A complex number z is said to be unimodular if . Suppose `z_1` and `z_2` are complex numbers such that `(z_1-2z_2)/(2-z_1z_2)` is unimodular and `z_2` is not unimodular. Then the point `z_1` lies on a :

Options

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

Answer

C

Explanation

If is unimodular then |z|=1 also use property of modules I,e `zz=|z|^2`
Given `z_2` is not unimodular I,e `|z_2|ne 1` and `(z_1-2z_2)/(2-z_1barz_2)|` is unimodular
`rArr |(z_1-2z_2)/(2-z_1barz_2)|=1 rArr |z_1-2z_2|^2=|2-z_1barz_2|^2`
`rArr ( z_1-2z_2)(barz_1-2barz_2)=(2-z_1bar z_2)(2-barz_1z_2)" "[zbarz=|z|^2]`
`rArr |z|^2+4|^2-2barz_1z_2-2z_1bar z_2`
`rArr 4+|z_1|^2|z_2|^2-2barz_1z_2-2z_1barz_2rArr(|z_2 |^2-1)(|z_1|^2-4)=0`
`because " "|z_2|ne1`
`therefore |z_1|=2`
Let `z_1=x+iy rArr x^2+y^2=(2)^2`
`therefore`Point `z_1`lies on circle of radius