Let the length of the latus rectum of an ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1,(ab), be 30. If its eccentricity is the maximum value of the function f(t)=- 3 4 +2 t-t^ 2 , then (a^ 2 +b^ 2 ) is equal to

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Let the length of the latus rectum of an ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1,(a>b), be 30. If its eccentricity is the maximum value of the function f(t)=- 3 4 +2 t-t^ 2 , then (a^ 2 +b^ 2 ) is equal to

Quantrex Academy · Accepted Answer

Correct answer: D. 496

f(t) = -t^2 + 2t - 3 4 = -(t-1)^2 + 1 4 . Maximum value = 1 4 , so e = 1 4 . b^2 = a^2(1 - e^2) = a^2 (1 - 1 16 ) = 15a^2 16 . Latus rectum = 2b^2 a = 2 15a^2/16 a = 15a 8 = 30 a = 16. b^2 = 15 256 16 = 240. a^2 + b^2 = 256 + 240 = 496.

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