Answer
$r^{2}\sin (2\theta)=1$
Work Step by Step
Substitute $x=r\cos\theta$, $y=r\sin\theta$ to obtain:
$2xy=1\\
2(r\cos\theta)(r\sin\theta)=1\\
r^{2}\times2\sin\theta\cos\theta=1$
Since $2\sin\theta \cos\theta=\sin (2\theta)$, we have
$r^{2}\times2\sin\theta\cos\theta=1\\
r^{2}\sin (2\theta)=1$
