Complex number loci on Argand diagrams

A question is this type if and only if it asks to sketch or describe loci such as |z - a| = r, arg(z - a) = θ, or Re(z) = k on an Argand diagram, possibly finding intersections.

1 questions

CAIE P3 2015 November Q9
9
  1. It is given that \(( 1 + 3 \mathrm { i } ) w = 2 + 4 \mathrm { i }\). Showing all necessary working, prove that the exact value of \(\left| w ^ { 2 } \right|\) is 2 and find \(\arg \left( w ^ { 2 } \right)\) correct to 3 significant figures.
  2. On a single Argand diagram sketch the loci \(| z | = 5\) and \(| z - 5 | = | z |\). Hence determine the complex numbers represented by points common to both loci, giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).