Let each of the two ellipses E _ 1 : x^ 2 a^ 2 + y^ 2 b^ 2 =1,(a>b) and E _ 2 : x^ 2 ~A ^ 2 + y^ 2 ~B ^ 2 =1,( ~A < B ) have eccentricity 4 5 . Let the lengths of the latus recta of E_ 1 and E_ 2 be l_ 1 and l_ 2 , respectively, such that 2 l_ 1 ^ 2 =9 l_ 2 . If the distance between the foci of E_ 1 is 8, then the distance between the foci of E_ 2 is
A. 32 5
Both ellipses have e = 4/5. For E_1 (a > b): b^2 = a^2(1-e^2) = 9a^2/25. Latus rectum l_1 = 2b^2/a = 18a/25. Distance between foci of E_1: 2ae = 8a/5 = 8 a = 5, so l_1 = 18/5. For E_2 (A A^2 = B^2(1-e^2) = 9B^2/25. Latus rectum l_2 = 2A^2/B = 18B/25. Given 2l_1^2 = 9l_2: 2 324 25 = 9 18B 25 648 = 162B B = 4. Distance between foci of E_2 = 2Be = 2 4 4 5 = 32 5
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