Question

The locus represented by the complex equation $|z-2-i|=|z|\sin \left(\frac{\pi }{4}-\arg z\right)$ is the part of

• A. a pair of straight lines
• B. a circle
• C. a parabola
• D. a rectangular hyperbola

Answer 1

Answer: (C)

Let $z=x+iy=r(\cos \theta +i\sin \theta )$, 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 $x-y=0.$