The question is simply having to determine the locus (not the equation, just the conic) of $z\in\mathbb{C}$ where $z$ satisfies $|z-(2+i)|=|z|\sin\left(\frac{\pi}{4}-\arg(z)\right)$. One method is that involving brute force, writing $z=x+iy ,x,y\in\mathbb{R}-\{0\}$ and simplifying the expression.
Is there any other simpler way to go about this problem. Any hints are appreciated. Thanks.
$$|z-(2+i)|=|z|\sin(\pi/4-Arg z) \implies \sqrt{ (x-2)^2+(y-1)^2}=\frac{x+y}{\sqrt{2}}\implies PS=e PM, P(x,y)$$ Here $M$ is the foot of perpendicular at the directrix. This is a conic with one focus at S(2,1) and directrx as $(x+y)/\sqrt{2}$ and its excentricity $e= 1$. Hence it is a parabola.