8 Two complex numbers are given by \(\alpha = 2 - \mathrm { j }\) and \(\beta = - 1 + 2 \mathrm { j }\).
- Find \(\alpha + \beta , \alpha \beta\) and \(\frac { \alpha } { \beta }\) in the form \(a + b \mathrm { j }\), showing your working.
- Find the modulus of \(\alpha\), leaving your answer in surd form. Find also the argument of \(\alpha\).
- Sketch the locus \(| z - \alpha | = 2\) on an Argand diagram.
- On a separate Argand diagram, sketch the locus \(\arg ( z - \beta ) = \frac { 1 } { 4 } \pi\).