- The point \(P\) represents the complex number \(z\) on an Argand diagram, where
$$| z - \mathrm { i } | = 2$$
The locus of \(P\) as \(z\) varies is the curve \(C\).
- Find a cartesian equation of \(C\).
- Sketch the curve \(C\).
A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z + \mathrm { i } } { 3 + \mathrm { i } z } , \quad z \neq 3 \mathrm { i }$$
The point \(Q\) is mapped by \(T\) onto the point \(R\). Given that \(R\) lies on the real axis,
- show that \(Q\) lies on \(C\).