Question

If z1 and z2 both satisfy z + Bar z = 2| z −1| arg (z1 – z2) = π/4, then find Im (z1 + z2).

Answer 1

Let z = x + iy, z1 = x1 + iy1 and z2 = x2 + iy2.

Since z1 and z2 both satisfy (1), we have

Therefore,

From (2), we get 2 = y1 + y2, i.e., Im (z1 + z2) = 2