Question

If $\omega =\frac{z}{z-\frac{1}{3}i}$ and $|\omega |=1$, then z lies on

• A. A straight line
• B. A parabola
• C. An ellipse
• D. A circle

Answer 1

The correct option is A A straight line

$\omega =\frac{2}{z-\frac{1}{3}}$ applying mod on both sides
$1=\frac{|z|}{|z-\frac{i}{3}|}$
$\Rightarrow \sqrt{x^2+(y-{\frac{1}{3}}^2)}=\sqrt{x^2+y^2}$
$\Rightarrow x^2+y^2-\frac{2y}{3}+\frac{1}{9}=x^2+y^2$
$\Rightarrow \frac{2y}{3}=\frac{1}{9}\Rightarrow y=\frac{1}{6}\to$ straight line