Answer
$r^2 = \sin 2\theta$
Work Step by Step
we know $x = r \cos \theta$ and $y = r \sin \theta$
putting the values in equation we get
${(x^2 +y^2)}^2 = 2xy$
$=> {({(r \cos \theta)}^2 + {(r \sin \theta)}^2)}^2 =2r^2 \cos \theta \sin\theta$
$=> r^4{({ \cos^2 \theta} + {\sin^2 \theta})}^2=2r^2 \cos \theta \sin\theta$
We know $({ \cos^2 \theta} + {\sin^2 \theta}) = 1$
and $2\cos \theta \sin \theta = \sin 2\theta$
$=> r^4 = r^2 \sin 2\theta$
$=> r^2 = \sin 2\theta$
The graph of the above equation is two-leaved rose as shown