Question

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

Answer 1

$2a+2b=4(2\sqrt2+\sqrt14)\ \ ...(1)$

$1+\frac{b^2}{a^2}=\frac{11}{4}\ \ ....(2)$

$⇒\frac{b^2}{a^2}=\frac{7}{4}\ \ ...(3)$

$a+b=4\sqrt2+2\sqrt\ \ ....(4)$

By $(3)$ and $(4)$

$a+\frac {\sqrt 7}{2}a = 4\sqrt 2+2\sqrt {14}$

$\frac {a(a+\sqrt 7)}{2}= 2\sqrt 2(2+\sqrt 7)$

$a=4\sqrt2$

⇒$$a^2 = 32 and $b^2 = 56$

$⇒ a^2 + b^2 = 32 + 56$

$⇒ a^2 + b^2= 88$

So, the answer is $88$.