Question

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

• A. A straight line parallel to x-axis
• B. A parabola
• C. A circle of radius 1
• D. None of these

Answer 1

Answer: (C)

A circle of radius 1

Answer 2

Sol. Let u = $\frac{z–1}{e^{\theta i}}$ Ž $\frac{e^{\theta i}}{z–1}$= $\frac{1}{4}$.

Now $\left( u + \frac{1}{u} \right)$– $\left( \bar{u} + \frac{1}{\bar{u}} \right)$ = 0

Ž (u –$\bar{u}$) $\left( 1–\frac{1}{u\bar{u}} \right)$ = 0

If u is not purely real, then $u\bar{u}$= 1

Ž $\left| \frac{z–1}{e^{\theta i}} \right|$= 1 Ž |z – 1| = 1