The eccentricity of an ellipse E with centre at the origin O is 3 2 and its directrices are x = 4 6 3 . Let H: x^2 a^2 - y^2 b^2 = 1 be a hyperbola whose eccentricity is equal to the length of semi-major axis of E , and whose length of latus rectum is equal to the length of minor axis of E . Then the distance between the foci of H is :

Step-by-step solution

For the ellipse E , the eccentricity is e_E = 3 2 and the directrices are x = a_E e_E = 4 6 3 . The semi-major axis a_E is given by: a_E = e_E 4 6 3 = 3 2 4 6 3 = 12 2 6 = 2 2 The semi-minor axis b_E is given by: b_E^2 = a_E^2(1 - e_E^2) = (2 2 )^2 (1 - 3 4 ) = 8 1 4 = 2 b_E = 2 For the hyperbola H: x^2 a^2 - y^2 b^2 = 1 , its eccentricity e_H is equal to the semi-major axis of E : e_H = a_E = 2 2 The length of the latus rectum of H is equal to the length of the minor axis of E ( 2b_E ): 2b^2 a = 2b_E = 2 2 b^2 = 2