9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{77d00a35-e947-41ef-8d80-5a573702ed39-14_643_1274_251_342}
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\caption{Figure 1}
\end{figure}
Figure 1 shows the curve \(C _ { 1 }\) with polar equation \(r = 2 a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and the circle \(C _ { 2 }\) with polar equation \(r = a , 0 \leqslant \theta \leqslant 2 \pi\), where \(a\) is a positive constant.
- Find, in terms of \(a\), the polar coordinates of the points where the curve \(C _ { 1 }\) meets the circle \(C _ { 2 }\)
The regions enclosed by the curve \(C _ { 1 }\) and the circle \(C _ { 2 }\) overlap and the common region \(R\) is shaded in Figure 1.
- Find the area of the shaded region \(R\), giving your answer in the form \(\frac { 1 } { 12 } a ^ { 2 } ( p \pi + q \sqrt { 3 } )\), where \(p\) and \(q\) are integers to be found.