Let H : x 2 a 2 - y 2 b 2 = 1 , a 0 , b 0 , be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is 4 2 2 + 14 . If the eccentricity H is 11 2 , then value of a 2 + b 2 is equal to ______.

Question

Let H : x 2 a 2 - y 2 b 2 = 1 , a > 0 , b > 0 , be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is 4 2 2 + 14 . If the eccentricity H is 11 2 , then value of a 2 + b 2 is equal to ______.,

MARKS App · Accepted Answer

Given x 2 a 2 - y 2 b 2 = 1 , 2 a + 2 b = 4 2 2 + 14 and e = 11 2 We know e 2 = 1 + b 2 a 2 ⇒ 11 4 = 1 + b 2 a 2 ⇒ b 2 = 7 4 a 2 ∴ x 2 a 2 - y 2 7 2 a 2 = 1 Now 2 a + 2 · 7 a 2 = 4 2 2 + 14 a 2 + 7 = 4 2 2 + 7 a = 4 2 ⇒ a 2 = 32 b 2 = 7 4 × 16 × 2 = 56 Hence a 2 + b 2 = 88

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