A complex number z is said to be unimodular if `abs(z)=1`. Suppose `z_(1)` and `z_(2)` are complex numbers such that `(z_(1)-2z_(2))/(2-z_(1)z_(2)^_)` is unimodular and `z_(2)` is not unimodular. Then the point `z_(1)` lies on a

Options

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

Answer

A

Explanation

We have, `|(z_(1)-2z_(2))/(2-z_(1)barz_(2))|=1` and `|z_(2)| ne 1`.
Now, `|(z_(1)-2z_(2))/(2-z_(1)barz_(2))|=1` and `|z_(2)| ne 1`.
Now, `(|z_(1)-2z_(2)|)/(2-z_(1)barz_(2))|=1`
`rArr |z_(1)-2z_(2)|=|2-z_(1)barz_(2)|`
`rArr |z_(1)-2z_(2)|^(2)=|2-z_(1)barz_(2)|^(2)`
`rArr |z_(1)|^(2)+|2z_(2)|^(2)-2(z_(1)barz_(2))-2(barz_(1)z_(2))=4+|z_(1)|^(2)+|z_(2)|^(2)-2(barz_(1)z_(2))-2(z_(1)barz_(2))`
`rArr |z_(1)|^(2)+4|z_(2)|^(2)-4-|z_(1)|^(2)|z_(2)|^(2)=0`
`rArr |z_(1)|^(2)(1-|z_(2)^(2))-4(1-|z_(2)|^(2))=0`
`rArr |z_(1)|=2` `[therefore 1-|z_(2)|^(2) ne 0]`
`rArr z_(1)` lies on `|z|=2`, i.e., a circle of radius 2.