Past exam of the mathematical methods I course of the University of Oxford / 2015 resit / 1 / iii / Solution

For $z=x+iy$, the equation becomes $\sqrt{(x-2{)}^2+y^2}=x$. Its right side must be nonnegative. Squaring then gives $y^2=4(x-1)$, whose points already have $x\ge 1$, so no extraneous branch has been introduced:This is a parabola opening along the positive real axis, with vertex $(1,0)$, focus $(2,0)$ and directrix $x=0$. The original modulus equation equates distance from the focus with distance from that directrix.