What will be the graph of the straight line Arg (z-a)= $\pi$ + $\theta$.

Question

I am not able of gaining proper geometry of the above equation, where a=m+ $\iota$ n, where m & n both are positive.

Answer 1

The number $a\in \mathbb{C}$ represents a point $A$ in the first quadrant, and we are looking for the points $M(z)$ where $z$ verifies $Arg(z-a)=\pi+\theta.$

In other words, the oriented angle of vectors $\vec{OA}$ and $\vec{OM}$ has the measure $\pi+\theta.$ Therefore, the set of points $M$ is the open half-line starting at $O$ that creates an angle $\pi+\theta$ (or, better, $-\theta$) with the half line $OA.$