The curve \(C\) has cartesian equation \(\left(x^{2}+y^{2}\right)^{2}=2 a^{2} x y\), where \(a\) is a positive constant. Show that the polar equation of \(C\) is \(r^{2}=a^{2} \sin 2 \theta\).

Sketch \(C\) for \(-\pi\lt \theta \leqslant \pi\).

Find the area enclosed by one loop of \(C\).