Let H be a hyperbola and H' be its conjugate hyperbola. The sum of the lengths of the latus rectums of H and H' is 35 . If the eccentricity of H is 13 3 , then the area of the quadrilateral whose vertices are the foci of H and H' is

Step-by-step solution

Let the hyperbola H be x^2 a^2 - y^2 b^2 = 1 . The eccentricity of H is given by e = 1 + b^2 a^2 . Given e = 13 3 , we have 1 + b^2 a^2 = 13 9 b^2 a^2 = 4 9 b = 2 3 a . The lengths of the latus rectums of H and H' are l = 2b^2 a and l' = 2a^2 b respectively. Given l + l' = 35 , we substitute b = 2 3 a : 2 ( 4 9 a^2 ) a + 2a^2 2 3 a = 35 8 9 a + 3a = 35 35 9 a = 35 a = 9 . Then b = 2 3 (9) = 6 . The foci of H are at ( ae, 0) and the foci of H' are at (0, be') . For both hyperbolas, the distance from the center to a