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 :
D. 8 7
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 a Using the standard relation for a hyperbola b^2 = a^2(e_H^2 - 1), we substitute b^2 and e_H: 2 a = a^2((2 2 )^2 - 1) 2 a = a^2(8 - 1) = 7a^2 Since a > 0, dividing by a gives: a = 2 7 The distance between the foci of the hyperbola H is 2ae_H: 2ae_H = 2 2 7 2 2 = 8 7 Answer: 8 7
Related: Mathematics — Ellipse · All PYQ Banks