Answer
$r^{2}\sin 2\theta=4$
Work Step by Step
Conversion formulas: $\left\{\begin{array}{ll}
(x,y)=(r\cos\theta,r\sin\theta) & \\
r^{2}=x^{2}+y^{2}, & \tan\theta=\frac{y}{x}
\end{array}\right.$
$x\cdot y \rightarrow (r\cos\theta)\cdot(r\sin\theta)=r^{2}(\sin\theta\cos\theta)$
Apply the double angle identity for sine, $ \sin 2\theta=2\sin\theta\cos\theta$
$=\displaystyle \frac{r^{2}\sin 2\theta}{2}$
Rewrite the equation in terms of $r$ and $\theta$.
$\displaystyle \frac{r^{2}\sin 2\theta}{2}=2$
$r^{2}\sin 2\theta=4$
