A complex number z is said to be unimodular if |z|=1. Suppose z1 and z2 are complex numbers such that z1z2 - z1*z2 is unimodular and z2 is not unimodular. Then the point z1 lies on a straight line parallel to x-axis, straight line parallel to y-axis, circle of radius √2, or circle of radius 2.

|z1z2 - z1z2| = 1 => |z1z2|^2 = |2 - z1z2|^2
Using the property, |a|^2 = a x a*, => (z1z2)(z1z2) = (2 - z1z2)(2 - z1z2)
=> |z1|^2 + 4|z2|^2 - z1z2 - z1z2 = 4 - z1z2 - z1z2 + |z1|^2|z2|^2
=> |z1|^2 + 4|z2|^2 - |z1|^2|z2|^2 = 4
=> |z1|^2(1 - |z2|^2) = 4(1 - |z2|^2)
=> (|z1|^2 - 4)(1 - |z2|^2) = 0
=> |z1|^2 = 4 or |z2|^2 = 1
Since z2 is not unimodular, |z2| ≠ 1
Therefore, |z1|^2 = 4 => |z1| = 2
Clearly this is locus of circle with radius 2