Question

Area Of The Quadrilateral Formed With The Foci Of The Hyperbola $\left(\frac{X_2}{A_2}\right)-\left(\frac{Y_2}{B_2}\right)=1$ And $\left(\frac{X_2}{A_2}\right)-\left(\frac{Y_2}{B_2}\right)=-1$ Is

Answer 1

The foci are

$\left(\pm ae,0\right)$ and $\left(0,\pm ae\right)$

The area $=\left(\frac{1}{2}\right)\times d_1\times d_2$

$$\begin{gathered} =\left(\frac{1}{2}\right)\times \left(2ae\right)\times \left(2ae\right) \\ =2a^{2}e \end{gathered}$$

As we know,

$$\begin{gathered} b^2=a^2\left(e^2-1\right) \\ {b}^2=a^2{e}^2-a^2 \end{gathered}$$

$b^2+a^2=a^2{e}^2$

Therefore, the area $=2\left(a^2+b^2\right)$