Solution
= Solution
The denominator requires $z\ne0$. Write $z^2=u+iv$, where $u=x^2-y^2$ and $v=2xy$. Multiplication by the nonzero denominator gives
$$
v=-i(u+iv)=v-iu,
$$
so $u=0$. Conversely, if $u=0$ and $z\ne0$, then $v\ne0$ and $v/(iv)=-i$. Thus the required <Argand diagram> consists exactly of
$$
\boxed{y=x\quad\text{or}\quad y=-x,\qquad (x,y)\ne(0,0).}
$$
These are two perpendicular <straight lines>, with the origin removed as shown in the figure in part (i).