9. The diagram is a sketch of the two curves\\
$C _ { 1 }$ and $C _ { 2 }$ with polar equations\\
$C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi$\\
$\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi$.\\
\includegraphics[max width=\textwidth, alt={}, center]{8646b60a-3822-4d41-8978-1ccad1e216d6-2_318_776_1567_1082}

The curves meet at the pole $O$, and at the points $A$ and $B$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$, the polar coordinates of the points $A$ and $B$.
\item Show that the length of the line $A B$ is $\frac { 3 \sqrt { } 3 } { 2 } a$.

The region inside $C _ { 2 }$ and outside $C _ { 1 }$ is shown shaded in the diagram above.
\item Find, in terms of $a$, the area of this region.

A badge is designed which has the shape of the shaded region.\\
Given that the length of the line $A B$ is 4.5 cm ,
\item calculate the area of this badge, giving your answer to three significant figures.\\
(Total 16 marks)
\end{enumerate}

\hfill \mbox{\textit{Edexcel FP2 2004 Q9 [16]}}