IAL 2021 Jan Q7 | 國際數學站

Figure 1

Figure 1 shows a sketch of curve $C$ with polar equation

\begin{align*} r = 3\sin 2\theta \qquad 0 \leqslant \theta \leqslant \frac{\pi}{2} \end{align*}

The point $P$ on $C$ has polar coordinates $(R,\phi)$ . The tangent to $C$ at $P$ is perpendicular to the initial line.

(a) Show that $\tan\phi = \frac{1}{\sqrt{2}}$

(4)

(b) Determine the exact value of $R$ .

(2)

The region $S$ , shown shaded in Figure 1, is bounded by $C$ and the line $OP$ , where $O$ is the pole.

\begin{align*} p\arctan\frac{1}{\sqrt{2}} + q\sqrt{2} \end{align*}

where $p$ and $q$ are constants to be determined.

Solutions relying entirely on calculator technology are not acceptable.

(7)