8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{709ed2f1-f81c-4820-ac31-e1f86baae9d7-28_552_759_246_660}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C\) with equation
$$r = 6 ( 1 + \cos \theta ) \quad 0 \leqslant \theta \leqslant \pi$$
Given that \(C\) meets the initial line at the point \(A\), as shown in Figure 1,
- write down the polar coordinates of \(A\).
The line \(l _ { 1 }\), also shown in Figure 1, is the tangent to \(C\) at the point \(B\) and is parallel to the initial line.
- Use calculus to determine the polar coordinates of \(B\).
The line \(l _ { 2 }\), also shown in Figure 1, is the tangent to \(C\) at \(A\) and is perpendicular to the initial line.
The region \(R\), shown shaded in Figure 1, is bounded by \(C , l _ { 1 }\) and \(l _ { 2 }\)
- Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(p \sqrt { 3 } + q \pi\) where \(p\) and \(q\) are constants to be determined.