If z≠1, z2z-1 is real, then point represented by the complex number z lies

As z2z-1 is real, we get            z2z−1=z¯2z¯−1⇔z2(z¯−1)=z¯2(z−1)⇔zz¯(z−z¯)−(z−z¯)(z+z¯)=0⇔(z−z¯)(zz¯−z−z¯)=0⇒z−z¯=0 or zz¯−z−z¯=0⇒  z lies on the real axis or z lies on a circle through the origin.