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

1) an ellipse
2) a circle
3) a straight line
4) a parabola
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(3)

Explanation

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 $\mathrm{z}$ 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.

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