IAL 2021 Oct Q6 | 國際數學站

The complex number $z$ on an Argand diagram is represented by the point $P$ where

\begin{align*} |z + 1 - 13\mathrm{i}| =\,& 3|z - 7 - 5\mathrm{i}|\\[2mm] \end{align*}

Given that the locus of $P$ is a circle,

(a) determine the centre and radius of this circle.

(5)

The complex number $w$ , on the same Argand diagram, is represented by the point $Q$ , where

\begin{align*} \arg(w - 8 - 6\mathrm{i}) =\,& -\frac{3\pi}{4}\\[2mm] \end{align*}

Given that the locus of $P$ intersects the locus of $Q$ at the point $R$ ,

(b) determine the complex number representing $R$ .

(4)