If z1 and z2  are two complex numbers such that z1−z2z1+z2=1, then

Note that at least one of z1,z2 is different from 0. Suppose z2≠0 we can writez1−z2z1+z2=1, as  z1/z2−1z1/z2+1=1 or as z1z2−1=z1z2+1This shows that z1/z2 lies on the perpendicular bisector of the segment joiningA(−1+i0) andB(1+i0) [See Theory].Thus, z1/z2 lies on the imaginary axis.∴ z1z2=ia for some a∈R. ⇒z2z1=1ia=−ia⇒ z2=ikz1 for some k∈R