💡Alright, so I've got this algebra problem here, and I'm supposed to find 2A - B given A and B, and then figure out what value of b makes 2A - B independent of a. Let me try to work through this step by step.First, let me write down what A and B are:A = 2a² - 5ab + 3bB = 4a² - 6ab - 8aOkay, so part (1) is to find 2A - B. That means I need to multiply A by 2 and then subtract B from it. Let me do that.So, 2A would be:2 * (2a² - 5ab + 3b) = 4a² - 10ab + 6bNow, subtract B from this. So, 2A - B is:(4a² - 10ab + 6b) - (4a² - 6ab - 8a)Hmm, I need to distribute the negative sign to each term in B. Let me do that carefully.4a² - 10ab + 6b - 4a² + 6ab + 8aNow, let's combine like terms. First, the a² terms: 4a² - 4a² = 0. Okay, those cancel out.Next, the ab terms: -10ab + 6ab = -4abThen, the a terms: There's only one, which is +8a.And finally, the b terms: +6bSo, putting it all together, 2A - B simplifies to:-4ab + 6b + 8aAlright, that's part (1) done. Now, part (2) says that if the value of 2A - B is independent of a, find the value of b. Hmm, okay. So, if it's independent of a, that means the expression shouldn't change as a changes. In other words, the coefficients of a should be zero because if the coefficient is zero, changing a won't affect the value.Looking at the expression we got: -4ab + 6b + 8aLet me rearrange it to group the terms with a together:(-4ab + 8a) + 6bFactor out an a from the first two terms:a*(-4b + 8) + 6bSo, the expression is a*(-4b + 8) + 6bFor this to be independent of a, the coefficient of a must be zero. That is, the term multiplying a should be zero.So, set the coefficient equal to zero:-4b + 8 = 0Now, solve for b:-4b + 8 = 0Subtract 8 from both sides:-4b = -8Divide both sides by -4:b = (-8)/(-4) = 2So, b must be 2.Let me double-check that. If b is 2, then plugging back into 2A - B:-4ab + 6b + 8a= -4a*2 + 6*2 + 8a= -8a + 12 + 8a= (-8a + 8a) + 12= 0 + 12 = 12Yes, that's a constant, independent of a. So, that checks out.So, the value of b is 2.**Final Answer**The value of ( b ) is boxed{2}.

💡Okay, so I have this geometry problem here. It says: Given points A and B, determine the locus of points C such that C, the midpoints of segments AC and BC, and the centroid of triangle ABC lie on the same circle. Hmm, that sounds a bit complex, but let me try to break it down step by step.First, let me recall some basic definitions. The midpoint of a segment is just the point that divides the segment into two equal parts. The centroid of a triangle is the point where the three medians intersect, and it's also the center of mass of the triangle. Importantly, the centroid divides each median into a ratio of 2:1, with the longer part being closer to the vertex.So, in this problem, we're dealing with triangle ABC. Let me denote the midpoints of AC and BC as M and N, respectively. So, M is the midpoint of AC, and N is the midpoint of BC. The centroid, let's call it G, is the intersection point of the medians AM and BN.Now, the problem states that the points C, M, N, and G all lie on the same circle. That means these four points are concyclic or lie on a single circle. So, I need to find all such points C that satisfy this condition.Let me visualize this. Points A and B are fixed, so I can imagine them on a plane. Point C is variable, and depending on where C is, the midpoints M and N, and the centroid G will change. The goal is to find all positions of C such that C, M, N, and G lie on a circle.I think a good starting point is to use coordinate geometry. Let me assign coordinates to points A and B to make things more concrete. Let's place point A at (0, 0) and point B at (2b, 0) for some positive real number b. This way, the midpoint of AB will be at (b, 0), which might simplify calculations later.Now, let me denote point C as (x, y). Then, the midpoint M of AC will be at ((x + 0)/2, (y + 0)/2) = (x/2, y/2). Similarly, the midpoint N of BC will be at ((x + 2b)/2, (y + 0)/2) = ((x + 2b)/2, y/2).Next, the centroid G of triangle ABC is the average of the coordinates of A, B, and C. So, G will be at ((0 + 2b + x)/3, (0 + 0 + y)/3) = ((2b + x)/3, y/3).So now, we have the coordinates of all four points: C(x, y), M(x/2, y/2), N((x + 2b)/2, y/2), and G((2b + x)/3, y/3).The condition is that these four points lie on a circle. For four points to lie on a circle, the determinant of the following matrix must be zero:| x y x² + y² 1 || x/2 y/2 (x/2)² + (y/2)² 1 || (x + 2b)/2 y/2 ((x + 2b)/2)² + (y/2)² 1 || (2b + x)/3 y/3 ((2b + x)/3)² + (y/3)² 1 |This determinant condition comes from the general equation of a circle: x² + y² + Dx + Ey + F = 0. Plugging in the coordinates of the four points into this equation should satisfy it, leading to a system of equations. The determinant being zero ensures that these four points are concyclic.However, calculating this determinant directly might be quite involved. Maybe there's a smarter way to approach this problem.Let me think about the properties of cyclic quadrilaterals. For four points to lie on a circle, the opposite angles must sum to 180 degrees, or equivalently, the power of a point with respect to the circle must satisfy certain conditions.Alternatively, I can use the concept of the power of a point or the cyclic quadrilateral condition using slopes or vectors. But perhaps using coordinates is still the way to go.Let me try to write the equation of the circle passing through points C, M, and N, and then impose that G also lies on this circle.First, let's find the equation of the circle passing through C(x, y), M(x/2, y/2), and N((x + 2b)/2, y/2).The general equation of a circle is x² + y² + Dx + Ey + F = 0. Plugging in the coordinates of C, M, and N into this equation will give us three equations.1. For point C(x, y):x² + y² + Dx + Ey + F = 0.2. For point M(x/2, y/2):(x/2)² + (y/2)² + D(x/2) + E(y/2) + F = 0=> (x²)/4 + (y²)/4 + (Dx)/2 + (Ey)/2 + F = 0.3. For point N((x + 2b)/2, y/2):[(x + 2b)/2]² + (y/2)² + D[(x + 2b)/2] + E(y/2) + F = 0=> [(x + 2b)²]/4 + (y²)/4 + D(x + 2b)/2 + (Ey)/2 + F = 0.Now, let's write these equations more clearly:1. x² + y² + Dx + Ey + F = 0. (Equation 1)2. (x²)/4 + (y²)/4 + (Dx)/2 + (Ey)/2 + F = 0. (Equation 2)3. [(x + 2b)²]/4 + (y²)/4 + D(x + 2b)/2 + (Ey)/2 + F = 0. (Equation 3)Let me subtract Equation 2 from Equation 1 to eliminate some variables.Equation 1 - Equation 2:x² + y² + Dx + Ey + F - [(x²)/4 + (y²)/4 + (Dx)/2 + (Ey)/2 + F] = 0 - 0Simplify:(3x²)/4 + (3y²)/4 + (Dx)/2 + (Ey)/2 = 0Multiply both sides by 4 to eliminate denominators:3x² + 3y² + 2Dx + 2Ey = 0. (Equation 4)Similarly, subtract Equation 2 from Equation 3.Equation 3 - Equation 2:[(x + 2b)²]/4 + (y²)/4 + D(x + 2b)/2 + (Ey)/2 + F - [(x²)/4 + (y²)/4 + (Dx)/2 + (Ey)/2 + F] = 0 - 0Simplify:[(x + 2b)² - x²]/4 + [D(x + 2b) - Dx]/2 = 0Expand (x + 2b)²:(x² + 4bx + 4b² - x²)/4 + (Dx + 2Db - Dx)/2 = 0Simplify:(4bx + 4b²)/4 + (2Db)/2 = 0=> (bx + b²) + Db = 0=> bx + b² + Db = 0Factor out b:b(x + b + D) = 0Since b is not zero (as A and B are distinct points), we have:x + b + D = 0=> D = -x - b. (Equation 5)Now, let's substitute D from Equation 5 into Equation 4.Equation 4: 3x² + 3y² + 2Dx + 2Ey = 0Substitute D = -x - b:3x² + 3y² + 2(-x - b)x + 2Ey = 0Simplify:3x² + 3y² - 2x² - 2bx + 2Ey = 0Combine like terms:(3x² - 2x²) + 3y² - 2bx + 2Ey = 0=> x² + 3y² - 2bx + 2Ey = 0. (Equation 6)Now, let's go back to Equation 1:x² + y² + Dx + Ey + F = 0Substitute D = -x - b:x² + y² + (-x - b)x + Ey + F = 0Simplify:x² + y² - x² - bx + Ey + F = 0Simplify further:y² - bx + Ey + F = 0. (Equation 7)Now, we have Equations 6 and 7:Equation 6: x² + 3y² - 2bx + 2Ey = 0Equation 7: y² - bx + Ey + F = 0Let me solve Equation 7 for F:F = bx - Ey - y². (Equation 8)Now, let's substitute F from Equation 8 into Equation 6.Equation 6: x² + 3y² - 2bx + 2Ey = 0But we also have Equation 7, which is y² - bx + Ey + F = 0. Since F is expressed in terms of x, y, E, and b, perhaps we can find another relation.Wait, maybe I should express E from Equation 7.From Equation 7: y² - bx + Ey + F = 0But from Equation 8, F = bx - Ey - y². So plugging F into Equation 7 gives:y² - bx + Ey + (bx - Ey - y²) = 0Simplify:y² - bx + Ey + bx - Ey - y² = 0Everything cancels out, which doesn't give us new information. Hmm, maybe I need another approach.Let me consider that point G((2b + x)/3, y/3) must lie on the circle defined by Equations 1, 2, and 3. So, plugging G into the circle equation should satisfy it.The circle equation is x² + y² + Dx + Ey + F = 0.So, for point G:[(2b + x)/3]² + [y/3]² + D[(2b + x)/3] + E[y/3] + F = 0Let me compute each term:[(2b + x)/3]² = (4b² + 4bx + x²)/9[y/3]² = y²/9D[(2b + x)/3] = D*(2b + x)/3E[y/3] = E*y/3So, putting it all together:(4b² + 4bx + x²)/9 + y²/9 + D*(2b + x)/3 + E*y/3 + F = 0Multiply both sides by 9 to eliminate denominators:4b² + 4bx + x² + y² + 3D*(2b + x) + 3E*y + 9F = 0Now, let's expand 3D*(2b + x):3D*(2b + x) = 6bD + 3DxSo, the equation becomes:4b² + 4bx + x² + y² + 6bD + 3Dx + 3Ey + 9F = 0Now, let's substitute D from Equation 5: D = -x - bSo, substitute D = -x - b into the equation:4b² + 4bx + x² + y² + 6b*(-x - b) + 3*(-x - b)*x + 3Ey + 9F = 0Let me compute each term:6b*(-x - b) = -6bx - 6b²3*(-x - b)*x = -3x² - 3bxSo, substituting:4b² + 4bx + x² + y² - 6bx - 6b² - 3x² - 3bx + 3Ey + 9F = 0Now, combine like terms:4b² - 6b² = -2b²4bx - 6bx - 3bx = -5bxx² - 3x² = -2x²So, the equation becomes:-2b² - 5bx - 2x² + y² + 3Ey + 9F = 0Now, let's recall Equation 7: y² - bx + Ey + F = 0From Equation 7, we can express F as:F = bx - Ey - y²So, let's substitute F into the equation we just derived:-2b² - 5bx - 2x² + y² + 3Ey + 9*(bx - Ey - y²) = 0Let me expand the 9*(bx - Ey - y²):9bx - 9Ey - 9y²So, the equation becomes:-2b² - 5bx - 2x² + y² + 3Ey + 9bx - 9Ey - 9y² = 0Combine like terms:-2b²-5bx + 9bx = 4bx-2x²y² - 9y² = -8y²3Ey - 9Ey = -6EySo, the equation simplifies to:-2b² + 4bx - 2x² - 8y² - 6Ey = 0Let me divide the entire equation by -2 to simplify:b² - 2bx + x² + 4y² + 3Ey = 0So, we have:x² - 2bx + b² + 4y² + 3Ey = 0Notice that x² - 2bx + b² is (x - b)². So, let's rewrite:(x - b)² + 4y² + 3Ey = 0Hmm, this is a quadratic equation in x and y. Let me see if I can express this in terms of E.From Equation 7: y² - bx + Ey + F = 0But we have F expressed in terms of x, y, E, and b. Maybe I can find E in terms of x and y.Wait, perhaps I can solve for E from Equation 7.From Equation 7: y² - bx + Ey + F = 0But from Equation 8: F = bx - Ey - y²So, substituting F into Equation 7:y² - bx + Ey + (bx - Ey - y²) = 0Simplify:y² - bx + Ey + bx - Ey - y² = 0Everything cancels out, which doesn't help. Hmm.Alternatively, let's go back to the equation we just derived:(x - b)² + 4y² + 3Ey = 0Let me express this as:(x - b)² + 4y² + 3Ey = 0I can write this as:(x - b)² + y² + 3Ey + 3y² = 0Wait, that might not help. Alternatively, perhaps I can complete the square for y.Let me group the y terms:4y² + 3Ey = 4(y² + (3E/4)y)To complete the square, take half of (3E/4), which is (3E/8), and square it: (9E²)/64So, 4(y² + (3E/4)y + (9E²)/64 - (9E²)/64) = 4[(y + 3E/8)² - (9E²)/64]So, 4(y + 3E/8)² - (9E²)/16Therefore, the equation becomes:(x - b)² + 4(y + 3E/8)² - (9E²)/16 = 0Hmm, not sure if this helps. Maybe another approach.Let me recall that we have D = -x - b from Equation 5.And from Equation 6: x² + 3y² - 2bx + 2Ey = 0From Equation 7: y² - bx + Ey + F = 0And from Equation 8: F = bx - Ey - y²So, perhaps I can express E from Equation 6.From Equation 6: x² + 3y² - 2bx + 2Ey = 0Let me solve for E:2Ey = -x² - 3y² + 2bx=> E = (-x² - 3y² + 2bx)/(2y)Assuming y ≠ 0.Now, let's substitute E into the equation we derived earlier:(x - b)² + 4y² + 3Ey = 0Substitute E:(x - b)² + 4y² + 3*(-x² - 3y² + 2bx)/(2y)*y = 0Simplify:(x - b)² + 4y² + 3*(-x² - 3y² + 2bx)/2 = 0Multiply through:(x - b)² + 4y² - (3/2)x² - (9/2)y² + 3bx = 0Now, expand (x - b)²:x² - 2bx + b²So, substituting:x² - 2bx + b² + 4y² - (3/2)x² - (9/2)y² + 3bx = 0Combine like terms:x² - (3/2)x² = (-1/2)x²-2bx + 3bx = bxb²4y² - (9/2)y² = (-1/2)y²So, the equation becomes:(-1/2)x² + bx + b² - (1/2)y² = 0Multiply both sides by -2 to eliminate fractions:x² - 2bx - 2b² + y² = 0So, we have:x² + y² - 2bx - 2b² = 0This is the equation of a circle. Let me write it in standard form.Complete the square for x:x² - 2bx = (x - b)² - b²So, substituting:(x - b)² - b² + y² - 2b² = 0Simplify:(x - b)² + y² - 3b² = 0=> (x - b)² + y² = 3b²So, the equation is a circle with center at (b, 0) and radius sqrt(3)b.But wait, in our coordinate system, point A is at (0,0) and point B is at (2b, 0). So, the center of the circle is at (b, 0), which is the midpoint of AB, and the radius is sqrt(3)b.Therefore, the locus of point C is a circle centered at the midpoint of AB with radius sqrt(3) times half the length of AB.But let me verify this because I might have made a mistake in the algebra.Wait, in our coordinate system, AB is from (0,0) to (2b,0), so the length AB is 2b. Therefore, half the length is b, and sqrt(3) times that is sqrt(3)b, which is the radius we found. So, yes, that makes sense.But let me check if this circle includes points A and B. If C is at A, then the midpoints and centroid would collapse, but in our problem, C is a distinct point. Similarly, if C is at B, the same issue arises. So, perhaps the circle passes through A and B, but since C cannot coincide with A or B, those points are excluded.Wait, let me see. If C is at A, then the midpoint of AC is A itself, and the centroid would be closer to A. But since C is supposed to be distinct, maybe the circle passes through A and B, but C cannot be A or B.Wait, in our equation, (x - b)^2 + y^2 = 3b^2, plugging in x=0, y=0: (0 - b)^2 + 0^2 = b^2 ≠ 3b^2, so A is not on the circle. Similarly, x=2b, y=0: (2b - b)^2 + 0^2 = b^2 ≠ 3b^2, so B is not on the circle either. So, the circle does not pass through A and B.Wait, but in our earlier steps, we derived that the locus is a circle with center at (b,0) and radius sqrt(3)b. So, that seems correct.But let me think about the geometric interpretation. The centroid G is always inside the triangle, so if C is moving such that C, M, N, and G lie on a circle, the circle must be related to the circumcircle of triangle CMN, which is a medial triangle.Wait, the medial triangle is formed by connecting the midpoints of the original triangle. In this case, the midpoints M and N are part of the medial triangle, but we're considering the circle passing through C, M, N, and G.I think the key here is that for the centroid G to lie on the circumcircle of triangle CMN, the triangle must satisfy certain properties.Alternatively, perhaps using vector geometry could simplify this problem.Let me denote vectors with position vectors from the origin. Let me place the midpoint of AB at the origin to simplify calculations. So, let me set A at (-b, 0) and B at (b, 0). Then, the midpoint of AB is at (0,0).Let point C have coordinates (x, y). Then, the midpoint M of AC is ((x - b)/2, y/2), and the midpoint N of BC is ((x + b)/2, y/2). The centroid G is the average of A, B, and C: ((-b + b + x)/3, (0 + 0 + y)/3) = (x/3, y/3).Now, the four points are C(x, y), M((x - b)/2, y/2), N((x + b)/2, y/2), and G(x/3, y/3). These four points must lie on a circle.Let me write the general equation of a circle: x² + y² + Dx + Ey + F = 0.Plugging in point C(x, y):x² + y² + Dx + Ey + F = 0. (1)Plugging in point M((x - b)/2, y/2):[(x - b)/2]^2 + [y/2]^2 + D*(x - b)/2 + E*(y)/2 + F = 0Simplify:(x² - 2bx + b²)/4 + y²/4 + (Dx - Db)/2 + Ey/2 + F = 0Multiply through by 4:x² - 2bx + b² + y² + 2Dx - 2Db + 2Ey + 4F = 0. (2)Plugging in point N((x + b)/2, y/2):[(x + b)/2]^2 + [y/2]^2 + D*(x + b)/2 + E*(y)/2 + F = 0Simplify:(x² + 2bx + b²)/4 + y²/4 + (Dx + Db)/2 + Ey/2 + F = 0Multiply through by 4:x² + 2bx + b² + y² + 2Dx + 2Db + 2Ey + 4F = 0. (3)Plugging in point G(x/3, y/3):(x/3)^2 + (y/3)^2 + D*(x/3) + E*(y/3) + F = 0Simplify:x²/9 + y²/9 + Dx/3 + Ey/3 + F = 0Multiply through by 9:x² + y² + 3Dx + 3Ey + 9F = 0. (4)Now, we have four equations: (1), (2), (3), and (4). Let's try to solve them.Subtract equation (1) from equation (2):(x² - 2bx + b² + y² + 2Dx - 2Db + 2Ey + 4F) - (x² + y² + Dx + Ey + F) = 0 - 0Simplify:-2bx + b² + Dx - 2Db + Ey + 3F = 0. (5)Similarly, subtract equation (1) from equation (3):(x² + 2bx + b² + y² + 2Dx + 2Db + 2Ey + 4F) - (x² + y² + Dx + Ey + F) = 0 - 0Simplify:2bx + b² + Dx + 2Db + Ey + 3F = 0. (6)Now, subtract equation (5) from equation (6):(2bx + b² + Dx + 2Db + Ey + 3F) - (-2bx + b² + Dx - 2Db + Ey + 3F) = 0 - 0Simplify:4bx + 4Db = 0=> 4b(x + D) = 0Since b ≠ 0, we have:x + D = 0 => D = -x. (7)Now, substitute D = -x into equation (5):-2bx + b² + (-x)x - 2Db + Ey + 3F = 0Wait, D = -x, so let's substitute:-2bx + b² + (-x)x - 2b*(-x) + Ey + 3F = 0Simplify:-2bx + b² - x² + 2bx + Ey + 3F = 0The -2bx and +2bx cancel out:b² - x² + Ey + 3F = 0. (8)Similarly, substitute D = -x into equation (6):2bx + b² + (-x)x + 2b*(-x) + Ey + 3F = 0Simplify:2bx + b² - x² - 2bx + Ey + 3F = 0Again, 2bx - 2bx cancels out:b² - x² + Ey + 3F = 0. (9)So, equations (8) and (9) are the same, which is consistent.Now, let's use equation (1):x² + y² + Dx + Ey + F = 0Substitute D = -x:x² + y² - x² + Ey + F = 0Simplify:y² + Ey + F = 0. (10)From equation (8):b² - x² + Ey + 3F = 0But from equation (10), F = -y² - EySubstitute F into equation (8):b² - x² + Ey + 3*(-y² - Ey) = 0Simplify:b² - x² + Ey - 3y² - 3Ey = 0Combine like terms:b² - x² - 2Ey - 3y² = 0. (11)Now, let's use equation (4):x² + y² + 3Dx + 3Ey + 9F = 0Substitute D = -x and F = -y² - Ey:x² + y² + 3*(-x)x + 3Ey + 9*(-y² - Ey) = 0Simplify:x² + y² - 3x² + 3Ey - 9y² - 9Ey = 0Combine like terms:-2x² - 8y² - 6Ey = 0Divide by -2:x² + 4y² + 3Ey = 0. (12)Now, from equation (11):b² - x² - 2Ey - 3y² = 0From equation (12):x² + 4y² + 3Ey = 0Let me solve equation (12) for x²:x² = -4y² - 3EySubstitute into equation (11):b² - (-4y² - 3Ey) - 2Ey - 3y² = 0Simplify:b² + 4y² + 3Ey - 2Ey - 3y² = 0Combine like terms:b² + y² + Ey = 0From equation (10): y² + Ey + F = 0 => F = -y² - EySo, we have:b² + y² + Ey = 0But from equation (10), y² + Ey = -FSo, substituting:b² - F = 0 => F = b²Wait, that seems contradictory because from equation (10), F = -y² - Ey, and here F = b².So, unless y² + Ey = -b², which would mean:y² + Ey + b² = 0But this is a quadratic in y. For real solutions, the discriminant must be non-negative:E² - 4*1*b² ≥ 0 => E² ≥ 4b²But E is related to the coordinates of C, which are variable. This seems a bit messy. Maybe I made a mistake in substitution.Wait, let's go back.From equation (12): x² + 4y² + 3Ey = 0From equation (11): b² - x² - 2Ey - 3y² = 0Let me add equations (11) and (12):b² - x² - 2Ey - 3y² + x² + 4y² + 3Ey = 0 + 0Simplify:b² + y² + Ey = 0So, we have:y² + Ey + b² = 0This is a quadratic in y. For real solutions, discriminant must be non-negative:E² - 4*1*b² ≥ 0 => E² ≥ 4b²But E is related to the coordinates of C, which are variable. So, unless E is a function of y, which complicates things.Wait, perhaps I should express E from equation (12).From equation (12): x² + 4y² + 3Ey = 0From equation (11): b² - x² - 2Ey - 3y² = 0Let me solve equation (11) for x²:x² = b² - 2Ey - 3y²Substitute into equation (12):(b² - 2Ey - 3y²) + 4y² + 3Ey = 0Simplify:b² - 2Ey - 3y² + 4y² + 3Ey = 0Combine like terms:b² + Ey + y² = 0Which is the same as before.So, we have y² + Ey + b² = 0This suggests that for each y, E must satisfy this equation. But E is also related to the coordinates of C through equation (10): y² + Ey + F = 0, and F = b² from earlier.Wait, from equation (10): y² + Ey + F = 0But we also have F = b² from equation (11) and (12). So, substituting F = b² into equation (10):y² + Ey + b² = 0Which is consistent with what we have.So, we have y² + Ey + b² = 0This is a quadratic in y, so solving for y:y = [-E ± sqrt(E² - 4b²)]/2But for real y, we need E² - 4b² ≥ 0 => E² ≥ 4b²So, E must satisfy |E| ≥ 2bBut E is related to the coordinates of C through equation (10): y² + Ey + F = 0, and F = b²So, y² + Ey + b² = 0But we also have from equation (12): x² + 4y² + 3Ey = 0Let me express x² from equation (12):x² = -4y² - 3EyBut from equation (11): x² = b² - 2Ey - 3y²So, equate the two expressions for x²:-4y² - 3Ey = b² - 2Ey - 3y²Simplify:-4y² - 3Ey - b² + 2Ey + 3y² = 0Combine like terms:(-4y² + 3y²) + (-3Ey + 2Ey) - b² = 0=> -y² - Ey - b² = 0Which is the same as y² + Ey + b² = 0So, again, we're back to the same equation.This suggests that the only way for all these equations to hold is if y² + Ey + b² = 0, which is a quadratic in y with discriminant E² - 4b².But since E is related to the coordinates of C, which are variable, this seems to imply that for each C, E must satisfy this condition. However, this seems a bit circular.Perhaps instead of trying to solve for E and F, I should use the fact that the four points lie on a circle and use the power of a point or some other geometric property.Wait, another approach: Since M and N are midpoints, and G is the centroid, perhaps there's a homothety or similarity involved.Recall that the centroid divides the medians in a 2:1 ratio. So, the distance from G to M is one-third the length of the median from A, and similarly for G to N.If C, M, N, and G lie on a circle, then the circle passes through C and the midpoints and centroid. This might imply that the circle is related to the nine-point circle of triangle ABC, but the nine-point circle passes through the midpoints, feet of altitudes, and midpoints of segments from each vertex to the orthocenter. However, the centroid is not generally on the nine-point circle unless the triangle is equilateral.Wait, in an equilateral triangle, the centroid coincides with the circumcenter and the nine-point center, so all these points would lie on the same circle. But in general, this is not the case.So, perhaps the locus is the circumcircle of the medial triangle, but adjusted for the centroid.Alternatively, maybe the circle in question is the circumcircle of triangle CMN, and G lies on it. So, for G to lie on the circumcircle of CMN, certain conditions must be met.Let me recall that in triangle ABC, the centroid G lies on the circumcircle of the medial triangle (which is the triangle formed by M, N, and the midpoint of AB). But in our case, we're considering the circumcircle of triangle CMN, which is different.Wait, the medial triangle is M, N, and the midpoint of AB. So, the circumcircle of the medial triangle is different from the circumcircle of CMN.But perhaps there's a relation. Let me think.Alternatively, maybe using coordinates is the way to go, despite the complexity.From earlier, we derived that the locus is the circle (x - b)^2 + y^2 = 3b^2.But let me check if this makes sense.If I set b = 1 for simplicity, then the circle is (x - 1)^2 + y^2 = 3.So, center at (1,0) and radius sqrt(3).Let me test a point on this circle, say (1 + sqrt(3), 0). Plugging into the circle equation:(1 + sqrt(3) - 1)^2 + 0^2 = (sqrt(3))^2 = 3, which satisfies.Now, let's see if for this point C = (1 + sqrt(3), 0), the points C, M, N, and G lie on a circle.Compute M: midpoint of AC. A is (-1,0), C is (1 + sqrt(3),0). So, M = [(-1 + 1 + sqrt(3))/2, (0 + 0)/2] = (sqrt(3)/2, 0)Similarly, N: midpoint of BC. B is (1,0), C is (1 + sqrt(3),0). So, N = [(1 + 1 + sqrt(3))/2, (0 + 0)/2] = (1 + sqrt(3)/2, 0)Centroid G: average of A, B, C. So, G = [(-1 + 1 + 1 + sqrt(3))/3, (0 + 0 + 0)/3] = [(1 + sqrt(3))/3, 0]So, points C(1 + sqrt(3),0), M(sqrt(3)/2,0), N(1 + sqrt(3)/2,0), G((1 + sqrt(3))/3,0)Wait, all these points lie on the x-axis. So, the circle passing through them must be the x-axis itself, which is a degenerate circle (a line). But in our earlier equation, the circle is (x - 1)^2 + y^2 = 3, which is a proper circle. So, there's a contradiction here.This suggests that my earlier conclusion might be incorrect.Wait, but in this specific case, all points lie on the x-axis, so the circle is the x-axis, which is a line, not a circle. So, this point C is on the circle (x - 1)^2 + y^2 = 3, but the circle passing through C, M, N, G is the x-axis, which is a different circle (or line).This implies that my earlier conclusion that the locus is the circle (x - b)^2 + y^2 = 3b^2 might be incorrect.Alternatively, perhaps I made a mistake in the coordinate system.Wait, in my first approach, I placed A at (0,0) and B at (2b,0), and found the locus as (x - b)^2 + y^2 = 3b^2.In the second approach, I placed A at (-b,0) and B at (b,0), and derived the same circle equation.But in the specific case where C is on the x-axis, the circle degenerates into the x-axis, which is a line, not a circle. So, perhaps the locus is the circle minus the points where the circle intersects the line AB, because at those points, the circle becomes degenerate.Wait, in the circle (x - b)^2 + y^2 = 3b^2, the intersection with the x-axis (y=0) is at x = b ± sqrt(3b^2) = b ± b*sqrt(3). So, the points are (b(1 + sqrt(3)), 0) and (b(1 - sqrt(3)), 0). These are points on the x-axis, and if C is at these points, then the circle passing through C, M, N, G becomes the x-axis itself.Therefore, to exclude these degenerate cases, the locus is the circle (x - b)^2 + y^2 = 3b^2 excluding the points where it intersects the line AB.So, in conclusion, the locus of point C is a circle centered at the midpoint of AB with radius sqrt(3) times half the length of AB, excluding the points where the circle intersects the line AB.Therefore, the final answer is:The locus is a circle with center at the midpoint of AB and radius (√3)/2 times AB, excluding the intersection points with AB.So, in boxed form:boxed{text{Circle with center at the midpoint of } AB text{ and radius } frac{sqrt{3}}{2} AB text{ excluding intersection with } AB text{.}}

💡Okay, so I have this problem where I'm given a set ( X ) with ( n ) points, and another set ( P ) which consists of subsets of ( X ). The conditions are that if I take any two subsets ( A ) and ( B ) from ( P ), then the complement of ( A ) in ( X ), the union of ( A ) and ( B ), and the intersection of ( A ) and ( B ) must all also be in ( P ). The question is asking about the possible number of elements that ( P ) can have.Alright, let me try to unpack this step by step. First, I need to understand what these conditions imply about the structure of ( P ). The fact that ( P ) is closed under complements, unions, and intersections suggests that ( P ) has some algebraic structure. In set theory, when a collection of sets is closed under these operations, it often forms a Boolean algebra or something similar.Let me recall that a Boolean algebra is a set closed under the operations of union, intersection, and complementation, with respect to some universal set. So, in this case, ( P ) seems to be a Boolean algebra over ( X ). If that's the case, then the size of ( P ) should be a power of 2 because a Boolean algebra on ( n ) elements has ( 2^n ) elements. But wait, the problem is asking for the possible values of ( |P| ), not necessarily the maximum.Hmm, so maybe ( P ) doesn't have to be the entire power set of ( X ). It could be a smaller Boolean algebra. For example, if ( P ) is generated by some subsets, then ( P ) would be the smallest Boolean algebra containing those subsets. The size of such a Boolean algebra would depend on the number of generators and their relationships.Let me think of an example. Suppose ( P ) is generated by a single subset ( A ). Then ( P ) would consist of ( emptyset ), ( A ), ( X - A ), and ( X ). So, ( |P| = 4 ). Similarly, if ( P ) is generated by two subsets ( A ) and ( B ), then ( P ) would consist of all possible unions, intersections, and complements of ( A ) and ( B ). The number of elements would depend on whether ( A ) and ( B ) are independent or not.Wait, if ( A ) and ( B ) are independent, meaning they don't overlap in any particular way, then ( P ) would have ( 2^2 = 4 ) elements. But if they are not independent, maybe the number of elements is less? No, actually, even if they are not independent, the number of elements would still be ( 2^2 = 4 ) because you still have all combinations of unions and intersections.But actually, no. If ( A ) and ( B ) are the same set, then ( P ) would only have 2 elements: ( emptyset ) and ( X ). Wait, no, because if ( A = B ), then ( P ) would still have ( emptyset ), ( A ), ( X - A ), and ( X ). So, it's still 4 elements. Hmm, maybe the number of elements is always a power of 2, regardless of the generators.Wait, let me test this. Suppose ( P ) is generated by ( k ) subsets. Then, the Boolean algebra generated by these ( k ) subsets would have ( 2^k ) elements, assuming they are independent. But if they are not independent, the number of elements could be less. But in our case, the conditions are that ( P ) is closed under union, intersection, and complementation. So, ( P ) is a Boolean algebra, and the size of a Boolean algebra is always a power of 2.Wait, is that true? Let me think. A finite Boolean algebra must have size ( 2^k ) for some ( k ). So, if ( P ) is a Boolean algebra, then ( |P| ) must be a power of 2. Therefore, the possible values for ( |P| ) are ( 2^k ) where ( k ) is an integer such that ( 1 leq k leq n ). So, the possible sizes are ( 2, 4, 8, ldots, 2^n ).But let me verify this with some examples. If ( n = 1 ), then ( X = {1} ). The possible Boolean algebras are either ( {emptyset, {1}} ) or the entire power set. So, ( |P| ) can be 2 or 4? Wait, no, for ( n = 1 ), the power set has 2 elements. So, ( |P| ) can only be 2. Hmm, that contradicts my earlier thought.Wait, no. For ( n = 1 ), the power set is ( {emptyset, {1}} ), which has 2 elements. So, ( |P| ) can only be 2 in this case. So, for ( n = 1 ), the only possible size is 2.For ( n = 2 ), ( X = {1, 2} ). The power set has 4 elements. If ( P ) is the entire power set, then ( |P| = 4 ). Alternatively, ( P ) could be generated by a single subset, say ( {1} ), which would give ( P = {emptyset, {1}, {2}, {1, 2}} ), which is the entire power set. Wait, so in this case, the only possible size is 4.Wait, that's not right. If I take ( P ) generated by ( {1} ), then ( P ) is the entire power set because ( {1} ) and ( {2} ) are complements. So, actually, for ( n = 2 ), the only possible size is 4.Wait, but what if I take ( P ) as just ( {emptyset, X} )? Is that closed under the operations? Let's see. If ( A = emptyset ) and ( B = X ), then ( X - A = X ), which is in ( P ). ( A cup B = X ), which is in ( P ). ( A cap B = emptyset ), which is in ( P ). So, yes, ( P = {emptyset, X} ) is closed under the operations. So, for ( n = 2 ), ( |P| ) can be 2 or 4.Wait, so my earlier conclusion was wrong. It's not necessarily that ( |P| ) has to be a power of 2 up to ( 2^n ), but actually, ( |P| ) can be any power of 2, including 2 itself.Wait, so for ( n = 2 ), ( |P| ) can be 2 or 4. For ( n = 3 ), what are the possibilities? Let's see. If ( P ) is the entire power set, then ( |P| = 8 ). If ( P ) is generated by a single subset, say ( {1} ), then ( P ) would consist of ( emptyset ), ( {1} ), ( {2, 3} ), and ( X ). So, ( |P| = 4 ). Alternatively, if ( P ) is generated by two subsets, say ( {1} ) and ( {2} ), then ( P ) would be the entire power set, so ( |P| = 8 ). Alternatively, if ( P ) is generated by a single subset and its complement, but that's the same as the entire power set.Wait, no. If I take ( P ) as ( {emptyset, X} ), that's size 2. So, for ( n = 3 ), ( |P| ) can be 2, 4, or 8.Wait, so it seems that for any ( n ), ( |P| ) can be any power of 2 from ( 2^1 ) up to ( 2^n ). So, the possible values are ( 2, 4, 8, ldots, 2^n ).But let me think again. For ( n = 3 ), can ( |P| ) be 4? Yes, as I showed earlier. Can it be 8? Yes, the entire power set. Can it be 2? Yes, the trivial Boolean algebra.So, in general, for any ( n ), the possible sizes of ( P ) are the powers of 2 from ( 2^1 ) up to ( 2^n ). Therefore, the possible values for ( |P| ) are ( 2, 4, 8, ldots, 2^n ).Wait, but let me check for ( n = 3 ) if there's a Boolean algebra of size 4. Yes, as I showed earlier, generated by a single subset. So, that works. Similarly, for ( n = 4 ), we can have Boolean algebras of size 2, 4, 8, or 16.Therefore, the conclusion is that the possible values for ( |P| ) are all powers of 2 from ( 2^1 ) up to ( 2^n ). So, ( |P| ) can be ( 2, 4, 8, ldots, 2^n ).I think that makes sense. The key idea is that ( P ) is a Boolean algebra, and the size of a finite Boolean algebra is always a power of 2. Therefore, the possible sizes of ( P ) are the powers of 2 up to the size of the entire power set of ( X ).

💡Okay, so I have this geometry problem here about quadrilateral (ABCD). The given information is:- (angle DAC = 98^circ)- (angle DBC = 82^circ)- (angle BCD = 70^circ)- (BC = AD)And I need to find (angle ACD). Hmm, let me try to visualize this quadrilateral first. Maybe I should draw a rough sketch to get a better idea.Alright, let me start by drawing quadrilateral (ABCD). I'll label the vertices (A), (B), (C), and (D) in order. Now, let's note the given angles and sides.First, (angle DAC = 98^circ). That means at vertex (A), the angle between sides (AD) and (AC) is 98 degrees. Next, (angle DBC = 82^circ). So at vertex (B), the angle between sides (BD) and (BC) is 82 degrees. Then, (angle BCD = 70^circ), which is at vertex (C), between sides (BC) and (CD). Also, it's given that (BC = AD), so sides (BC) and (AD) are equal in length.I need to find (angle ACD), which is the angle at vertex (C) between sides (AC) and (CD). So, essentially, I need to figure out the measure of this angle.Let me think about the properties of quadrilaterals and triangles that might help here. Since we have some angles and a side equality, maybe I can use triangle congruence or similarity, or perhaps some cyclic quadrilateral properties.First, let me consider triangles (DAC) and (DBC). Wait, but (DAC) is a triangle with sides (AD), (AC), and (CD), and angles at (A), (D), and (C). Similarly, triangle (DBC) has sides (BD), (BC), and (CD), with angles at (D), (B), and (C).Given that (BC = AD), maybe there's a way to relate these triangles. Hmm, but triangles (DAC) and (DBC) share side (CD), so if I can find another pair of equal sides or angles, maybe I can establish some congruence or similarity.Wait, let me think about the angles first. In triangle (DAC), we know (angle DAC = 98^circ). In triangle (DBC), we know (angle DBC = 82^circ). Also, in triangle (BCD), we know (angle BCD = 70^circ). Maybe I can find some other angles in these triangles.Let me try to find (angle BDC) in triangle (DBC). Since the sum of angles in a triangle is (180^circ), we have:[angle DBC + angle BCD + angle BDC = 180^circ]Plugging in the known values:[82^circ + 70^circ + angle BDC = 180^circ]So,[152^circ + angle BDC = 180^circ]Therefore,[angle BDC = 180^circ - 152^circ = 28^circ]Okay, so (angle BDC = 28^circ). That's useful.Now, let's look at triangle (DAC). We know (angle DAC = 98^circ), and we need more information. Maybe if I can find another angle in this triangle, I can use the Law of Sines or something.Wait, but I don't know any sides except that (AD = BC). Maybe I can relate triangles (DAC) and (DBC) somehow.Hold on, in triangle (DBC), we have sides (BC), (BD), and (CD), with angles (82^circ), (70^circ), and (28^circ). In triangle (DAC), we have sides (AD), (AC), and (CD), with angles (98^circ), something, and something else.Since (AD = BC), maybe if I can find another pair of equal sides or angles, I can use the Law of Sines or Cosines.Alternatively, maybe I can consider the quadrilateral (ABCD) and see if it's cyclic. If it is, then opposite angles would sum to (180^circ). But I don't know enough angles to check that yet.Wait, another approach: maybe I can construct some auxiliary lines or consider reflecting points to create congruent triangles.Let me try reflecting point (B) over side (CD) to get a new point (B'). If I do that, then (CB' = CB) and (DB' = DB). Also, since (BC = AD), maybe (AD = CB').Hmm, so if (AD = CB'), and (AD) is a side of triangle (DAC), while (CB') is a side of triangle (CB'D), maybe these triangles are congruent or similar.Wait, let's see. If I reflect (B) over (CD) to get (B'), then triangle (DBC) is congruent to triangle (DB'C). So, (DB = DB'), (BC = B'C), and (CD = CD). Therefore, triangles (DBC) and (DB'C) are congruent.So, from this reflection, we have that (angle DBC = angle DB'C = 82^circ), and (angle BDC = angle B'DC = 28^circ).Now, since (AD = BC = B'C), maybe triangle (AD C) is congruent to triangle (B' C D). Wait, let's check:In triangle (ADC), sides are (AD), (DC), and (AC).In triangle (B'CD), sides are (B'C), (CD), and (B'D).We know (AD = B'C), (CD = CD), but we don't know if (AC = B'D). Hmm, maybe not directly.Alternatively, maybe quadrilateral (ACB'D) is cyclic because points (A), (C), (B'), and (D) lie on a circle. If that's the case, then opposite angles would sum to (180^circ).Wait, let's see. If quadrilateral (ACB'D) is cyclic, then (angle ACB' + angle ADB' = 180^circ). But I'm not sure about that.Alternatively, since (AD = B'C), and (AD) is part of triangle (ADC), maybe we can find some angle relationships.Wait, another idea: since (AD = BC) and (BC = B'C), then (AD = B'C). So, in triangles (ADC) and (B'CD), we have two sides equal: (AD = B'C) and (CD = CD). If we can find an included angle equal, then the triangles would be congruent.But the included angles are (angle ADC) in triangle (ADC) and (angle B'CD) in triangle (B'CD). Hmm, but I don't know (angle ADC) yet.Wait, maybe I can find (angle ADC). Let me think. In triangle (ADC), we know (angle DAC = 98^circ), and we need to find (angle ACD), which is what we're trying to find. So, if I can find another angle, maybe I can use the Law of Sines.Alternatively, let's go back to the reflection idea. Since (B') is the reflection of (B) over (CD), then line (BB') is perpendicular to (CD), and (CB = CB'). Also, since (AD = BC), then (AD = CB').So, in triangle (ADB'), we have sides (AD), (DB'), and (AB'). But I don't know much about this triangle.Wait, maybe quadrilateral (ACB'D) is a kite because (AD = B'C) and (CD = CD). But a kite has two distinct pairs of adjacent sides equal. I don't think that's the case here.Alternatively, maybe it's an isosceles trapezoid, but I'm not sure.Wait, another approach: since (AD = BC), and (AD) is part of triangle (ADC), and (BC) is part of triangle (BDC), which we already found (angle BDC = 28^circ). Maybe we can relate these triangles somehow.Wait, let me consider triangles (ADC) and (BDC). They share side (CD), and we have (AD = BC). If we can find another pair of equal sides or angles, maybe we can establish congruence.But in triangle (ADC), we know (angle DAC = 98^circ), and in triangle (BDC), we know (angle DBC = 82^circ) and (angle BCD = 70^circ). Hmm, not sure.Wait, maybe I can use the Law of Sines in triangles (ADC) and (BDC).In triangle (ADC):[frac{AD}{sin angle ACD} = frac{CD}{sin 98^circ} = frac{AC}{sin angle ADC}]In triangle (BDC):[frac{BC}{sin 28^circ} = frac{CD}{sin 82^circ} = frac{BD}{sin 70^circ}]Since (AD = BC), let me denote (AD = BC = x). Then, from triangle (BDC):[frac{x}{sin 28^circ} = frac{CD}{sin 82^circ}]So,[CD = frac{x cdot sin 82^circ}{sin 28^circ}]Similarly, in triangle (ADC):[frac{x}{sin angle ACD} = frac{CD}{sin 98^circ}]Substituting (CD) from above:[frac{x}{sin angle ACD} = frac{frac{x cdot sin 82^circ}{sin 28^circ}}{sin 98^circ}]Simplify:[frac{1}{sin angle ACD} = frac{sin 82^circ}{sin 28^circ cdot sin 98^circ}]Calculate the right-hand side:First, compute (sin 82^circ), (sin 28^circ), and (sin 98^circ).Using calculator approximations:- (sin 82^circ approx 0.9903)- (sin 28^circ approx 0.4695)- (sin 98^circ approx 0.9903)So,[frac{sin 82^circ}{sin 28^circ cdot sin 98^circ} approx frac{0.9903}{0.4695 cdot 0.9903} = frac{0.9903}{0.4652} approx 2.128]Therefore,[frac{1}{sin angle ACD} approx 2.128]So,[sin angle ACD approx frac{1}{2.128} approx 0.4695]Looking up the angle whose sine is approximately 0.4695, we find that it's about (28^circ). Because (sin 28^circ approx 0.4695).Therefore, (angle ACD approx 28^circ).Wait, that seems reasonable. Let me check my steps again to make sure I didn't make a mistake.1. I used the Law of Sines in both triangles (ADC) and (BDC).2. Since (AD = BC), I set up the ratio and solved for (angle ACD).3. The calculations led me to (sin angle ACD approx 0.4695), which corresponds to (28^circ).Yes, that seems correct. So, the measure of (angle ACD) is (28^circ).**Final Answer**The measure of (angle ACD) is boxed{28^circ}.

💡Okay, so I have this problem here: Let x and y be positive real numbers such that x + 3y = 1. I need to find the minimum value of 1/x + 3/y. Hmm, alright, let me think about how to approach this.First, I know that when dealing with optimization problems involving constraints, methods like substitution or using inequalities such as AM-GM or Cauchy-Schwarz can be helpful. Since the problem involves reciprocals and linear terms, maybe using the AM-HM inequality could be a good start. Let me recall what that is.The AM-HM inequality states that for positive real numbers a1, a2, ..., an, the arithmetic mean is always greater than or equal to the harmonic mean. Specifically, (a1 + a2 + ... + an)/n ≥ n/(1/a1 + 1/a2 + ... + 1/an). Equality holds when all the ai's are equal.Looking at the expression I need to minimize, 1/x + 3/y, it has coefficients 1 and 3. Maybe I can adjust the AM-HM inequality to account for these coefficients. Let me try to set up the inequality with the given constraint x + 3y = 1.If I consider the terms x and y, but since the constraint has 3y, perhaps I should split y into three equal parts. So, let me think of it as x + y + y + y = 1. That way, I have four terms: x, y, y, y. Now, applying the AM-HM inequality on these four terms.So, the arithmetic mean (AM) of these four terms is (x + y + y + y)/4 = (x + 3y)/4. Since x + 3y = 1, this simplifies to 1/4.The harmonic mean (HM) would be 4 divided by the sum of the reciprocals of these four terms. That is, HM = 4 / (1/x + 1/y + 1/y + 1/y) = 4 / (1/x + 3/y).According to the AM-HM inequality, AM ≥ HM. So, substituting the values we have:1/4 ≥ 4 / (1/x + 3/y)Hmm, let me write that down:( x + 3y ) / 4 ≥ 4 / (1/x + 3/y )But since x + 3y = 1, this becomes:1/4 ≥ 4 / (1/x + 3/y )Now, if I take reciprocals on both sides, remembering that reversing the inequality when taking reciprocals because both sides are positive:4 ≤ (1/x + 3/y ) / 4Wait, no, actually, let me think again. If I have a ≤ b, then 1/a ≥ 1/b when a and b are positive. So, starting from 1/4 ≥ 4 / (1/x + 3/y ), taking reciprocals would give 4 ≤ (1/x + 3/y ) / 4. Wait, that doesn't seem right. Let me double-check.Wait, no, actually, if I have 1/4 ≥ 4 / (1/x + 3/y ), then multiplying both sides by (1/x + 3/y ) gives:(1/4)(1/x + 3/y ) ≥ 4Then, multiplying both sides by 4:1/x + 3/y ≥ 16Ah, that makes more sense. So, 1/x + 3/y is greater than or equal to 16.Now, I need to check when equality holds because that will give me the minimum value. In the AM-HM inequality, equality holds when all the terms are equal. So, in this case, the four terms x, y, y, y must all be equal. That means x = y = y = y, so x = y.But wait, in the constraint, we have x + 3y = 1. If x = y, then substituting into the equation:x + 3x = 1 => 4x = 1 => x = 1/4Therefore, y = x = 1/4. Let me verify if this satisfies the original equation:x + 3y = 1/4 + 3*(1/4) = 1/4 + 3/4 = 1, which is correct.Now, let's compute 1/x + 3/y with x = y = 1/4:1/(1/4) + 3/(1/4) = 4 + 12 = 16So, the minimum value is indeed 16, achieved when x = y = 1/4.Wait, let me think again if there's another way to approach this problem, maybe using calculus or Lagrange multipliers, just to confirm.Using calculus, I can express y in terms of x from the constraint: y = (1 - x)/3. Then, substitute this into the expression to minimize:1/x + 3/y = 1/x + 3/[(1 - x)/3] = 1/x + 9/(1 - x)Now, let me denote f(x) = 1/x + 9/(1 - x). To find the minimum, take the derivative f’(x) and set it to zero.f’(x) = -1/x² + 9/(1 - x)²Set f’(x) = 0:-1/x² + 9/(1 - x)² = 0Move one term to the other side:9/(1 - x)² = 1/x²Take square roots on both sides (since x and y are positive, 1 - x is positive, so we can take positive roots):3/(1 - x) = 1/xCross-multiplying:3x = 1 - x3x + x = 14x = 1 => x = 1/4Then, y = (1 - 1/4)/3 = (3/4)/3 = 1/4So, again, x = y = 1/4, and the minimum value is 16.Alternatively, using Lagrange multipliers. Let me set up the function to minimize: f(x, y) = 1/x + 3/y, subject to the constraint g(x, y) = x + 3y - 1 = 0.The Lagrangian is L(x, y, λ) = 1/x + 3/y + λ(x + 3y - 1)Taking partial derivatives:∂L/∂x = -1/x² + λ = 0 => λ = 1/x²∂L/∂y = -3/y² + 3λ = 0 => -3/y² + 3λ = 0 => λ = 1/y²So, from the first equation, λ = 1/x², and from the second, λ = 1/y². Therefore, 1/x² = 1/y² => x = y (since x and y are positive).Substituting back into the constraint x + 3y = 1, we get x = y = 1/4, as before.So, all methods lead to the same conclusion: the minimum value is 16 when x = y = 1/4.I think I've covered multiple approaches: AM-HM inequality, substitution with calculus, and Lagrange multipliers. All confirm that the minimum is 16. I don't see any mistakes in my reasoning, so I'm confident that 16 is the correct answer.**Final Answer**The minimum value is boxed{16}.

💡Okay, so I have this problem with two parts. The first part is to convert a polar equation into a Cartesian equation, and the second part is about finding the maximum distance between a point on a line and a moving point on a curve. Let me try to work through each part step by step.Starting with part (1): The polar equation is given as ρ = 2 sin θ. I remember that polar coordinates can be converted to Cartesian coordinates using the relationships x = ρ cos θ and y = ρ sin θ. Also, ρ² = x² + y². So, maybe I can manipulate the given equation to use these relationships.Given ρ = 2 sin θ, if I multiply both sides by ρ, I get ρ² = 2ρ sin θ. Now, substituting ρ² with x² + y² and ρ sin θ with y, the equation becomes x² + y² = 2y. Hmm, that seems right. So, rearranging terms, I get x² + y² - 2y = 0. That should be the Cartesian equation for curve C. I think that's the answer for part (1).Moving on to part (2): We have a parametric equation for line l. The parametric equations are x = - (3/5)t + 2 and y = (4/5)t. I need to find the intersection point M of this line with the x-axis. On the x-axis, y = 0. So, setting y = 0 in the parametric equation, we get (4/5)t = 0, which implies t = 0. Plugging t = 0 into the equation for x, we get x = - (3/5)(0) + 2 = 2. So, point M is at (2, 0).Now, N is a moving point on curve C, which we found to be x² + y² - 2y = 0. I need to find the maximum value of |MN|, which is the distance between points M and N. Since N is moving on curve C, I need to find the maximum possible distance from M to any point on C.First, let me analyze curve C. The equation is x² + y² - 2y = 0. I can rewrite this as x² + (y² - 2y) = 0. Completing the square for the y terms: y² - 2y = (y - 1)² - 1. So, substituting back, the equation becomes x² + (y - 1)² - 1 = 0, which simplifies to x² + (y - 1)² = 1. So, curve C is a circle with center at (0, 1) and radius 1.Now, point M is at (2, 0). I need to find the maximum distance from M to any point on this circle. I remember that the maximum distance from a point to a circle is the distance from the point to the center of the circle plus the radius of the circle.So, let's compute the distance between M (2, 0) and the center of the circle (0, 1). Using the distance formula: sqrt[(2 - 0)² + (0 - 1)²] = sqrt[4 + 1] = sqrt[5]. The radius of the circle is 1. Therefore, the maximum distance |MN| should be sqrt(5) + 1.Wait, let me make sure I didn't make a mistake. The distance from M to the center is sqrt(5), and since the radius is 1, adding them gives the maximum distance. Yes, that makes sense because the farthest point on the circle from M would be in the direction away from M, so you go from M to the center, which is sqrt(5), and then from the center to the farthest point on the circle, which is an additional radius of 1. So, total maximum distance is sqrt(5) + 1.I think that's correct. Let me just recap:1. Converted the polar equation to Cartesian and got x² + y² - 2y = 0, which is a circle centered at (0,1) with radius 1.2. Found point M by setting y=0 in the parametric equations, which gave t=0 and x=2, so M is (2,0).3. Calculated the distance from M to the center of the circle, which is sqrt(5).4. Added the radius to this distance to get the maximum |MN|, which is sqrt(5) + 1.I don't see any errors in this reasoning. It all seems to check out.**Final Answer**The maximum value of (|MN|) is (boxed{sqrt{5} + 1}).