- The point \(P\) in the complex plane represents a complex number \(z\) such that
$$| z + 9 | = 4 | z - 12 i |$$
Given that, as \(z\) varies, the locus of \(P\) is a circle,
- determine the centre and radius of this circle.
- Shade on an Argand diagram the region defined by the set
$$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$