If $\omega=\frac{z}{z-\frac{1}{3} i}$ and $|\omega|=1$, then $z$ lies on
If $\omega=\frac{z}{z-\frac{1}{3} i}$ and $|\omega|=1$, then $z$ lies on
an ellipse
a circle
a straight line
a parabola.
Solution
As given $w=\frac{z}{z-\frac{1}{3} i} \Rightarrow|w|=\frac{|z|}{\left|z-\frac{1}{3} i\right|}=1 \Rightarrow$ distance of $z$ from origin and point
$\left(0, \frac{1}{3}\right)$ is same hence $z$ lies on bisector of the line joining points $(0,0)$ and $(0,1 / 3)$.
Hence $z$ lies on a straight line.