Argument of fraction of complex numbers is constant How can I find all  $z \in \Bbb C$ such that:
$$ \arg\left(\frac{z-a}{z-b}\right) = \text{constant value}  $$
Where $a,b \in \mathbb{C}  $ are constants.

Write $\frac{z-a}{z-b} =  re^{\alpha i}$ ($r,\alpha$ a real numbers) and we have $\text{arg}\left(\frac{z-a}{z-b}\right) = \alpha =$ constant, hence $$z = \frac{b re^{\alpha i} - a}{re^{\alpha i} -1},$$ where $\alpha$ is constant and $r$ is variable.
