Let the hyperbola H : x 2 a 2 - y 2 = 1 and the ellipse E : 3 x 2 + 4 y 2 = 12 be such that the length of latus rectum of H is equal to the length of latus rectum of E . If e H and e E are the eccentricities of H and E respectively, then the value of 12 e H 2 + e E 2 is equal to _____.

Step-by-step solution

Given E ≡ 3 x 2 + 4 y 2 = 12     ⇒   x 2 4 + y 2 3 = 1 Now, e E = 1 - b 2 a 2 = 1 - 3 4 = 1 2 Length of L.R. = 2 b 2 a = 2 × 3 2 = 3 Now H ≡    x 2 a 2 - y 2 1 = 1 Length of L.R. = 2 b 2 a = 2 × 1 a Given length of L.R. of E and H are equal So 2 a = 3    ⇒    a = 2 3 Now e H = b 2 a 2 + 1 = 1 2 3 2 + 1     ⇒     e H = 9 4 + 1 = 13 4 So 12 e H 2 + e E 2 = 12 13 4 + 1 4 = 12 14 4 = 14 × 3 = 42