Answer
The equation is:
$$
(x^{2}+y^{2})^{3}=4x^{2}y^{2}
$$
using polar coordinates, the given equation becomes:
$$
r= \pm \sin ( 2\theta).
$$
and is sketched in the figure.
Work Step by Step
Sketch the curve
$$
(x^{2}+y^{2})^{3}=4x^{2}y^{2}
$$
we know that
$$
x^{2}+y^{2}=r^{2}, \quad x=r \cos \theta \quad \text {and } y=r \sin \theta
$$
Substituting into the given equation:
$$
(x^{2}+y^{2})^{3}=4x^{2}y^{2}
$$
using polar coordinates, we get
$$
r^{6}=4 r^{2} \cos ^{2}(\theta) r^{2} \sin^{2}( \theta)
$$
$\Rightarrow $
$$
r^{2}=4 \cos ^{2}(\theta) \sin^{2}( \theta)
$$
$\Rightarrow $
$$
r= \pm 2 \cos (\theta) \sin ( \theta)= \pm \sin ( 2\theta).
$$
$$
r= \pm \sin ( 2\theta).
$$
and is sketched in the figure.
