A curve \(C\) has polar equation \(r=2a\cos\left(2\theta+\frac{\pi}{2}\right)\), for \(0\le\theta\lt2\pi\), where \(a\) is a positive constant.

(i) Show that \(r=-2a\sin2\theta\) and sketch \(C\).

(ii) Deduce that the Cartesian equation of \(C\) is \((x^2+y^2)^{3/2}=-4axy\).

(iii) Find the area of one loop of \(C\).

(iv) Show that, at the points other than the pole at which a tangent to \(C\) is parallel to the initial line, \(2\tan\theta=-\tan2\theta\).