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Q.
The locus represented by the complex equation $|z-2-i|=|z| \sin \left(\frac{\pi}{4}-\arg z\right)$ is the part of
Complex Numbers and Quadratic Equations
Solution:
Let $z=x+i y=r(\cos \theta+i \sin \theta)$, then the equation is
$|(x-2)+i(y-1)|$
$=r\left(\frac{1}{\sqrt{2}} \cos \theta-\frac{1}{\sqrt{2}} \sin \theta\right)$
$=\frac{1}{\sqrt{2}}(r \cos \theta-r \sin \theta)$
or, $\sqrt{(x-2)^{2}+(y-1)^{2}}$
$=\frac{1}{\sqrt{2}}(x-y)$
which is the part of a parabola with focus $(2,1)$ and directrix $x-y=0 .$