7.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{141c7b1b-4236-4433-84af-04fa9baa3d96-2_568_1431_1637_258}
\end{figure}
A logo is designed which consists of two overlapping closed curves.
The polar equations of these curves are \(r = \boldsymbol { a } ( \mathbf { 3 } + \mathbf { 2 } \cos \boldsymbol { \theta } )\) and
$$r = a ( 5 - 2 \cos \theta ) , \quad 0 \leq \theta < 2 \pi .$$
Figure 1 is a sketch (not to scale) of these two curves.
- Write down the polar corrdinates of the points \(A\) and \(B\) where the curves meet the initial line.(2)
- Find the polar coordinates of the points \(\boldsymbol { C }\) and \(\boldsymbol { D }\) where the two curves meet. (4)
- Show that the area of the overlapping region, which is shaded in the figure, is
$$\frac { a ^ { 2 } } { 3 } ( 49 \pi - 48 \sqrt { } 3 )$$