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If $\omega=\frac{z}{z-\frac{1}{3} i}$ and $|\omega|=1$, then $z$ lies on
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The correct answer is:
a straight line
a straight line
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.
$\left(0, \frac{1}{3}\right)$ 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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