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 value of $a ^{2}+ b ^{2}$ is equal to _______

Answer 1

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Given $e^{2}=1+\frac{b^{2}}{a^{2}} \Rightarrow \frac{11}{4}=1+\frac{b^{2}}{a^{2}} \Rightarrow b^{2}=\frac{7}{4} a^{2}$
$\therefore \frac{x^{2}}{(a)^{2}}-\frac{y^{2}}{\left(\frac{\sqrt{7}}{2} a\right)^{2}}=1$ Now given
$2 a +2 \cdot \frac{\sqrt{7} a }{2}=4(2 \sqrt{2}+\sqrt{14}) $
$ a (2+\sqrt{7})=4 \sqrt{2}(2+\sqrt{7}) $
$ a =4 \sqrt{2} \Rightarrow a ^{2}=32 $
$b ^{2}=\frac{7}{4} \times 16 \times 2=56$