Sketch given sets of complex numbers in the complex plane

For each of the following, make a sketch of all complex numbers $z$ in the complex plane which satisfy the given expression.

The complex numbers $z$ such that $|z| < 1$ are a solid disk of radius 1 centered at the origin:
We have
$\begin{align*} z + \overline{z} = 1 && \implies && x+ iy + x - iy &= 1 \\ && \implies && 2x &=1 \\ && \implies && x &= \frac{1}{2}. \end{align*}$

The plot in the complex plane is a vertical line at $x = \frac{1}{2}$ ,
We have
$\begin{align*} z - \overline{z} = i && \implies && x + iy - x + iy &= i \\ && \implies && y &= \frac{1}{2}. \end{align*}$

The plot in the complex plane is a horizontal line at $y = \frac{1}{2}$ .
We have
$\begin{align*} |z - 1| = |z + 1| && \implies && |x+iy -1| &= |x+iy + 1| \\ && \implies && |(x-1) + i y | &= |(x+1) + iy | \\ && \implies && \sqrt{(x-1)^2 + y^2} &= \sqrt{(x+1)^2 + y^2} \\ && \implies && x^2 - 2x + 1 + y^2 &= x^2 + 2x + 1 + y^2 \\ && \implies && 4x &= 0 \\ && \implies && x &= 0. \end{align*}$

The plot is a vertical line at $x = 0$ .
We have
$\begin{align*} |z - i| = |z + i| && \implies && |x + (y-1)i| &= |x + (y+1)i| \\ && \implies && \sqrt{x^2 + y^2 - 2y + 1} &= \sqrt{x^2 + y^2 + 2y + 1} \\ && \implies && x^2 + y^2 - 2y + 1 &= x^2 + y^2 + 2y + 1 \\ && \implies && y &= 0. \end{align*}$

The plot is a horizontal line at $y = 0$ .
We have
$\begin{align*} z + \overline{z} = |z|^2 && \implies && x + iy + x - iy &= x^2 + y^2 \\ && \implies && 2x &= x^2 + y^2 \\ && \implies && y = \pm \sqrt{x(x-2)}. \end{align*}$

The plot is a circle of radius one centered at $(1,0)$ ,

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