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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.

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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 $A,B.$

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