Answer
$$r^2\cos 2\theta =4$$
Work Step by Step
Since $x=r\cos \theta $ and $y=r\sin \theta $
\begin{align*}
x^{2}-y^{2}&=4\\
r^2\cos^2 \theta -r^2 \sin^2 \theta &=4 \\
r^2[\cos^2 \theta - \sin^2 \theta ]&=4\\
r^2\cos 2\theta&=4
\end{align*}