7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{17b48fd7-5e88-4a62-beb9-8590a363e70f-20_476_972_251_488}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The curve \(C\), shown in Figure 1, has polar equation
$$r = 2 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$
where \(a\) is a positive constant.
The tangent to \(C\) at the point \(A\) is parallel to the initial line.
- Determine the polar coordinates of \(A\).
The point \(B\) on the curve has polar coordinates \(\quad a ( 2 + \sqrt { 3 } ) , \frac { \pi } { 6 }\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\) and the line \(A B\).
- Use calculus to determine the exact area of the shaded region \(R\).
Give your answer in the form
$$\frac { a ^ { 2 } } { 4 } ( d \pi - e + f \sqrt { 3 } )$$
where \(d , e\) and \(f\) are integers.