Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 11 / b / ii

The denominator requires $z\ne 0$. Write $z^2=u+iv$, where $u=x^2-y^2$ and $v=2xy$. Multiplication by the nonzero denominator givesso $u=0$. Conversely, if $u=0$ and $z\ne 0$, then $v\ne 0$ and $v/(iv)=-i$. Thus the required Argand diagram consists exactly ofThese are two perpendicular straight lines, with the origin removed as shown in the figure in part (i).