Question

Area of the quadrilateral formed with the foci of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$ is $k\left(a^2+b^2\right)$ , then the value of $k$ is equal to

Answer 1

Answer: 2

The foci are $\left(\pm ae,0\right)\&\left(0,\pm ae\right)$ .

The above diagram is a rhombus. Its area $=\frac{1}{2}{d}_1{d}_2$
$=\frac{1}{2}\left(2ae\right)\left(2ae\right)$
$=2a^2{e}^2$
We know that, $b^2=a^2\left(e^2-1\right)$
$b^2=a^2{e}^2-a^2$
$b^2+a^2=a^2{e}^2$
So, area is $2\left(a^2+b^2\right)$ .