💡Okay, so I have this problem here where I need to find the value of y when x is 4 and z is 6. The problem says that 5y varies inversely as the square of x, and 3z varies directly as x. Hmm, okay, let me try to unpack that step by step.First, let's recall what it means for something to vary inversely or directly. If A varies inversely as B, that means A = k/B for some constant k. And if A varies directly as B, that means A = kB for some constant k. So, applying that to the problem:1. 5y varies inversely as the square of x. So, that should translate to 5y = k / x², where k is a constant.2. 3z varies directly as x. So, that should translate to 3z = mx, where m is another constant.Alright, so we have two relationships here: one involving y and x, and another involving z and x. The problem gives us some initial conditions: when x = 2 and y = 25, the relationship holds true. So, I can use this information to find the constant k for the first equation.Let me write down the first equation again: 5y = k / x². Plugging in x = 2 and y = 25:5 * 25 = k / (2)²125 = k / 4To solve for k, I can multiply both sides by 4:k = 125 * 4k = 500Okay, so now I know that 5y = 500 / x². That means y = (500 / x²) / 5, which simplifies to y = 100 / x². So, y is inversely proportional to the square of x with a constant of 100.Now, the problem also mentions that 3z varies directly as x, which is 3z = mx. I need to find m, but wait, do I have enough information to find m? The problem doesn't give me an initial condition for z, only for y. Hmm, maybe I don't need m because the question is asking for y when x = 4 and z = 6. Let me see.Wait, if 3z varies directly as x, then 3z = mx, so z = (m/3)x. But I don't have a value for z when x is something else, so maybe I don't need to find m because the question gives me z when x is 4. Let me think.The question is asking for y when x = 4 and z = 6. So, maybe I can use the relationship between z and x to find m, and then use that to find y? Or perhaps it's not necessary because y is already determined by x through the inverse variation.Wait, let me clarify. The problem states that 5y varies inversely as x², and 3z varies directly as x. So, these are two separate relationships, but they both involve x. So, when x changes, both y and z change accordingly. But in the problem, we are given that when x = 2, y = 25, and we need to find y when x = 4 and z = 6. So, maybe I can find y directly from the inverse variation without needing z, but since z is given, perhaps it's a way to check or maybe it's a red herring.Wait, no, maybe I need to use both relationships because z is given when x is 4. So, perhaps I can find m from the second equation using z = 6 and x = 4, and then use that to find y? Let me try that.So, from 3z = mx, when x = 4 and z = 6:3 * 6 = m * 418 = 4mm = 18 / 4m = 4.5So, m is 4.5. Therefore, the equation for z is 3z = 4.5x, or z = (4.5/3)x = 1.5x.But wait, do I need this for finding y? Because y is already determined by x through the inverse variation. So, when x = 4, y should be 100 / (4)^2 = 100 / 16 = 6.25. So, is that the answer? But let me make sure.Alternatively, maybe I need to consider both relationships together. Let me see.Wait, the problem says that when x = 2 and y = 25, the relationship holds true. So, that's how we found k = 500. Then, when x = 4 and z = 6, we need to find y. So, since y is determined solely by x through the inverse variation, and z is determined by x through the direct variation, but in this case, z is given as 6 when x is 4. So, maybe I can check if z = 6 when x = 4 is consistent with the direct variation.From earlier, we found that m = 4.5, so 3z = 4.5x. When x = 4, 3z = 4.5 * 4 = 18, so z = 6. That matches the given z = 6 when x = 4. So, that's consistent.Therefore, since y is determined by x through the inverse variation, and x is 4, y should be 100 / (4)^2 = 6.25. So, the answer should be 6.25, which is option A.Wait, but let me double-check. Maybe I made a mistake in calculating k.From 5y = k / x², when x = 2 and y = 25:5 * 25 = k / 4125 = k / 4k = 500Yes, that's correct. So, 5y = 500 / x², so y = 100 / x².When x = 4, y = 100 / 16 = 6.25. That seems right.Alternatively, maybe I should express y in terms of z. Let me see.From 3z = mx, we have z = (m/3)x. So, m = 4.5, so z = 1.5x.Therefore, x = z / 1.5.So, if z = 6, then x = 6 / 1.5 = 4. Which is consistent with the given x = 4.So, y = 100 / x² = 100 / 16 = 6.25.Yes, that seems consistent.Wait, but maybe I should consider if the relationships are connected in some way that I haven't considered. For example, maybe the constants k and m are related? Let me think.In the problem, it's stated that 5y varies inversely as x² and 3z varies directly as x. These are two separate relationships, so their constants k and m are independent. Therefore, I don't need to relate them unless the problem states that they are connected somehow, which it doesn't.So, I think I'm safe to proceed with y = 100 / x², and when x = 4, y = 6.25.Therefore, the answer is 6.25, which is option A.

💡Okay, so I have this problem here: "If x > 1, then x² + x > 2." And I need to find its contrapositive. The options are A, B, C, and D. Hmm, let me think about what a contrapositive is. From what I remember, the contrapositive of a statement "If P, then Q" is "If not Q, then not P." So, basically, you switch the hypothesis and the conclusion and negate both. That makes sense because the contrapositive is logically equivalent to the original statement, right?So, applying that to the given proposition. The original statement is "If x > 1, then x² + x > 2." Here, P is "x > 1" and Q is "x² + x > 2." So, to form the contrapositive, I need to negate both P and Q and then switch their positions.First, let's negate Q. The original Q is "x² + x > 2." The negation of that would be "x² + x ≤ 2." Okay, that makes sense because if it's not greater than 2, it must be less than or equal to 2.Next, let's negate P. The original P is "x > 1." The negation of that is "x ≤ 1." Got it. So, if it's not greater than 1, it's less than or equal to 1.Now, switching their positions. The contrapositive should be "If x² + x ≤ 2, then x ≤ 1." Let me check the options to see which one matches this.Option A: "If x > 1, then x² + x ≤ 2." Hmm, that's not the contrapositive. It seems like they just negated the conclusion but didn't switch the hypothesis and conclusion.Option B: "If x² + x ≤ 2, then x ≤ 1." Yes, that's exactly what I came up with. So, this looks correct.Option C: "If x² + x > 2, then x > 1." Wait, that's actually the converse of the original statement, not the contrapositive. The converse swaps the hypothesis and conclusion without negating them.Option D: "If x ≤ 1, then x² + x ≤ 2." That's the inverse of the original statement. The inverse negates both the hypothesis and conclusion but doesn't switch them. So, that's not the contrapositive either.So, going back, the contrapositive should be "If x² + x ≤ 2, then x ≤ 1," which is option B. Let me just double-check to make sure I didn't make any mistakes.Original statement: If P, then Q. Contrapositive: If not Q, then not P. So, P is "x > 1," Q is "x² + x > 2." Not Q is "x² + x ≤ 2," and not P is "x ≤ 1." So, contrapositive is "If x² + x ≤ 2, then x ≤ 1." Yep, that's definitely option B.I think I'm confident with that answer. It's easy to mix up contrapositive with converse or inverse, but as long as I remember to negate both parts and switch them, I should be fine.

💡First, I need to combine the fraction (frac{5 + 2d}{8}) with the integer 3 into a single fraction. To do this, I'll convert the integer 3 into a fraction with the same denominator as (frac{5 + 2d}{8}). Since 3 can be expressed as (frac{24}{8}), I can now add the two fractions together:[frac{5 + 2d}{8} + frac{24}{8} = frac{5 + 2d + 24}{8} = frac{29 + 2d}{8}]Therefore, the expression (frac{5 + 2d}{8} + 3) as a single fraction is:[boxed{frac{29 + 2d}{8}}]

💡Okay, so I have this hyperbola equation: (frac{x^2}{6} - frac{y^2}{3} = 1). I need to find the length of segment (AB) where a line passes through the left focus (F_1) and intersects the left branch of the hyperbola at points (A) and (B). The condition given is that (|AF_2| + |BF_2| = 2|AB|). First, I should recall some properties of hyperbolas. The standard form of a hyperbola is (frac{x^2}{a^2} - frac{y^2}{b^2} = 1), where (a) is the distance from the center to each vertex, and (b) is related to the conjugate axis. The foci are located at ((pm c, 0)) where (c^2 = a^2 + b^2).Looking at the given equation, (a^2 = 6) and (b^2 = 3). So, (a = sqrt{6}) and (b = sqrt{3}). Then, (c^2 = a^2 + b^2 = 6 + 3 = 9), so (c = 3). Therefore, the foci (F_1) and (F_2) are at ((-3, 0)) and ((3, 0)) respectively.Now, the line passes through (F_1 = (-3, 0)) and intersects the left branch of the hyperbola at points (A) and (B). So, both (A) and (B) are on the left side of the hyperbola, meaning their x-coordinates are less than or equal to (-a = -sqrt{6}).Given that (|AF_2| + |BF_2| = 2|AB|), I need to find (|AB|). Hmm, this seems to relate the distances from points (A) and (B) to the right focus (F_2) with the distance between (A) and (B).I remember that for hyperbolas, the difference of distances from any point on the hyperbola to the two foci is constant and equal to (2a). Specifically, for any point (P) on the hyperbola, (|PF_2| - |PF_1| = 2a). Since (A) and (B) are on the left branch, this difference would be (|AF_2| - |AF_1| = 2a) and similarly for (B).So, applying this to points (A) and (B):[|AF_2| - |AF_1| = 2a][|BF_2| - |BF_1| = 2a]Adding these two equations together:[(|AF_2| + |BF_2|) - (|AF_1| + |BF_1|) = 4a]But from the problem statement, we know that (|AF_2| + |BF_2| = 2|AB|). Let me substitute that in:[2|AB| - (|AF_1| + |BF_1|) = 4a]So,[2|AB| - (|AF_1| + |BF_1|) = 4sqrt{6}]Hmm, I need to find (|AB|), so I should express (|AF_1| + |BF_1|) in terms of (|AB|). Since the line passes through (F_1), which is one of the foci, and intersects the hyperbola at (A) and (B), (F_1) lies on segment (AB). Therefore, (|AF_1| + |BF_1| = |AB|). Wait, is that correct?Let me visualize this. If (F_1) is on the line (AB), then (A), (F_1), and (B) are colinear with (F_1) between (A) and (B). So, the distance from (A) to (B) is the sum of the distances from (A) to (F_1) and from (F_1) to (B). Therefore, (|AB| = |AF_1| + |BF_1|).So, substituting back into the equation:[2|AB| - |AB| = 4sqrt{6}]Simplifying:[|AB| = 4sqrt{6}]Wait, that seems straightforward. Let me double-check my reasoning.1. I found (a = sqrt{6}), (c = 3), so foci at ((-3, 0)) and ((3, 0)).2. Points (A) and (B) are on the left branch, so for each, (|PF_2| - |PF_1| = 2a).3. Summing for (A) and (B), I get (|AF_2| + |BF_2| - (|AF_1| + |BF_1|) = 4a).4. Given (|AF_2| + |BF_2| = 2|AB|), so substitution gives (2|AB| - (|AF_1| + |BF_1|) = 4sqrt{6}).5. Since (F_1) is on (AB), (|AB| = |AF_1| + |BF_1|).6. Therefore, (2|AB| - |AB| = |AB| = 4sqrt{6}).Yes, that seems correct. So, the length of segment (AB) is (4sqrt{6}).**Final Answer**The length of segment (AB) is (boxed{4sqrt{6}}).

💡Okay, so I have this problem about an ellipse with center P. The major axis is XY and the minor axis is WZ. There's a focus G, and the distance from P to G is 8. Also, the diameter of the inscribed circle of triangle PWG is 4. I need to find the product of the lengths of the major and minor axes, which are XY and WZ.First, let me recall some properties of an ellipse. The standard form of an ellipse is (x²/a²) + (y²/b²) = 1, where 2a is the length of the major axis and 2b is the length of the minor axis. The distance from the center to each focus is c, and it's related to a and b by the equation c² = a² - b².In this problem, P is the center, so the major axis XY is 2a, and the minor axis WZ is 2b. The distance from P to G is PG = c = 8. So, we have c = 8, which gives us c² = 64. Therefore, a² - b² = 64.Now, the problem also mentions triangle PWG. Let me visualize this triangle. Points P, W, and G. Since W is an endpoint of the minor axis, PW is equal to b. Similarly, since G is a focus, PG is equal to c, which is 8. So, triangle PWG has sides of length b, 8, and the third side, which is WG.Wait, what is WG? Since W is on the minor axis and G is a focus, which is on the major axis. So, the distance between W and G can be found using the distance formula. Let me think about the coordinates. If I place the ellipse at the origin, then P is (0,0). The major axis is along the x-axis, so G is at (c,0) = (8,0). Point W is on the minor axis, so its coordinates are (0,b) or (0,-b). Let's take (0,b) for simplicity.So, the distance between W (0,b) and G (8,0) is sqrt[(8 - 0)² + (0 - b)²] = sqrt(64 + b²). Therefore, the sides of triangle PWG are b, 8, and sqrt(64 + b²).But wait, triangle PWG is a right triangle? Let me check. If P is (0,0), W is (0,b), and G is (8,0), then the triangle is formed by the points (0,0), (0,b), and (8,0). So, yes, it's a right triangle with legs along the y-axis and x-axis, and hypotenuse from (0,b) to (8,0). So, the legs are of length b and 8, and the hypotenuse is sqrt(64 + b²).Now, the problem says the diameter of the inscribed circle of triangle PWG is 4. That means the radius is 2. I remember that for a right triangle, the inradius r can be calculated using the formula r = (a + b - c)/2, where a and b are the legs, and c is the hypotenuse.So, applying this formula, the inradius r is (b + 8 - sqrt(64 + b²))/2. But we know the radius is 2, so:(b + 8 - sqrt(64 + b²))/2 = 2Multiplying both sides by 2:b + 8 - sqrt(64 + b²) = 4Simplify:b + 8 - 4 = sqrt(64 + b²)So,b + 4 = sqrt(64 + b²)Now, let's square both sides to eliminate the square root:(b + 4)² = (sqrt(64 + b²))²Expanding the left side:b² + 8b + 16 = 64 + b²Subtract b² from both sides:8b + 16 = 64Subtract 16 from both sides:8b = 48Divide both sides by 8:b = 6So, b = 6. That means the minor axis WZ is 2b = 12.Earlier, we had a² - b² = 64, and since b = 6, b² = 36. So,a² - 36 = 64Adding 36 to both sides:a² = 100So, a = 10 (since a is positive). Therefore, the major axis XY is 2a = 20.Now, the product of the major axis and minor axis is XY * WZ = 20 * 12 = 240.Wait, let me just verify the steps again to make sure I didn't make a mistake.1. Identified that a² - b² = c² = 64.2. Considered triangle PWG as a right triangle with legs b and 8, hypotenuse sqrt(64 + b²).3. Used the inradius formula for right triangles: r = (a + b - c)/2, set it equal to 2 (since diameter is 4, radius is 2).4. Solved the equation and found b = 6.5. Plugged back into a² - b² = 64 to find a = 10.6. Calculated the lengths of the axes and their product.Everything seems to check out. So, the product (XY)(WZ) is 240.**Final Answer**boxed{240}

💡Okay, so I have this problem here about a function defined on complex numbers. The function is given by f(z) = (a + bi)z, where a and b are positive numbers. The key property is that for every complex number z, f(z) is equidistant from both z and the origin. Also, it's given that the magnitude of a + bi is 5. I need to find b squared.Alright, let's break this down. First, I know that in complex numbers, the magnitude |a + bi| is calculated as sqrt(a² + b²). Since it's given that |a + bi| = 5, that means sqrt(a² + b²) = 5. Squaring both sides, I get a² + b² = 25. So that's one equation involving a and b.Now, the function f(z) = (a + bi)z is a linear transformation on the complex plane. It's essentially a scaling and rotation. The problem states that for every z, f(z) is equidistant from z and the origin. So, mathematically, this means that the distance between f(z) and z is equal to the distance between f(z) and 0. In terms of complex numbers, distance is measured using the modulus, so |f(z) - z| = |f(z)|.Let me write that down:|f(z) - z| = |f(z)|Substituting f(z) with (a + bi)z, we get:|(a + bi)z - z| = |(a + bi)z|Factor out z on the left side:|(a + bi - 1)z| = |(a + bi)z|Since modulus is multiplicative, |ab| = |a||b|, so we can write:|a + bi - 1| * |z| = |a + bi| * |z|Hmm, interesting. Now, since this equation must hold for every complex number z, the only way this can be true is if the coefficients multiplying |z| are equal. Otherwise, for different z's, the equation might not hold. So, we can set:|a + bi - 1| = |a + bi|But we already know that |a + bi| = 5, so:|a + bi - 1| = 5Let me compute |a + bi - 1|. That's the modulus of (a - 1) + bi, which is sqrt[(a - 1)² + b²]. So:sqrt[(a - 1)² + b²] = 5Squaring both sides:(a - 1)² + b² = 25But we also have from earlier that a² + b² = 25. So now we have two equations:1. a² + b² = 252. (a - 1)² + b² = 25Let me subtract the second equation from the first to eliminate b²:a² + b² - [(a - 1)² + b²] = 25 - 25Simplify:a² - (a - 1)² = 0Expanding (a - 1)²:a² - (a² - 2a + 1) = 0Simplify inside the brackets:a² - a² + 2a - 1 = 0Which simplifies to:2a - 1 = 0So, solving for a:2a = 1 => a = 1/2Alright, so a is 1/2. Now, let's substitute a back into one of the original equations to find b². Let's use the first equation:a² + b² = 25Substitute a = 1/2:(1/2)² + b² = 25Which is:1/4 + b² = 25Subtract 1/4 from both sides:b² = 25 - 1/4Convert 25 to quarters to subtract:25 = 100/4So:b² = 100/4 - 1/4 = 99/4Therefore, b squared is 99/4.Wait, let me just verify that. If a is 1/2, then |a + bi| is sqrt[(1/2)² + b²] = sqrt[1/4 + b²] = 5. So, 1/4 + b² = 25, which gives b² = 25 - 1/4 = 99/4. Yep, that seems right.So, I think that's the answer. b squared is 99/4.**Final Answer**The value of ( b^2 ) is boxed{dfrac{99}{4}}.