Problem source

The complex number $w$ is such that $w^{2}-2w$ is real. 
\begin{questionparts}
\item Sketch the locus of $w$ in the Argand diagram. 
\item If $w^{2}=x+\mathrm{i}y,$ describe fully and sketch the locus of
points $(x,y)$ in the $x$-$y$ plane. 
\end{questionparts}
The complex number $t$ is such that $t^{2}-2t$ is imaginary. If
$t^{2}=p+\mathrm{i}q$, sketch the locus of points $(p,q)$ in the
$p$-$q$ plane.

Solution source

\begin{questionparts}
\item Suppose $w = u+ vi$ then $w^2 - 2w = u^2-v^2-2u+(2uv-2v)i$ so to be purely real we must have $2uv-2v = 2v(u-1) = 0$ ie either $v = 0$ or $u = 1$. Therefore the locus is the real axis and the line $1 + ti$:

\begin{center}
    \begin{tikzpicture}[scale=1]
        \draw[->] (-3, 0) -- (3,0) node [right] {$\textrm{Re}$};
        \draw[->] (0, -3) -- (0,3) node [above] {$\textrm{Im}$};

        \draw[ultra thick, red] (-3,0) -- (3,0);
        \draw[ultra thick, red] (1,-3) -- (1,3);
        
    \end{tikzpicture}
\end{center}

\item If $w^2 = x+yi$ then we must have $x = u^2-v^2$ and $y = 2uv$, so either $v = 0, y = 0, x = u^2-2u \geq -1$ or $u = 1, x = 1-v^2, y = 2v$ which is a parabola:

\begin{center}
    \begin{tikzpicture}[scale=1]
        \draw[->] (-3, 0) -- (3,0) node [right] {${x}$};
        \draw[->] (0, -3) -- (0,3) node [above] {${y}$};

        \draw[ultra thick, red] (-1,0) -- (3,0);
        \draw[domain = -1.4:1.4, samples=50, variable = \x, red, ultra thick]  plot ({1-\x*\x},{2*\x});
        
    \end{tikzpicture}
\end{center}

\end{questionparts}

If $t = u+iv$ then $t^2-2t = u^2-v^2-2u + (2uv-2v)i$. For this to be purely imaginary, we need $u^2-v^2 - 2u = 0 \Rightarrow (u-1)^2-v^2 = 1$, ie points on a hyperbola.  Then $p = u^2-v^2$ and $q = 2uv$. We can parameterise our hyperbola  as $u = 1 \pm \cosh s, v = \sinh s$ and so $p = 1 + 2 \cosh s$ and $q = \sinh 2s$ or $q = \pm (p-1) \sqrt{(\frac{p-1}{2})^2-1}$ where $p \geq 3$


\begin{center}
    \begin{tikzpicture}[scale=1]
        \draw[->] (-3, 0) -- (3,0) node [right] {$p$};
        \draw[->] (0, -3) -- (0,3) node [above] {$q$};

        \draw[domain = -1:1, samples=50, variable = \x, red, ultra thick]  plot ({(1+cosh(\x))},{sinh(2*\x)});
        \draw[domain = -1:1, samples=50, variable = \x, red, ultra thick]  plot ({(1-cosh(\x))},{sinh(2*\x)});
    \end{tikzpicture}
\end{center}