A curve \(C\) has polar equation \(r = 2 a \cos \left( 2 \theta + \frac { 1 } { 2 } \pi \right)\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.
- Show that \(r = - 2 a \sin 2 \theta\) and sketch \(C\).
- Deduce that the cartesian equation of \(C\) is
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = - 4 a x y .$$
- Find the area of one loop of \(C\).
- Show that, at the points (other than the pole) at which a tangent to \(C\) is parallel to the initial line,
$$2 \tan \theta = - \tan 2 \theta .$$