Question

If the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$ intersect each other orthogonally, then the value of $a^2$ is

Answer 1

If an ellipse and a hyperbola intersects each other orthogonally, then they are confocal .
Foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $(\pm 3,0)$
Let $e$ be the eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$
$\pm ae=\pm 3\Rightarrow ae=3\Rightarrow a^2{e}^2=9\Rightarrow a^2+4=9\Rightarrow a^2=5$