An ellipse E : x 2 a 2 + y 2 b 2 = 1 passes through the vertices of the hyperbola H : x 2 49 - y 2 64 = - 1 . Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H . Let the product of the eccentricities of E and H be 1 2 . If l is the length of the latus rectum of the ellipse E , then the value of 113 l is equal to _______.

Step-by-step solution

Given, Hyperbola: y 2 64 - x 2 49 = 1 And ellipse E : x 2 a 2 + y 2 b 2 = 1 passes through the vertices of the hyperbola H : x 2 49 - y 2 64 = - 1 , so vertices will be V ≡ 0 , ± 8 So b 2 = 64 Now eccentricity of hyperbola will be e H = 1 + a 2 b 2 = 1 + 49 64 And eccentricity of ellipse x 2 a 2 + y 2 b 2 = 1 will be e E = 1 - a 2 b 2 = 1 - a 2 64 And using b = 8 We get, e H × e E = 1 2 (given) ⇒ 1 - a 2 64 × 113 8 = 1 2 ⇒ 64 - a 2 × 113 = 32 ⇒ 64 - a 2 = 32 2 113 ⇒