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}{\left|t\right|^{2}}\right) = 0 $
$ \therefore 1 - \frac{1}{\left|t\right|^{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+ i sin \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