (2ac)For angle C: cos C = (a² + b² - c²) / (2ab)So, substituting these into the given equation:cos B / cos C = [ (a² + c² - b²) / (2ac) ] / [ (a² + b² - c²) / (2ab) ]Simplify this fraction:= [ (a² + c² - b²) / (2ac) ] * [ 2ab / (a² + b² - c²) ]= [ (a² + c² - b²) * 2ab ] / [ 2ac * (a² + b² - c²) ]= [ (a² + c² - b²) * b ] / [ c * (a² + b² - c²) ]So, according to the problem, this is equal to -b / (2a + c). So:[ (a² + c² - b²) * b ] / [ c * (a² + b² - c²) ] = -b / (2a + c)I can cancel out the b from both sides (assuming b ≠ 0, which it can't be in a triangle):[ (a² + c² - b²) ] / [ c * (a² + b² - c²) ] = -1 / (2a + c)Cross-multiplying:( a² + c² - b² ) * (2a + c) = -c * (a² + b² - c² )Let me expand both sides:Left side: (a² + c² - b²)(2a + c) = 2a(a² + c² - b²) + c(a² + c² - b²)= 2a³ + 2a c² - 2a b² + a² c + c³ - b² cRight side: -c(a² + b² - c²) = -a² c - b² c + c³So, set left side equal to right side:2a³ + 2a c² - 2a b² + a² c + c³ - b² c = -a² c - b² c + c³Let me bring all terms to the left side:2a³ + 2a c² - 2a b² + a² c + c³ - b² c + a² c + b² c - c³ = 0Simplify term by term:2a³ remains.2a c² remains.-2a b² remains.a² c + a² c = 2a² cc³ - c³ = 0-b² c + b² c = 0So, altogether:2a³ + 2a c² - 2a b² + 2a² c = 0Factor out a 2a:2a(a² + c² - b² + a c) = 0Since a is a side length, it can't be zero, so:a² + c² - b² + a c = 0Hmm, that's interesting. So:a² + c² + a c = b²Wait, that's a relation between the sides. Maybe I can use the Law of Cosines here. From the Law of Cosines, b² = a² + c² - 2ac cos B.But from above, we have b² = a² + c² + a c.So, equate these two expressions for b²:a² + c² + a c = a² + c² - 2ac cos BSubtract a² + c² from both sides:a c = -2 a c cos BDivide both sides by a c (assuming a and c are not zero, which they can't be in a triangle):1 = -2 cos BSo, cos B = -1/2What angle has cosine of -1/2? Well, in the range of 0 to π radians (since it's a triangle angle), cos B = -1/2 implies that B is 2π/3 radians, which is 120 degrees.So, angle B is 120 degrees or 2π/3 radians.Alright, that seems solid. Let me recap:1. Started with the given equation, expressed cos B and cos C using Law of Cosines.2. Substituted into the equation, simplified, and ended up with a relation between sides a, b, c.3. Compared with Law of Cosines to find cos B = -1/2, leading to B = 120 degrees.Good, that seems correct.Now, moving on to part (2). Given that b = sqrt(13) and a + c = 4, find a.From part (1), we know that angle B is 120 degrees, so we can use the Law of Cosines again.Law of Cosines: b² = a² + c² - 2ac cos BWe know b = sqrt(13), so b² = 13.Angle B is 120 degrees, so cos B = cos 120° = -1/2.So, plug into the formula:13 = a² + c² - 2ac (-1/2)13 = a² + c² + acAlso, we know that a + c = 4. Let me write that as equation (1): a + c = 4We need another equation to solve for a and c. From above, we have equation (2): a² + c² + a c = 13So, let's see. From equation (1), c = 4 - a. Let's substitute c into equation (2):a² + (4 - a)² + a(4 - a) = 13Let me compute each term:a² is just a².(4 - a)² = 16 - 8a + a²a(4 - a) = 4a - a²So, putting it all together:a² + (16 - 8a + a²) + (4a - a²) = 13Simplify term by term:a² + 16 - 8a + a² + 4a - a²Combine like terms:a² + a² - a² = a²-8a + 4a = -4aConstant term: 16So, overall:a² - 4a + 16 = 13Subtract 13 from both sides:a² - 4a + 3 = 0Now, solve this quadratic equation.Quadratic equation: a² - 4a + 3 = 0We can factor this:(a - 1)(a - 3) = 0So, a = 1 or a = 3.Therefore, the possible values for a are 1 and 3.But let's check if both solutions make sense in the context of the triangle.If a = 1, then c = 4 - 1 = 3If a = 3, then c = 4 - 3 = 1So, both solutions are valid because sides can be swapped, and the triangle remains the same with sides 1, 3, and sqrt(13). Let me verify using the Law of Cosines.Case 1: a = 1, c = 3Compute b²: 1² + 3² + 1*3 = 1 + 9 + 3 = 13, which matches.Case 2: a = 3, c = 1Compute b²: 3² + 1² + 3*1 = 9 + 1 + 3 = 13, which also matches.So, both are valid. Therefore, a can be either 1 or 3.Wait, but the problem says "find the value of a". So, both are possible? Or is there a constraint I'm missing?Looking back, the problem didn't specify any other constraints, so both solutions are acceptable. So, a can be 1 or 3.Wait, but in the context of triangles, sometimes side lengths have to satisfy triangle inequalities. Let's check.If a = 1, c = 3, b = sqrt(13) ≈ 3.605Check triangle inequalities:1 + 3 > sqrt(13): 4 > 3.605, which is true.1 + sqrt(13) > 3: 1 + 3.605 > 3, which is true.3 + sqrt(13) > 1: 3 + 3.605 > 1, which is true.Similarly, for a = 3, c = 1, same thing.So, both are valid.Therefore, the value of a is either 1 or 3.Wait, but in the problem statement, it's just asking for "the value of a". So, perhaps both are acceptable, but maybe in the context, a is the side opposite angle A, which is not specified. So, unless there's more information, both are possible.So, I think both 1 and 3 are correct.So, summarizing:(1) Angle B is 120 degrees or 2π/3 radians.(2) The value of a is either 1 or 3.**Final Answer**(1) The measure of angle ( B ) is (boxed{dfrac{2pi}{3}}).(2) The value of ( a ) is (boxed{1}) or (boxed{3}).

💡Okay, so I need to figure out the interval where the function ( f(x) = sin x + cos x ) is monotonically increasing on the interval ([0, pi]). Hmm, I remember that to find where a function is increasing or decreasing, I should look at its derivative. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing.Alright, let's start by finding the derivative of ( f(x) ). The derivative of ( sin x ) is ( cos x ), and the derivative of ( cos x ) is ( -sin x ). So, putting that together:[f'(x) = cos x - sin x]Okay, so ( f'(x) = cos x - sin x ). Now, I need to find where this derivative is positive because that's where the function is increasing. So, I need to solve the inequality:[cos x - sin x > 0]Hmm, how do I solve this inequality? Maybe I can rewrite it in a different form. I remember that ( cos x - sin x ) can be expressed as a single sine or cosine function using a phase shift. Let me think about that.I recall the identity:[a sin x + b cos x = sqrt{a^2 + b^2} sin(x + phi)]where ( phi ) is some phase shift. In this case, though, I have ( cos x - sin x ), which is similar to ( a cos x + b sin x ) but with a negative sign. Maybe I can factor out a negative sign or adjust the identity accordingly.Let me try expressing ( cos x - sin x ) as a single sine function. Let's see:[cos x - sin x = sqrt{2} left( frac{1}{sqrt{2}} cos x - frac{1}{sqrt{2}} sin x right)]I know that ( frac{1}{sqrt{2}} = sin frac{pi}{4} = cos frac{pi}{4} ), so this can be rewritten using the sine of a difference:[sqrt{2} left( sin frac{pi}{4} cos x - cos frac{pi}{4} sin x right) = sqrt{2} sin left( frac{pi}{4} - x right)]Wait, is that right? Let me double-check. The sine of a difference is:[sin(A - B) = sin A cos B - cos A sin B]So, if I set ( A = frac{pi}{4} ) and ( B = x ), then:[sin left( frac{pi}{4} - x right) = sin frac{pi}{4} cos x - cos frac{pi}{4} sin x]Yes, that's correct. So, ( cos x - sin x = sqrt{2} sin left( frac{pi}{4} - x right) ). Therefore, the derivative ( f'(x) ) can be written as:[f'(x) = sqrt{2} sin left( frac{pi}{4} - x right)]Now, I need to find where ( f'(x) > 0 ), which is equivalent to:[sqrt{2} sin left( frac{pi}{4} - x right) > 0]Since ( sqrt{2} ) is always positive, this simplifies to:[sin left( frac{pi}{4} - x right) > 0]Okay, so when is ( sin theta > 0 )? It's when ( theta ) is in the first or second quadrants, meaning ( 0 < theta < pi ). So, applying that to our inequality:[0 < frac{pi}{4} - x < pi]Let me solve this inequality for ( x ). First, subtract ( frac{pi}{4} ) from all parts:[- frac{pi}{4} < -x < frac{3pi}{4}]Now, multiply all parts by ( -1 ), remembering to reverse the inequality signs:[frac{pi}{4} > x > -frac{3pi}{4}]Which can be rewritten as:[- frac{3pi}{4} < x < frac{pi}{4}]But wait, our original interval is ( [0, pi] ). So, we need to find the overlap between ( - frac{3pi}{4} < x < frac{pi}{4} ) and ( [0, pi] ). The overlap is ( 0 leq x < frac{pi}{4} ).Therefore, ( f'(x) > 0 ) when ( x ) is in ( [0, frac{pi}{4}) ), which means the function ( f(x) ) is increasing on this interval.Let me just verify this result. If I plug in ( x = 0 ), ( f'(0) = cos 0 - sin 0 = 1 - 0 = 1 > 0 ). At ( x = frac{pi}{4} ), ( f'(frac{pi}{4}) = cos frac{pi}{4} - sin frac{pi}{4} = frac{sqrt{2}}{2} - frac{sqrt{2}}{2} = 0 ). So, the derivative is positive up to ( x = frac{pi}{4} ), and then it becomes zero. Beyond that, let's check ( x = frac{pi}{2} ): ( f'(frac{pi}{2}) = cos frac{pi}{2} - sin frac{pi}{2} = 0 - 1 = -1 < 0 ). So, indeed, the function is increasing up to ( frac{pi}{4} ) and then starts decreasing.Therefore, the interval where ( f(x) ) is monotonically increasing on ( [0, pi] ) is ( [0, frac{pi}{4}] ).**Final Answer**boxed{C}

💡Alright, so I have this problem here about a point M(x, y) on the coordinate plane and two fixed points, M₁(26, 1) and M₂(2, 1). The ratio of the distances from M to M₁ and M₂ is given as 5. I need to find the equation of the trajectory of point M and describe its shape. Then, in part II, I have to find the equation of a line passing through a specific point that intersects this trajectory and forms a segment of length 8.Starting with part I. The ratio of distances is 5, so |MM₁| / |MM₂| = 5. That means the distance from M to M₁ is five times the distance from M to M₂. I remember that when the ratio of distances from two fixed points is constant, the locus is a circle called the Apollonius circle. So, I think that's what this is.To find the equation, I can set up the distance formula. Let me write that down:√[(x - 26)² + (y - 1)²] / √[(x - 2)² + (y - 1)²] = 5To simplify this, I can square both sides to eliminate the square roots:[(x - 26)² + (y - 1)²] / [(x - 2)² + (y - 1)²] = 25Then, cross-multiplying:(x - 26)² + (y - 1)² = 25[(x - 2)² + (y - 1)²]Expanding both sides:Left side: (x² - 52x + 676) + (y² - 2y + 1) = x² - 52x + 676 + y² - 2y + 1Right side: 25[(x² - 4x + 4) + (y² - 2y + 1)] = 25x² - 100x + 100 + 25y² - 50y + 25So, putting it all together:x² - 52x + 676 + y² - 2y + 1 = 25x² - 100x + 100 + 25y² - 50y + 25Now, let's bring all terms to the left side:x² - 52x + 676 + y² - 2y + 1 - 25x² + 100x - 100 - 25y² + 50y - 25 = 0Combine like terms:(1 - 25)x² + (-52x + 100x) + (676 - 100 - 25) + (1 - 25)y² + (-2y + 50y) = 0Calculating each:-24x² + 48x + 551 + (-24y²) + 48y = 0Hmm, that seems a bit messy. Maybe I made a mistake in expanding or combining terms. Let me double-check.Wait, perhaps instead of expanding everything, I can rearrange the equation before expanding. Let me try that.Starting again:(x - 26)² + (y - 1)² = 25[(x - 2)² + (y - 1)²]Let me subtract 25[(x - 2)² + (y - 1)²] from both sides:(x - 26)² + (y - 1)² - 25(x - 2)² - 25(y - 1)² = 0Factor out the terms:[(x - 26)² - 25(x - 2)²] + [(y - 1)² - 25(y - 1)²] = 0Wait, that's not helpful. Maybe I should factor differently.Alternatively, let me denote A = (x - 26)² + (y - 1)² and B = (x - 2)² + (y - 1)². Then, A = 25B.So, A - 25B = 0.Expanding A and B:A = x² - 52x + 676 + y² - 2y + 1B = x² - 4x + 4 + y² - 2y + 1So, A - 25B = (x² - 52x + 676 + y² - 2y + 1) - 25(x² - 4x + 4 + y² - 2y + 1) = 0Expanding the 25B:= x² - 52x + 676 + y² - 2y + 1 - 25x² + 100x - 100 - 25y² + 50y - 25 = 0Combine like terms:(1 - 25)x² + (-52x + 100x) + (676 - 100 - 25) + (1 - 25)y² + (-2y + 50y) = 0Calculating each:-24x² + 48x + 551 -24y² + 48y = 0Hmm, same result. Maybe I can factor out -24:-24(x² - 2x) -24(y² - 2y) + 551 = 0Wait, let's see:-24x² + 48x -24y² + 48y + 551 = 0Divide both sides by -24:x² - 2x + y² - 2y - 551/24 = 0Hmm, that fraction is messy. Maybe I should complete the square instead.Looking back at the equation:-24x² + 48x -24y² + 48y + 551 = 0Let me factor out -24 from the x terms and y terms:-24(x² - 2x) -24(y² - 2y) + 551 = 0Now, complete the square inside the parentheses:x² - 2x = (x - 1)² - 1Similarly, y² - 2y = (y - 1)² - 1So, substituting back:-24[(x - 1)² - 1] -24[(y - 1)² - 1] + 551 = 0Expanding:-24(x - 1)² + 24 -24(y - 1)² + 24 + 551 = 0Combine constants:24 + 24 + 551 = 600 - 24 = 576? Wait, 24 + 24 is 48, plus 551 is 599.Wait, 24 + 24 = 48, 48 + 551 = 599.So:-24(x - 1)² -24(y - 1)² + 599 = 0Move constants to the other side:-24(x - 1)² -24(y - 1)² = -599Divide both sides by -24:(x - 1)² + (y - 1)² = 599/24Wait, that's approximately 24.958, which is close to 25. Hmm, maybe I made a calculation mistake earlier.Wait, let's go back to the step where I had:-24x² + 48x -24y² + 48y + 551 = 0Let me factor out -24:-24(x² - 2x + y² - 2y) + 551 = 0Then, complete the square:x² - 2x = (x - 1)^2 -1y² - 2y = (y - 1)^2 -1So, substituting:-24[(x - 1)^2 -1 + (y - 1)^2 -1] + 551 = 0Which is:-24(x - 1)^2 +24 -24(y - 1)^2 +24 +551 = 0Combine constants: 24 +24 +551 = 599So:-24(x - 1)^2 -24(y - 1)^2 +599 = 0Then, moving constants:-24(x - 1)^2 -24(y - 1)^2 = -599Divide both sides by -24:(x - 1)^2 + (y - 1)^2 = 599/24But 599 divided by 24 is approximately 24.958, which is almost 25. Maybe I made a mistake in earlier steps.Wait, let's check the initial equation after squaring:[(x - 26)^2 + (y - 1)^2] = 25[(x - 2)^2 + (y - 1)^2]Let me compute both sides numerically for a point that should lie on the circle, say, the center (1,1).Left side: (1-26)^2 + (1-1)^2 = (-25)^2 + 0 = 625Right side: 25[(1-2)^2 + (1-1)^2] =25[(-1)^2 +0] =25*1=25But 625 ≠25, so (1,1) is not on the circle. Wait, that can't be. Maybe I messed up the algebra.Wait, actually, if the ratio is 5, the center should be somewhere else. Maybe I need to use the formula for Apollonius circle.The general formula for Apollonius circle when the ratio is k is:(x - ( (k²x₂ - x₁)/(k² -1) ))² + (y - ( (k²y₂ - y₁)/(k² -1) ))² = (k/(k² -1))² * (distance between M₁ and M₂)^2Wait, let me recall the formula correctly.Given two points M₁(x₁,y₁) and M₂(x₂,y₂), the locus of points M such that |MM₁| / |MM₂| = k is a circle with center at ((k²x₂ - x₁)/(k² -1), (k²y₂ - y₁)/(k² -1)) and radius k/(k² -1) times the distance between M₁ and M₂ divided by 2.Wait, let me compute the distance between M₁ and M₂ first.M₁ is (26,1), M₂ is (2,1). So, distance is sqrt[(26-2)^2 + (1-1)^2] = sqrt[24² +0] =24.So, distance between M₁ and M₂ is 24.Given k=5, so the center is at ((5²*2 -26)/(5² -1), (5²*1 -1)/(5² -1)).Compute numerator for x-coordinate: 25*2 -26=50-26=24Denominator:25-1=24So, x-coordinate:24/24=1Similarly, y-coordinate: (25*1 -1)/24=(25-1)/24=24/24=1So, center is (1,1). Radius is k/(k² -1)*distance/2=5/(25-1)*24/2=5/24*12=5/24*12=5/2*1=2.5? Wait, no.Wait, radius formula: radius = (k / |k² -1|) * (distance between M₁ and M₂)/2So, distance between M₁ and M₂ is 24, so half of that is 12.So, radius=5/(25-1)*12=5/24*12=5/2=2.5Wait, but 5/24*12= (5*12)/24=60/24=2.5=5/2.But earlier, when I tried to compute the equation, I ended up with (x-1)^2 + (y-1)^2=599/24≈24.958, which is approximately 25, which is (5)^2. So, radius is 5, but according to the formula, it should be 5/2=2.5.Wait, that's conflicting. So, which one is correct?Wait, let's think again. Maybe I messed up the formula.Wait, the formula for the radius is |k/(k² -1)| * distance between M₁ and M₂ / 2.So, with k=5, distance=24.So, radius=5/(25-1)*24/2=5/24*12=5/2=2.5.But when I computed the equation, I got radius squared as 599/24≈24.958, which is about 25, so radius≈5.So, which is correct?Wait, let's plug in the center (1,1) into the equation I derived:(x-1)^2 + (y-1)^2=599/24≈24.958So, radius≈5, but according to the formula, radius should be 2.5.This inconsistency suggests I made a mistake in the algebra.Wait, let's go back to the equation after squaring:(x - 26)^2 + (y - 1)^2 =25[(x - 2)^2 + (y - 1)^2]Let me compute both sides for the center (1,1):Left side: (1-26)^2 + (1-1)^2= (-25)^2 +0=625Right side:25[(1-2)^2 + (1-1)^2]=25[(-1)^2 +0]=25*1=25But 625≠25, so (1,1) is not on the circle. But according to the formula, it should be the center.Wait, maybe I confused the formula. Let me check the formula again.The Apollonius circle for |MM₁| / |MM₂| =k has center at ((k²x₂ -x₁)/(k² -1), (k²y₂ - y₁)/(k² -1)) and radius k/(k² -1)*distance(M₁,M₂)/2.So, with k=5, M₁(26,1), M₂(2,1), distance=24.So, center x=(25*2 -26)/(25-1)=(50-26)/24=24/24=1Center y=(25*1 -1)/24=24/24=1So, center is (1,1). Radius=5/(25-1)*24/2=5/24*12=5/2=2.5.So, the equation should be (x-1)^2 + (y-1)^2=(2.5)^2=6.25.But when I derived it earlier, I got (x-1)^2 + (y-1)^2≈25, which is conflicting.Wait, perhaps I made a mistake in the algebra when expanding.Let me try again.Starting from:√[(x - 26)^2 + (y - 1)^2] / √[(x - 2)^2 + (y - 1)^2] =5Square both sides:[(x - 26)^2 + (y - 1)^2] =25[(x - 2)^2 + (y - 1)^2]Expand both sides:Left: (x² -52x +676) + (y² -2y +1)=x² -52x +676 + y² -2y +1Right:25[(x² -4x +4) + (y² -2y +1)]=25x² -100x +100 +25y² -50y +25Now, bring all terms to left:x² -52x +676 + y² -2y +1 -25x² +100x -100 -25y² +50y -25=0Combine like terms:(1-25)x² + (-52x +100x) + (676 -100 -25) + (1-25)y² + (-2y +50y)=0Calculating each:-24x² +48x +551 -24y² +48y=0Divide both sides by -24:x² -2x - y² +2y -551/24=0Wait, that's not helpful. Maybe I should complete the square.Starting from:-24x² +48x -24y² +48y +551=0Factor out -24:-24(x² -2x) -24(y² -2y) +551=0Complete the square:x² -2x=(x-1)^2 -1y² -2y=(y-1)^2 -1Substitute back:-24[(x-1)^2 -1] -24[(y-1)^2 -1] +551=0Expand:-24(x-1)^2 +24 -24(y-1)^2 +24 +551=0Combine constants:24+24=48; 48+551=599So:-24(x-1)^2 -24(y-1)^2 +599=0Move constants:-24(x-1)^2 -24(y-1)^2= -599Divide both sides by -24:(x-1)^2 + (y-1)^2=599/24≈24.958≈25So, radius squared≈25, so radius≈5.But according to the formula, radius should be 2.5. So, which is correct?Wait, maybe the formula is different. Let me check the formula again.Wait, the formula for the Apollonius circle is:If |MM₁| / |MM₂| =k, then the center is at ((k²x₂ -x₁)/(k² -1), (k²y₂ - y₁)/(k² -1)) and radius is (k/(k² -1))*distance(M₁,M₂)/2.So, with k=5, distance=24.Radius=5/(25-1)*24/2=5/24*12=5/2=2.5.But when I derived it, I got radius≈5. So, something is wrong.Wait, maybe I messed up the formula. Let me derive it from scratch.Given |MM₁| / |MM₂|=k.So, |MM₁|=k|MM₂|Square both sides:|MM₁|² =k²|MM₂|²Which is:(x -26)^2 + (y -1)^2 =k²[(x -2)^2 + (y -1)^2]Expand:x² -52x +676 + y² -2y +1 =k²x² -4k²x +4k² +k²y² -2k²y +k²Bring all terms to left:x² -52x +676 + y² -2y +1 -k²x² +4k²x -4k² -k²y² +2k²y -k²=0Factor:(1 -k²)x² + (-52 +4k²)x + (676 -4k²) + (1 -k²)y² + (-2 +2k²)y + (1 -k²)=0Wait, that seems complicated. Maybe I should rearrange differently.Wait, let me collect like terms:x²(1 -k²) + y²(1 -k²) +x(-52 +4k²) + y(-2 +2k²) + (676 +1 -4k² -k²)=0Simplify constants:676 +1=677; -4k² -k²=-5k²So:(1 -k²)(x² + y²) + (-52 +4k²)x + (-2 +2k²)y + (677 -5k²)=0Now, divide both sides by (1 -k²):x² + y² + [(-52 +4k²)/(1 -k²)]x + [(-2 +2k²)/(1 -k²)]y + (677 -5k²)/(1 -k²)=0This is the general equation of a circle.Now, let's plug in k=5:First, compute 1 -k²=1 -25=-24So, coefficients:x term: (-52 +4*25)/(-24)=(-52 +100)/(-24)=48/(-24)=-2y term: (-2 +2*25)/(-24)=(-2 +50)/(-24)=48/(-24)=-2Constant term: (677 -5*25)/(-24)=(677 -125)/(-24)=552/(-24)=-23So, equation becomes:x² + y² -2x -2y -23=0Which can be rewritten as:x² -2x + y² -2y =23Complete the square:(x -1)^2 -1 + (y -1)^2 -1=23So:(x -1)^2 + (y -1)^2 -2=23Thus:(x -1)^2 + (y -1)^2=25So, radius is 5, center at (1,1). So, the equation is (x-1)^2 + (y-1)^2=25.Wait, so earlier when I thought the radius was 2.5, that was wrong. The correct radius is 5. So, the formula I used earlier was incorrect. The correct radius is indeed 5.So, the equation is (x-1)^2 + (y-1)^2=25, which is a circle with center (1,1) and radius 5.Okay, so part I is solved.Now, part II. The trajectory C is this circle. A line l passes through point M(-2,3) and intersects C to form a segment of length 8. Need to find the equation of line l.So, line passes through (-2,3) and intersects the circle (x-1)^2 + (y-1)^2=25 such that the chord length is 8.First, let's recall that the length of a chord in a circle is related to the distance from the center to the line. The formula is:Length of chord=2√(r² -d²)Where r is radius, d is distance from center to the line.Given length=8, r=5.So,8=2√(25 -d²)Divide both sides by 2:4=√(25 -d²)Square both sides:16=25 -d²Thus,d²=25 -16=9So,d=3So, the distance from the center (1,1) to the line l must be 3.Now, the line passes through (-2,3). Let's denote the equation of line l as y=mx +c. But since it passes through (-2,3), we can write it as y -3 =m(x +2), so y=mx +2m +3.Alternatively, in standard form: mx - y + (2m +3)=0.The distance from center (1,1) to this line is |m*1 -1 +2m +3| / sqrt(m² +1)= |3m +2| / sqrt(m² +1)We know this distance must be 3.So,|3m +2| / sqrt(m² +1)=3Square both sides:(3m +2)² / (m² +1)=9Multiply both sides by (m² +1):(9m² +12m +4)=9(m² +1)Expand:9m² +12m +4=9m² +9Subtract 9m² from both sides:12m +4=9Thus,12m=5So,m=5/12Thus, the slope is 5/12. So, the equation is y -3=(5/12)(x +2)Multiply both sides by 12:12y -36=5x +10Bring all terms to left:5x -12y +46=0So, that's one possible line.But wait, when we squared both sides, we might have introduced an extraneous solution. Let's check.Compute |3m +2|=3*sqrt(m² +1)With m=5/12,Left side: |3*(5/12)+2|=|15/12 +24/12|=|39/12|=13/4=3.25Right side:3*sqrt((25/144)+1)=3*sqrt(169/144)=3*(13/12)=39/12=3.25So, it works.But wait, when we squared, we might have another solution where 3m +2 is negative.So, let's consider |3m +2|=3*sqrt(m² +1)Which implies 3m +2=±3*sqrt(m² +1)We considered the positive case, giving m=5/12.What about the negative case:3m +2= -3*sqrt(m² +1)Square both sides:9m² +12m +4=9(m² +1)Which simplifies to:9m² +12m +4=9m² +9Subtract 9m²:12m +4=912m=5m=5/12But plugging back into 3m +2= -3*sqrt(m² +1):3*(5/12)+2=15/12 +24/12=39/12=3.25-3*sqrt((25/144)+1)= -3*(13/12)= -39/12= -3.25So, 3.25≠-3.25, so no solution here.Thus, only one solution: m=5/12.But wait, another possibility is that the line is vertical, i.e., undefined slope. Let's check that.If the line is vertical, x=-2.Check if this line intersects the circle.Plug x=-2 into the circle equation:(-2 -1)^2 + (y -1)^2=259 + (y -1)^2=25(y -1)^2=16So, y -1=±4, so y=5 or y=-3.Thus, the points are (-2,5) and (-2,-3). The distance between these points is |5 - (-3)|=8, which is the required length.So, another line is x=-2.Thus, there are two lines: x=-2 and 5x -12y +46=0.So, the equations are x=-2 and 5x -12y +46=0.**Final Answer**The equations of line ( l ) are (boxed{x = -2}) and (boxed{5x - 12y + 46 = 0}).

💡Okay, so I have this problem about China's GDP from 2012 to 2018, and I need to find a linear regression equation to predict when China's GDP will surpass the United States' GDP in 2018, which is 20.5 trillion US dollars. Hmm, let's break this down step by step.First, I need to understand what a linear regression equation is. From what I remember, it's a way to model the relationship between two variables, in this case, the year (x) and GDP (y). The equation will look something like y = bx + a, where b is the slope and a is the y-intercept. This equation will help me predict future GDP values based on the year.The problem provides me with some reference data: the sum of y_i from 2012 to 2018 is 76.3, and the sum of y_i times x_i is 326.2. It also gives me the formulas for calculating the slope (b) and the intercept (a). Let me write those down:b = [sum(y_i * x_i) - n * mean(y) * mean(x)] / [sum(x_i^2) - n * (mean(x))^2]a = mean(y) - b * mean(x)Okay, so I need to calculate the means of x and y first. Let me list out the x values and y values:x: 1, 2, 3, 4, 5, 6, 7y: 8.5, 9.6, 10.4, 11, 11.1, 12.1, 13.6There are 7 data points, so n = 7.Calculating the mean of x:mean(x) = (1 + 2 + 3 + 4 + 5 + 6 + 7) / 7Let me add those up: 1+2=3, 3+3=6, 6+4=10, 10+5=15, 15+6=21, 21+7=28So, mean(x) = 28 / 7 = 4Calculating the mean of y:mean(y) = sum(y_i) / n = 76.3 / 7Let me do that division: 76.3 divided by 7. 7*10=70, so 76.3 - 70 = 6.3, which is 7*0.9, so 10 + 0.9 = 10.9So, mean(y) = 10.9Alright, now I need to calculate the numerator and denominator for the slope b.The numerator is sum(y_i * x_i) - n * mean(y) * mean(x)We have sum(y_i * x_i) = 326.2n = 7, mean(y) = 10.9, mean(x) = 4So, n * mean(y) * mean(x) = 7 * 10.9 * 4Let me compute that: 7 * 10.9 = 76.3, then 76.3 * 4 = 305.2So, numerator = 326.2 - 305.2 = 21Now, the denominator is sum(x_i^2) - n * (mean(x))^2I need to calculate sum(x_i^2). Let's list the x_i squared:x_i: 1, 2, 3, 4, 5, 6, 7x_i^2: 1, 4, 9, 16, 25, 36, 49Adding those up: 1 + 4 = 5, 5 + 9 = 14, 14 + 16 = 30, 30 + 25 = 55, 55 + 36 = 91, 91 + 49 = 140So, sum(x_i^2) = 140Now, n * (mean(x))^2 = 7 * (4)^2 = 7 * 16 = 112So, denominator = 140 - 112 = 28Therefore, the slope b = numerator / denominator = 21 / 28 = 0.75Okay, now that I have b, I can find a using the formula a = mean(y) - b * mean(x)mean(y) = 10.9, b = 0.75, mean(x) = 4So, a = 10.9 - 0.75 * 40.75 * 4 = 3So, a = 10.9 - 3 = 7.9Therefore, the linear regression equation is y = 0.75x + 7.9Let me double-check my calculations to make sure I didn't make any mistakes.First, mean(x) was 4, that's correct because the sum of x from 1 to 7 is 28, divided by 7 is 4.Mean(y) was 76.3 / 7 = 10.9, that's correct.Sum(y_i * x_i) was given as 326.2, and n * mean(y) * mean(x) was 7 * 10.9 * 4 = 305.2, so numerator was 21.Sum(x_i^2) was 140, and n * (mean(x))^2 was 112, so denominator was 28.21 / 28 is indeed 0.75.Then, a = 10.9 - 0.75 * 4 = 10.9 - 3 = 7.9.So, the regression equation is y = 0.75x + 7.9.Now, part 2 asks me to determine in which year China could surpass the GDP of the United States in 2018, which was 20.5 trillion US dollars.So, I need to find the value of x such that y = 20.5.Using the regression equation: 20.5 = 0.75x + 7.9Let me solve for x.Subtract 7.9 from both sides: 20.5 - 7.9 = 0.75x20.5 - 7.9 is 12.6, so 12.6 = 0.75xNow, divide both sides by 0.75: x = 12.6 / 0.75Calculating that: 12.6 divided by 0.75.Well, 0.75 goes into 12.6 how many times?0.75 * 16 = 12, so 16 * 0.75 = 1212.6 - 12 = 0.60.6 / 0.75 = 0.8So, total x = 16 + 0.8 = 16.8So, x = 16.8But x represents the year code, where x=1 is 2012, x=2 is 2013, and so on.So, x=16.8 would be the year 2012 + 16.8 - 1 = 2027.8Wait, let me think about that.If x=1 is 2012, then x=16 would be 2012 + 15 = 2027, and x=17 would be 2028.But since x=16.8 is almost 17, so it would be in the year 2028.But let me verify.Wait, if x=1 is 2012, then x=16 is 2012 + 15 = 2027, and x=17 is 2028.But 16.8 is 0.8 of the way from x=16 to x=17, which would correspond to 0.8 of a year into 2028, so effectively, in 2028, China's GDP would surpass the US GDP.But let me check my calculation for x.I had y = 0.75x + 7.9Set y = 20.520.5 = 0.75x + 7.9Subtract 7.9: 12.6 = 0.75xDivide by 0.75: x = 12.6 / 0.75 = 16.8Yes, that's correct.So, x=16.8 corresponds to the year 2012 + 16.8 -1 = 2027.8Wait, why did I subtract 1? Because x=1 is 2012, so x=16 would be 2012 + 15 = 2027, and x=17 is 2028.So, x=16.8 is 0.8 of the way from 2027 to 2028, so it's 2027.8, which is approximately 2028.Therefore, China's GDP would surpass the US GDP in 2028.But let me think about this again. Is this a realistic projection?Given that China's GDP is increasing linearly at a rate of 0.75 trillion per year, starting from 7.9 in year 1 (2012). So, in 2012, it's 8.5, which is close to the regression line's prediction of 0.75*1 + 7.9 = 8.65, which is pretty close.In 2018, which is x=7, the regression line predicts y = 0.75*7 + 7.9 = 5.25 + 7.9 = 13.15, which is close to the actual value of 13.6.So, the regression line seems to be a good fit.Therefore, projecting forward, in x=16.8, which is 2028, China's GDP would reach 20.5 trillion, surpassing the US GDP of 20.5 trillion in 2018.But wait, the US GDP in 2018 is 20.5, but the US GDP is not static; it grows as well. However, the problem only asks to use the regression equation to determine when China's GDP will surpass the 2018 US GDP, not considering future growth of the US GDP.So, assuming the US GDP remains at 20.5 trillion, China would surpass it in 2028.But in reality, the US GDP would also grow, so the actual year might be later. But since the problem specifies to use the regression equation and the 2018 US GDP, I think 2028 is the answer they're looking for.Let me just recap:1. Calculated mean(x) = 4, mean(y) = 10.92. Calculated slope b = 0.753. Calculated intercept a = 7.94. Regression equation: y = 0.75x + 7.95. Solved for x when y = 20.5, got x = 16.86. Converted x to year: 2012 + 16.8 -1 = 2027.8, which is approximately 2028So, the final answers are:1. The linear regression equation is y = 0.75x + 7.92. China will surpass the US GDP in 2028