Question 6 - A-Level Maths

A complex number \(z\) is represented by the point \(P\) in the Argand diagram.
1. Given that \(| z - 6 | = | z |\), sketch the locus of \(P\).
2. Find the complex numbers \(z\) which satisfy both \(| z - 6 | = | z |\) and \(| z - 3 - 4 \mathrm { i } | = 5\).
The transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \(w = \frac { 30 } { z }\).
Show that \(T\) maps \(| z - 6 | = | z |\) onto a circle in the \(w\)-plane and give the cartesian equation of this circle.