If z=1 and z≠±1, then all the values of z1-z2 lie on:

As z=1, we get zz¯=1.Let w=z1-z2, thenw+w¯=z1−z2+z¯1−z¯2=z1−z2+1/z1−(1/z)2            =z1−z2+zz2−1=0⇒   2Re(w)=0   ⇒  Re(w)=0Thus, w lies on the y- axis.Alternate Solutionw=z1−z2=zzz¯−z2=1z¯−z=1−2iIm⁡(z)⇒ w is purely imaginary, that is, w lies on the y-axis.