Question

If $|z-2-i|=|z|\left|\sin (\frac{\pi }{4}-\arg (z\left)\right)\right|$, where $z$ is in first quadrant, then the locus of $z$ is

• A. a ray.
• B. a part of a circle.
• C. a part of a parabola.
• D. a part of an ellipse.

Answer 1

The correct option is C a part of a parabola.

Given : $|z-2-i|=|z|\left|\sin (\frac{\pi }{4}-\arg (z\left)\right)\right|$

Let $z=x+iy$ and $\arg \left(z\right)=\theta \Rightarrow \tan \theta =\frac{y}{x}$, so
$|x-2+i(y-1)|=|z|\left|\sin (\frac{\pi }{4}-\arg (z\left)\right)\right|\Rightarrow \sqrt{(x-2{)}^2+(y-1{)}^2}=\sqrt{x^2+y^2}\times \frac{|x-y|}{\sqrt{2}\sqrt{x^2+y^2}}\Rightarrow \sqrt{(x-2{)}^2+(y-1{)}^2}=\frac{|x-y|}{\sqrt{2}}$

This represents a parabola, whose focus is $(2,1)$ and equation of directrix is $y=x$

Hence, the locus of $z$ is a part of a parabola.