Question

Let $S_1$ and $S_2$ be the foci of the hyperbola whose length of transverse axis is $6$ and length of conjugate axis is $10$. If $S_3$ and $S_4$ are the foci of the conjugate hyperbola of the given hyperbola, then the area of the quadrilateral $S_1{S}_2{S}_3{S}_4$ is

Answer 1

Figure
$e^2=1+\frac{b^2}{a^2}=1+\frac{5^2}{3^2}=\frac{34}{9}\Rightarrow e=\frac{\sqrt{34}}{3}$
We know that $\frac{1}{e^2}+\frac{1}{e^{\prime 2}}=1\Rightarrow e^{\prime }=\frac{\sqrt{34}}{5}$
So,
area of quadrilateral $S_1{S}_2{S}_3{S}_4=4\times$ area of $△S_{1}O{S}_3=4\times \frac{1}{2}\times ae\times a{e}^{\prime }=2\times 3\times \frac{\sqrt{34}}{3}\times 5\times \frac{\sqrt{34}}{5}=68$