Past exam of the mathematical methods I course of the University of Oxford / 2016 resit / 1 / ii / Solution

Put $z=x+iy$ and $z^{\ast }=x-iy$. The first two squared terms are complex conjugates, so their sum is $2[(x-2{)}^2-(y+1{)}^2]$. Also $z{z}^{\ast }=x^2+y^2$. The equation therefore reduces toThus the locus is the upward-opening parabolawith vertex $1+2i$ and axis $x=1$ in the Argand diagram.