Let $H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1, a0, b0$, be a hyperbola such that the sum of le

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

Professor PC Thomas Classes · Answer

(88) : Given that, sum of lengths of the transverse and the conjugate axes is $4(2 \sqrt{2}+\sqrt{14})$.\ $\Rightarrow \quad 2 a+2 b=4(2 \sqrt{2}+\sqrt{14})$ $$\begin{equation*}\Rightarrow a+b=2(2 \sqrt{2}+\sqrt{14}) 	ag{i} \end{equation*}$$ Since, $e=\frac{\sqrt{11}}{2}$\ $\Rightarrow 1+\frac{b^{2}}{a^{2}}=\frac{11}{4} \Rightarrow \frac{b^{2}}{a^{2}}=\frac{7}{4}$ $$\begin{equation*}\Rightarrow \quad b=\frac{\sqrt{7}}{2} a 	ag{ii} \end{equation*}$$ By (i) and (ii), we have\ $a+\frac{\sqrt{7}}{2} a=4 \sqrt{2}+2 \sqrt{14}$\ $\Rightarrow \quad \frac{2 a+\sqrt{7} a}{2}=2 \sqrt{2}(2+\sqrt{7})$\ $\Rightarrow \quad a(2+\sqrt{7})=4 \sqrt{2}(2+\sqrt{7})$\ $\Rightarrow a=4 \sqrt{2} \Rightarrow a^{2}=32$ and $b^{2}=56$\ $\Rightarrow \quad a^{2}+b^{2}=88$\