Question

The locus of the point $z$ satisfying the condition $arg\frac{z-1}{z+1}=\frac{\pi }{3}$ is

• A. a straight line
• B. circle
• C. a parabola
• D. none of these

Answer 1

The correct option is B circle

We have, $arg\frac{z-1}{z+1}=\frac{\pi }{3}\Rightarrow arg\frac{x+iy-1}{x+iy+1}=\frac{\pi }{3}$ $[$ Putting $z=x+iy]$

$\Rightarrow {\tan }^{-1}\frac{y}{x-1}-{\tan }^{-1}\frac{y}{x+1}=\frac{\pi }{3}(\because arg\frac{z_1}{z_2}=arg{z}_1-arg{z}_2)$

$\Rightarrow {\tan }^{-1}\left(\frac{\frac{y}{x-1}-\frac{y}{x+1}}{1+\frac{y^2}{x^2-1}}\right)=\frac{\pi }{3}$

$\Rightarrow \frac{2y}{x^2+y^2-1}=\tan \frac{\pi }{3}=\sqrt{3}$

$\Rightarrow x^2+y^2-\frac{2}{\sqrt{3}}y-1=0$ which is a circle.