Question

If $z\ne 1$ and $\frac{z^2}{z-1}$ 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 B

on a circle with centre at the origin

Explanation for the correct option.

Step 1 : Find the value of $\frac{z^2}{z-1}$

Let $z=x+iy$, now complex $z^2$ is defined as:

$\begin{array}{rcl}{z}^2& =& {\left(x+iy\right)}^2\\ & =& x^2+2xyi+i^2{y}^2\\ & =& x^2+2xyi-y^2\left[i^2=-1\right]\\ & =& \left(x^2-y^2\right)+2xyi\end{array}$

Now, simplify the complex number $\frac{z^2}{z-1}$.

$\begin{array}{rcl}\frac{z^2}{z-1}& =& \frac{\left(x^2-y^2\right)+2xyi}{x+iy-1}\\ & =& \frac{\left(x^2-y^2\right)+2xyi}{(x-1)+iy}\\ & =& \frac{\left[\left(x^2-y^2\right)+2xyi\right]\left[\left(x-1\right)-iy\right]}{\left[(x-1)+iy\right]\left[(x-1)-iy\right]}\\ & =& \frac{\left(x^2-y^2\right)\left(x-1\right)-\left(x^2-y^2\right)iy+2xy\left(x-1\right)i-2x{y}^2{i}^2}{{\left(x-1\right)}^2-i^2{y}^2}\\ & =& \frac{\left(x^2-y^2\right)\left(x-1\right)-\left(x^{2}y-y^3\right)i+\left(2x^{2}y-2xy\right)i+2x{y}^2}{{\left(x-1\right)}^2+y^2}\left[i^2=-1\right]\\ & =& \frac{\left(x^2-y^2\right)\left(x-1\right)+2x{y}^2}{{\left(x-1\right)}^2+y^2}+\frac{\left(-x^{2}y+y^3+2x^{2}y-2xy\right)}{{\left(x-1\right)}^2+y^2}i\\ & =& \frac{\left(x^2-y^2\right)\left(x-1\right)+2x{y}^2}{{\left(x-1\right)}^2+y^2}+\frac{\left(y^3+x^{2}y-2xy\right)}{{\left(x-1\right)}^2+y^2}i\end{array}$

Step 2 : Find the locus of complex number $z$.

It is given that the complex number $\frac{z^2}{z-1}$ is real. So its imaginary part is zero. So,

$$\begin{gathered} \begin{array}{rcl}& & \frac{\left(y^3+x^{2}y-2xy\right)}{{\left(x-1\right)}^2+y^2}\end{array}=0 \\ \Rightarrow y^3+x^{2}y-2xy=0 \\ \Rightarrow y\left(x^2+y^2-2x\right)=0 \end{gathered}$$

So either $y=0$ which represents the real axis, or,

$x^2+y^2-2x=0$ which represents a circle which passes through origin.

Thus, the complex number $z$ lies either on the real axis or on a circle passing through the origin.

Hence, the correct option is A.