How to describe the locus of $z$ where $\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta$.

Question

I need to describe the locus of $z$ which stisfies $\displaystyle\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta$.

$\displaystyle\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta \Rightarrow \arg(z-z_1)-\arg(z-z_2)=\theta$

So I got that ,

$\arg(z-z_1)=\arg(z-z_2)+\theta$

So $z$ lies on an arc.

If $0<\theta \leq\frac{\pi}{2}$ , $z$ lies on the major arc of a circle which passes through $z_1$ and $z_2.$ But there are two such circles ! How can we identify the correct circle ?

If $\frac{\pi}{2}<\theta < \pi$ , $z$ lies on the major arc of a circle which passes through $z_1$ and $z_2.$ But there are two such circles ! How can we identify the correct circle ?

Answer 1

Since $\theta$ is a given real number, the arc is oriented. Only one of two possible circles has the right orientation of the angle from $z_1$ via $z$ to $z_2.$