If `z_1` and `z_2` are two complex numbers such that `(z_1-2z_2)/(2-z_1bar(z_2))` is unimodular whereas `z_1` is not unimodular then `|z_1|`=

Options

A. Straight line parallel to x-axis
B. sraight line parallel to y-axis
C. circle of radius 2
D. circle of radius `sqrt(2)`

Answer

C

Explanation

`|z_(1)|= 1, |z_(2)| ne 1`
Given `|(z_(1)-2z_(2))/(2-z_(1)barz_(2))|=1`
`therefore |z_(1) -2z_(2)|^(2)= |2-z_(1)barz_(1)|^(2)`
`rArr |z_(1)|^(2) + 4 |z_(2)|^(2)-2barz_(1)z_(2) - 2z_(1)barz_(2) = 4 + |z_(1)|^(2) |z_(2)|^(2)-2barz_(1)z_(2) - 2z_(1)barz_(2)`
`rArr (|z_(2)|^(2)-1)(|z_(1)|^(2) - 4)=0`
`becuase |z_(1)| ne 1`
`therefore |Z_(2)| = 2`