A complex number z is said to be unimodular, if $| z | = 1.$ If $z_{1} and z_{2}$ are complex numbers such that $\frac{z_{1} - 2 z_{2}}{2 - z_{1} \bar{z_{2}}} is$ unimodular and Z₂ is not unimodular. Then the point Z₁ lies on a

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straight line parallel to Y – axis

straight line parallel to X – axis

circle of radius 2

circle of radius $\sqrt{2}$

answer is C.

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Detailed Solution

$\begin{array}{l} |\frac{z_{1} - 2 z_{2}}{2 - z_{1} \bar{z_{2}}}| = 1 \Rightarrow {|z_{1} - 2 z_{2}|}^{2} = {|2 - z_{1} \bar{z_{2}}|}^{2} \\ \Rightarrow (z_{1} - 2 z_{2}) (\bar{z} - 2 z_{2}) = (2 - z_{1} \bar{z_{2}}) (2 - \bar{z_{1}} z_{2}) \\ = 4 + {|z_{1}|}^{2} {|z_{2}|}^{2} - 2 \bar{z_{1}} z_{2} - 2 z_{1} \bar{z_{2}} \Rightarrow ({|z_{2}|}^{2} - 1) ({|z_{1}|}^{2} - 4) = 0 \end{array}$