Question

$z\ne 1and\frac{z^2}{z-1}$ is real, then the point represented by the complex number z lies

• A. either on the real axis or on a circle passing through the origin.
• B. on a circle with centre at the origin.
• C. either on the real axis or on a circle not passing through the origin.
• D. on the imaginary axis.

Answer 1

The correct option is A either on the real axis or on a circle passing through the origin.

$\frac{z^2}{z-1}ispurelyreal\therefore \frac{z^2}{z-1}=\frac{{\overline{z}}^2}{\overline{z}-1}\Rightarrow z\overline{z}z-z^2=z\overline{z}\overline{z}-{\overline{z}}^2\Rightarrow |\overline{z}{|}^2(z-\overline{z})-(z-\overline{z})(z+\overline{z})=0\Rightarrow (z-\overline{z})(|z{|}^2-(z+\overline{z}\left)\right)=0Eitherz=\overline{z}\Rightarrow zliesonrealaxisor|z{|}^2=z+\overline{z}\Rightarrow z\overline{z}-z-\overline{z}=0\Rightarrow x^2+y^2-2x=0$
Which represents a circle passing through origin.