Question

If the imaginary part of the expression
$\frac{z-1}{e^{i\theta }}+\frac{e^{i\theta }}{z-1}$ be zero, then locus of $z$ is :

• A. st. line
• B. parabola
• C. unit circle
• D. none

Answer 1

The correct option is C unit circle

Given $I.P.$ of $U+\frac{1}{U}=0$ where $U=\frac{z-1}{e^{i\theta }}$.
Now, $I.P.$ of $Z=0\Rightarrow Z-\overline{Z}=2iY=0$
$\therefore$ $(U+\frac{1}{U})-\left(\overline{U+\frac{1}{U}}\right)=0$
or $(U+\frac{1}{U})-(\overline{U}+\frac{1}{\overline{U}})=0$
or $(U-\overline{U})-(\frac{1}{U}-\frac{1}{\overline{U}})=0$
or $(U-\overline{U})-\frac{U-\overline{U}}{U\overline{U}}=0$
or $(U-\overline{U})-(1-\frac{1}{{\left|U\right|}^2})=0$
or ${\left|U\right|}^2=1$ as $U\ne \overline{U}$
${\left|\frac{z-1}{e^{i\theta }}\right|}^2=\frac{{|z-1|}^2}{1}=1$ as $\left|e^{i\theta }\right|=1$.
$\therefore |z-1|=1$ which represent a unit circle.