Let the length of the latus rectum of an ellipse x² a² + y² b² =1,(a>b) , be 30. If its eccentricity is the maximum value of the function f(t)=- 3 4 +2 t-t² , then ( a²+b² ) is equal to

Step-by-step solution

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 .