how to plot the set $M:= \{z \in \mathbb{C}; Im \left( \frac{z+i}{z-2} \right) = 0\}$ in the complex plane? $Im(z)$ means the imaginary part of a complex number.
I have defined $z \in \mathbb{C}: z = x+iy, \quad x,y \in \mathbb{R}$
$\begin{align} \frac{z+i}{z-2}=\frac{x+iy+i}{x+iy-2}&= \frac{[x+iy+i][x-2-iy]}{[x-2+iy][x-2-iy]}\\ &= \frac{x^2-2x-iyx+iyx-2iy-i^2y^2+ix-2i-i^2y}{x^2-2x-iyx-2x+4+2iy+iyx-2iy-i^2y^2} \\ &= \frac{x^2-2x-2iy+y^2+ix-2i+y}{x^2-4x+4+y^2}\\ &= \frac{x^2-2x+y^2+y-2iy+ix-2i}{x^2-4x+4+y^2} \\ \end{align}$
How should I go on ? And more important: How to draw that number?
$$\Im\frac{x^2-2x+y^2+y-2iy+ix-2i}{x^2-4x+4+y^2}=0\iff-2y+x-2=0$$