Question

If $Im(z)$ of $\frac{e^{i\theta }}{z-1}+\frac{z-1}{e^{i\theta }}$ is zero, then $z$ lies on

• A. parabola
• B. hyperbola
• C. unit circle
• D. ellipse

Answer 1

Answer: (C)

Let $t=\frac{e^{i\theta }}{z-1}$
Given $Im(z)$ of $\left(t+\frac{1}{t}\right)=0$
$\therefore \left(t+\frac{1}{t}\right)-\left(\overline{t+\frac{1}{t}}\right)=0$
$\left(\because z=x+iy=x,\therefore \bar{z}=x\Rightarrow z-\bar{z}=0\right)$
$\Rightarrow \left(t-\bar{t}\right)+\left(\frac{1}{t}-\frac{1}{t}\right)=0$
$\Rightarrow \left(t-\bar{t}\right)+\frac{\bar{t}-t}{t\bar{t}}=0$
$\Rightarrow \left(t-\bar{t}\right)\left(1-\frac{1}{{|t|}^2}\right)=0$
$\therefore 1-\frac{1}{{|t|}^2}=0$ or $t=\bar{t}$
$\Rightarrow {\left|t\right|}^2=1\Rightarrow {\left|\frac{e^{i\theta }}{z-1}\right|}^2=1$
$\Rightarrow {\left|z-1\right|}^2=1$
$\left(\because \left|e^{i\theta }\right|=\left|cos\theta +isin\theta \right|=1\right)$
$\Rightarrow \left|z-1\right|=1$ or ${\left(x-1\right)}^2+y^2=1$
which represent a circle of unit radius