Let H : x 2 a 2 - y 2   b 2 = 1 , a > 0 ,   b > 0 , be a hyperbola such that the sum of…

Given $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, $2a+2b=4\left(2\sqrt{2}+\sqrt{14}\right)$ and $e=\frac{\sqrt{11}}{2}$

We know  $e^2=1+\frac{b^2}{a^2}\Rightarrow \frac{11}{4}=1+\frac{b^2}{a^2}\Rightarrow b^2=\frac{7}{4}{a}^2$

$\therefore \frac{x^2}{{\left(a\right)}^2}-\frac{y^2}{{\left(\frac{\sqrt{7}}{2}a\right)}^2}=1$ 

Now $2a+2·\frac{\sqrt{7}a}{2}=4\left(2\sqrt{2}+\sqrt{14}\right)$

$a\left(2+\sqrt{7}\right)=4\sqrt{2}\left(2+\sqrt{7}\right)$

$a=4\sqrt{2}\Rightarrow a^2=32$

$b^2=\frac{7}{4}\times 16\times 2=56$

Hence $a^2+b^2=88$