IAL 2021 Oct Q8 | 國際數學站

Figure 1

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

\begin{align*} r =\,& 1 + \sin \theta \qquad -\frac{\pi}{2} < \theta \leqslant \frac{\pi}{2}\\[2mm] \end{align*}

The point $P$ lies on $C$ such that the tangent to $C$ at $P$ is perpendicular to the initial line.

(a) Use calculus to determine the polar coordinates of $P$ .

(5)

The tangent to $C$ at the point $Q$ where $\theta = \frac{\pi}{2}$ is parallel to the initial line.

The tangent to $C$ at $Q$ meets the tangent to $C$ at $P$ at the point $S$ , as shown in Figure 1.

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

(b) Use algebraic integration to show that the area of $R$ is

\begin{align*} \frac{1}{32} (a\sqrt{3} + b\pi)\\[2mm] \end{align*}

where $a$ and $b$ are integers to be determined.

(6)