- A 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
$$w = \frac { z - 3 } { 2 \mathrm { i } - z } \quad z \neq 2 \mathrm { i }$$
The line in the \(z\)-plane with equation \(y = x + 3\) is mapped by \(T\) onto a circle \(C\) in the \(w\)-plane.
- Determine
- the coordinates of the centre of \(C\)
- the exact radius of \(C\)
The region \(y > x + 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
- On a single Argand diagram
- sketch the circle \(C\)
- shade and label the region \(R\)