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) $\arg(z-(-1-4i)) + \arg(z-(5+8i)) =\frac{ \pi}{2}$
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 $\arg(z-(-1-4i)) - \arg(z-(5+8i)) = \frac{\pi}{2}$ is a semicircle, and for other angles, say $\frac{\pi}{3}$ or $\frac{\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 $A,B.$