Question

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

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

Answer 1

Answer: (C)

a parabola

Let $z=x+iy=r\cos \theta +ri\sin \theta$

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 $\sqrt{{(x-2)}^2+{(y-1)}^2}=\frac{1}{\sqrt{2}}(x-y)$

Parabola