IAL 2025 June Q3 | 國際數學站

A complex number $z$ is represented by the point $P$ on an Argand diagram where $|z| = 1$

(a) Sketch the locus of $P$ as $z$ varies.

(1)

The transformation $T$ from the $z$ -plane, where $z = x + \mathrm{i}y$ , to the $w$ -plane, where $w = u + \mathrm{i}v$ , is given by

\begin{align*} w = \frac{9\mathrm{i}z - \mathrm{i}}{z + 1} \qquad z \neq -1 \end{align*}

Given that the image under $T$ of the locus of $P$ in the $z$ -plane, where $z \neq -1$ , is the line $l$ in the $w$ -plane,

(b) determine, in simplest form, a Cartesian equation for $l$

(5)

解答

(a)

The locus of $P$ is a circle centered at the origin $(0,0)$ with a radius of $1$ . (Draw a circle on the Argand diagram with center $O$ passing through $1, \mathrm{i}, -1, -\mathrm{i}$ )

(b)

\begin{align*} w = &\,\frac{9\mathrm{i}z - \mathrm{i}}{z + 1}\\[4mm] w(z + 1) = &\,9\mathrm{i}z - \mathrm{i}\\[4mm] wz + w = &\,9\mathrm{i}z - \mathrm{i}\\[4mm] z(w - 9\mathrm{i}) = &\,-w - \mathrm{i}\\[4mm] z = &\,\frac{-w - \mathrm{i}}{w - 9\mathrm{i}} \end{align*}

Since the locus is $|z| = 1$ :

\begin{align*} \left|\frac{-w - \mathrm{i}}{w - 9\mathrm{i}}\right| = &\,1\\[4mm] |-w - \mathrm{i}| = &\,|w - 9\mathrm{i}|\\[4mm] |w + \mathrm{i}| = &\,|w - 9\mathrm{i}| \end{align*}

Let $w = u + \mathrm{i}v$ :

\begin{align*} |u + \mathrm{i}(v + 1)| = &\,|u + \mathrm{i}(v - 9)|\\[4mm] u^2 + (v + 1)^2 = &\,u^2 + (v - 9)^2\\[4mm] v^2 + 2v + 1 = &\,v^2 - 18v + 81\\[4mm] 20v = &\,80\\[4mm] v = &\,4 \end{align*}

Thus, the Cartesian equation for $l$ is $v = 4$ (or $y = 4$ ).