IAL 2020 Oct Q5 | 國際數學站

The transformation $T$ from the $z$ -plane to the $w$ -plane is given by

\begin{align*} w = \frac{z - 3\mathrm{i}}{z + 2\mathrm{i}} \qquad z \neq -2\mathrm{i} \end{align*}

The circle with equation $|z| = 1$ in the $z$ -plane is mapped by $T$ onto the circle $C$ in the $w$ -plane.

Determine

(i) the centre of $C$ ,

(ii) the radius of $C$ .

(7)

解答

\begin{align*} w(z + 2\mathrm{i}) = &\,z - 3\mathrm{i}\\[4mm] z(w - 1) = &\,-\mathrm{i}(2w + 3)\\[4mm] z = &\,\frac{\mathrm{i}(2w + 3)}{1 - w} \end{align*}

Now let $w = u + \mathrm{i}v$ . Since $|z| = 1$ ,

\begin{align*} \left| \mathrm{i}(2w + 3) \right| = &\,|1 - w|\\[4mm] (2u + 3)^2 + 4v^2 = &\,(1 - u)^2 + v^2\\[4mm] 4u^2 + 4v^2 + 12u + 9 = &\,1 - 2u + u^2 + v^2\\[4mm] 3u^2 + 3v^2 + 14u + 8 = &\,0\\[4mm] u^2 + v^2 + \frac{14}{3}u + \frac{8}{3} = &\,0\\[4mm] \left(u + \frac{7}{3}\right)^2 + v^2 = &\,\frac{25}{9} \end{align*}

(i) Centre $\left(-\dfrac{7}{3}, 0\right)$

(ii) Radius $\dfrac{5}{3}$