10 The curve \(C\) has polar equation
$$r = a \sin 3 \theta$$
where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
- Show that the area of the region enclosed by \(C\) is \(\frac { 1 } { 12 } \pi a ^ { 2 }\).
- Show that, at the point of \(C\) at maximum distance from the initial line,
$$\tan 3 \theta + 3 \tan \theta = 0 .$$
- Use the formula
$$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$
to find this maximum distance.
- Draw a sketch of \(C\).