$\displaystyle r^2\sin2\vartheta=-2$
can someone help me solve this?
I assume you want to convert the polar equation into a rectangular one.
$\displaystyle r^2\sin 2\theta=-2$
Use the double angle identity:
$\displaystyle \begin{aligned}
r^2 (2\sin \theta \cos \theta) &= -2 \\
2(r \sin \theta)(r \cos \theta) &= -2 \\
2yx &= -2 \\
xy &= -1
\end{aligned}$