- A complex number \(z\) is represented by the point \(P\) in the complex plane.
Given that \(z\) satisfies
$$| z - 6 | = 2 | z + 3 i |$$
- show that the locus of \(P\) passes through the origin and the points - 4 and - 8 i
- Sketch on an Argand diagram the locus of \(P\) as \(z\) varies.
- On your sketch, shade the region which satisfies both
$$| z - 6 | \geqslant 2 | z + 3 i | \text { and } | z | \leqslant 4$$