Let E : x^2 a^2 + y^2 b^2 = 1 ( a > b ) and H : x^2 A^2 - y^2 B^2 = 1 be confocal conics such that the distance between their foci is 2 10 . The length of the minor axis of E is equal to the length of the conjugate axis of H . If the line y = 2x + 10 is a tangent to the hyperbola H , then the sum of the lengths of the latus rectums of E and H is equal to

Step-by-step solution

For the ellipse E , the distance from the center to a focus is c = a^2 - b^2 . For the hyperbola H , the distance from the center to a focus is c' = A^2 + B^2 . Since E and H are confocal, they share the same foci, so c = c' . The distance between the foci is 2c = 2 10 c = 10 c^2 = 10 . Thus, we have: a^2 - b^2 = 10 A^2 + B^2 = 10 It is given that the length of the minor axis of E equals the length of the conjugate axis of H : 2b = 2B b = B b^2 = B^2 . The line y = 2x + 10 is tangent to the hyperbola H . Using the