What does $\arg(z-z_1)-\arg(z-z_2)=\phi$ represents.

Question

What does $\arg(z-z_1)-\arg(z-z_2)=\phi$ represents.

where $z$ is point in argand plane

My Doubt: I am attaching a Image for the doubt because I don't know How to draw in latex. I'll be grateful if you can tell me how to do that.

I am considering $z_1$ is right of $z_2$

I've drawn $4$ diagram for locus. Can you tell me Are the arrow marked for angle by me are correct?

Also you can give me more information If you have.

Also locus of $z$ will be full circle or half circle if $\phi$ is $\dfrac{\pi}{2}$. If it is full circle what will be arrow mark for angle.

Answer 1

I am adding My Own Answer

Here I'm Attaching two Images in which I've plotted Locus of $z$ because I don't know how to plot them Digitally. Arrow Must end at $(z-z_1)$ and must originate from $z-z_2$.

Also I am considering Principle Argument that is $(-\pi,\pi]$.