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 : (1) straight line parallel to x-axis (2) straight line parallel to y-axis (3) circle of radius 2 (4) circle of radius `sqrt(2)`

Explanation

`|(z_1 - 2z_2)/(2 - z_1 bar (z_2))| = 1`

`|z_1 - 2 z_2| = |2 - z_1 bar (z_2)|`

squaring both sides

`(z_1 - 2z_2)(bar (z_1) - 2 bar (z_2))= ( 2 - z_1bar(z_2)) xx ( 2 - bar(z_1) z_2)`

`z_1 bar (z_2) - 2 z_2 bar(z_1) - 2z_1bar(z_2) + 4 z_2 bar(z_2) = 4 - 2 bar(z_1)z_2 - 2z_1 bar(z_2) + z_1bar(z_1) z_2 bar(z_2) `

`|z_1|^2 + 4|z_2|^2 = 4 + |z_1|^2|z_2|^2`

`|z_1|^2 - |z_1|^2|z_2|^2 + 4|z_2|^2 -4 = 0`

`(|z|^2 - 4)(1-|z_2|^2)= 0`

...