Let H: x ^ 2 a ^ 2 - y ^ 2 b ^ 2 = 1, a > 0, b > 0 , be a hyperbola...

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

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

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

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

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

$a = 4\sqrt{2} ⇒ {a}^{2} = 32$

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

Hence ${a}^{2} + {b}^{2} = 88$