8. In the Argand diagram the point \(P\) represents the complex number \(z\).
Given that arg \(\left( \frac { z - 2 \mathrm { i } } { z + 2 } \right) = \frac { \pi } { 2 }\),
- sketch the locus of \(P\),
- deduce the value of \(| \mathrm { z } + 1 - \mathrm { i } |\).
The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is defined by
$$w = \frac { 2 ( 1 + \mathrm { i } ) } { z + 2 } , \quad z \neq - 2$$
- Show that the locus of \(P\) in the \(z\)-plane is mapped to part of a straight line in the \(w\)-plane, and show this in an Argand diagram.
(6)(Total 12 marks)