11 The curve \(C\) has polar equation
$$r = \frac { a } { 1 + \theta }$$
where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
- Show that \(r\) decreases as \(\theta\) increases.
- The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\),
$$\tan \theta = 1 + \theta$$
and verify that this equation has a root between 1.1 and 1.2.
- Draw a sketch of \(C\).
- Find the area of the region bounded by the initial line, the line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).