9. A complex number \(z\) is represented by the point \(P\) in the Argand diagram. Given that
$$| z - 3 \mathrm { i } | = 3$$
- sketch the locus of \(P\).
- Find the complex number \(z\) which satisfies both \(| z - 3 i | = 3\) and \(\arg ( z - 3 i ) = \frac { 3 } { 4 } \pi\).
The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { 2 \mathrm { i } } { z }$$
- Show that \(T\) maps \(| z - 3 i | = 3\) to a line in the \(w\)-plane, and give the cartesian equation of this line.
(5)(Total 11 marks)