IAL 2022 Oct Q7 | 國際數學站

Figure 1

The curve $C$ , shown in Figure 1, has polar equation

\begin{align*} r =\,& 2a(1+\cos \theta) \qquad 0 \leqslant \theta \leqslant \pi\\[2mm] \end{align*}

where $a$ is a positive constant.

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

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

(6)

The point $B$ on the curve has polar coordinates

\begin{align*} \left( a(2+\sqrt{3}), \frac{\pi}{6} \right)\\[2mm] \end{align*}

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

(b) Use calculus to determine the exact area of the shaded region $R$ .

Give your answer in the form

\begin{align*} \frac{a^2}{4}(d\pi - e + f\sqrt{3})\\[2mm] \end{align*}

where $d$ , $e$ and $f$ are integers.

(7)