# Complex Loci with Arguments

I need to find the locus of points (on an Argand diagram) such that:

(i) $$\arg(z-(-1-4i)) + \arg(z-(5+8i)) =0$$

(ii) $$\arg(z-(-1-4i)) + \arg(z-(5+8i)) = \pi/2$$

I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.

I am aware that $$\arg(z-(-1-4i)) - \arg(z-(5+8i)) = \pi/2$$ is a semicircle, and for other angles, say $$\pi/3$$ or $$\pi/4$$, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).

I am also interested in whether problems (i) and (ii) can be generalised to any angle between $$0$$ to $$\pi$$. Any help here would be greatly appreciated.

Denote the points $$A(-1-4i),\; B(5+8i),\; M(z).$$
(i) is equivalent to $$\arg(z-(-1-4i)) =- \arg(z-(5+8i))$$ This signifies that the direction from $$A$$ towards $$M$$ is opposite to the one from $$B$$ towards $$M.$$ The locus of points $$M(z)$$ is the segment $$AB$$ except $$A,B.$$