A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac{z}{z-i}, \quad z \neq i.$$
The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).
- Show that \(C\) is a circle and find its centre and radius. [8]
The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
- Shade the region \(R\) on an Argand diagram. [2]