IAL 2022 Jan Q7 | 國際數學站

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

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

The transformation maps points on the imaginary axis in the $z$ -plane onto a line in the $w$ -plane.

(a) Find an equation for this line.

(2)

The transformation maps points on the real axis in the $z$ -plane onto a circle in the $w$ -plane.

(b) Find the centre and radius of this circle.

(6)