5 The variable complex number \(z\) is given by
$$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$
where \(\theta\) takes all values in the interval \(- \pi < \theta \leqslant \pi\).
- Show that \(| z - \mathrm { i } | = 2\), for all values of \(\theta\). Hence sketch, in an Argand diagram, the locus of the point representing \(z\).
- Prove that the real part of \(\frac { 1 } { z + 2 - \mathrm { i } }\) is constant for \(- \pi < \theta < \pi\).