6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff9ff379-78d8-41c0-a177-ec346e359249-20_497_1196_260_520}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
The curve shown in Figure 1 has polar equation
$$r = 4 a ( 1 + \cos \theta ) \quad 0 \leqslant \theta < \pi$$
where \(a\) is a positive constant.
The tangent to the curve at the point \(A\) is parallel to the initial line.
- Show that the polar coordinates of \(A\) are \(\left( 6 a , \frac { \pi } { 3 } \right)\)
The point \(B\) lies on the curve such that angle \(A O B = \frac { \pi } { 6 }\)
The finite region \(R\), shown shaded in Figure 1, is bounded by the line \(A B\) and the curve. - Use calculus to determine the area of the shaded region \(R\), giving your answer in the form \(a ^ { 2 } ( n \pi + p \sqrt { 3 } + q )\), where \(n , p\) and \(q\) are integers.