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An ellipse E and a hyperbola H are confocal (they share the same foci). The distance between their foci is 10 . If the eccentricity of E is 4 5 and the length of the latus rectum of H is equal to the length of the latus rectum of E , then the length of the conjugate axis of H is equal to ____.

Correct answer

6

Step-by-step solution

Let the distance between the foci be 2c = 10 , so c = 5 . For the ellipse E , the distance from the center to a focus is a_E e_E = c = 5 . Given e_E = 4 5 , we have a_E ( 4 5 ) = 5 a_E = 25 4 . The semi-minor axis squared is b_E^2 = a_E^2(1 - e_E^2) = ( 25 4 )^2 (1 - 16 25 ) = 625 16 9 25 = 225 16 . The length of the latus rectum of E is 2b_E^2 a_E = 2 ( 225 16 ) 25 4 = 225 8 4 25 = 9 2 . For the hyperbola H , let its equation be x^2 A^2 - y^2 B^2 = 1 . It shares the same foci, so A^2 + B^2 = c^2 = 25 , which gives

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