4. The curve \(C\) has polar equation \(r = 3 a \cos \theta , - \frac { \pi } { 2 } \leq \frac { \pi } { 2 }\). The curve \(D\) has polar equation \(r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\). Given that \(a\) is a positive constant,
- sketch, on the same diagram, the graphs of \(C\) and \(D\), indicating where each curve cuts the initial line.
The graphs of \(C\) intersect at the pole \(O\) and at the points \(P\) and \(Q\).
- Find the polar coordinates of \(P\) and \(Q\).
- Use integration to find the exact value of the area enclosed by the curve \(D\) and the lines \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\).
The region \(R\) contains all points which lie outside \(D\) and inside \(C\).
Given that the value of the smaller area enclosed by the curve \(C\) and the line \(\theta = \frac { \pi } { 3 }\) is
$$\frac { 3 a ^ { 2 } } { 16 } ( 2 \pi - 3 \sqrt { } 3 )$$
- show that the area of \(R\) is \(\pi a ^ { 2 }\).
(4)
[0pt]
[P4 January 2002 Qn 8]