Q97 / 225 · JEE Main 2005 · Mathematics

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

  • A
    an ellipse
  • B
    a circle
  • Correct answer
    C
    a straight line
  • D
    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.

Hence, the correct answer is option: 'C'.

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