Let x 2 a 2 + y 2 b 2 = 1 a > b be a given ellipse, length of whose latus rectum is 10 . If its eccentricity is the maximum value of the function, ϕ t = 5 12 + t - t 2 , then a 2 + b 2 is equal to :

Step-by-step solution

L R = 2 b 2 a = 10 ⇒     b 2 = 5 a ϕ t = 5 12 − t 2 - t + 1 4 − 1 4 = 5 12 + 1 4 − t − 1 2 2 = 2 3 − t − 1 2 2 max ϕ t = 2 3 = e b 2 = a 2 1 − e 2 5 a = a 2 1 − 4 9 ⇒    5 = 5 9 a ⇒     a 2 = 81 ,    b 2 = 45 a 2 + b 2 = 126 .