IAL 2021 June Q3 | 國際數學站

The curve $C$ , with pole $O$ , has polar equation

\begin{align*} r = 1 + \cos\theta, \qquad 0 \leqslant \theta \leqslant \frac{\pi}{2} \end{align*}

At the point $A$ on $C$ , the tangent to $C$ is parallel to the initial line.

(a) Find the polar coordinates of $A$ .

(4)

(b) Find the finite area enclosed by the initial line, the line $OA$ and the curve $C$ , giving your answer in the form $a\pi + b\sqrt{3}$ , where $a$ and $b$ are rational constants to be found.

(6)