What will be the locus of $|z-2-i|=|z||\sin(\frac{\pi}{4}-\arg z)|$?

Question

I am looking for alternative methods for this question. I tried taking $z=x+iy$ and then squared both sides and got the locus as a parabola, but since complex numbers carry a lot of beautiful methods, I want to look alternative methods into this. Is there any other way to reach the same answer?

Answer 1

Thanks to Dustan Levenstein for giving me a hint in the comments.

$|z-2-i|=|z||\sin(\frac{\pi}{4}-\arg z)|=|z||\sin(\arg z-\frac{\pi}{4})|=|z||\operatorname{Im}(\frac{z}{1+i})||\frac{1+i}{z}|=|(1+i)(\operatorname{Im}(\frac{z}{1+i}))|=|\operatorname{Im}(z)i|=|z-\operatorname{Re}(z)|$

$|z-2-i|$ gives the distance of the point $z$ from focus $(2,1)$ while $|z-\operatorname{Re}(z)|$ gives the distance of the point $z$ from directrix $y=0$.

Hence the locus will be a parabola.