IAL 2022 June Q6

The curve shown in Figure 1 has polar equation

where $a$ is a positive constant.

The tangent to the curve at the point $A$ is parallel to the initial line.

(a) Show that the polar coordinates of $A$ are $\left( 6a, \frac{\pi}{3} \right)$

The point $B$ lies on the curve such that angle $AOB = \frac{\pi}{6}$

The finite region $R$ , shown shaded in Figure 1, is bounded by the line $AB$ and the curve.

(b) 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.