Express the equation $\;$ $r^2 \sin 2 \theta = a^2$ $\;$ in Cartesian coordinates.
The relation between Cartesian coordinates $\left(x, y\right)$ and polar coordinates $\left(r, \theta\right)$ is
$x = r \cos \theta$, $\;\;$ $y = r \sin \theta$
$\implies$ $\cos \theta = \dfrac{x}{r}$, $\;\;$ $\sin \theta = \dfrac{y}{r}$
Given equation is $\;\;$ $r^2 \sin 2\theta = a^2$
i.e. $\;$ $2 r^2 \sin \theta \cos \theta = a^2$
i.e. $\;$ $2 r^2 \times \dfrac{y}{r} \times \dfrac{x}{r} = a^2$
i.e. $\;$ $2 x y = a^2$
$\therefore \;$ The given equation in Cartesian coordinates is $\;$ $2xy = a^2$