Question

A complex number $z$ is such that $arg\left(\frac{z-2}{z+2}\right)=\frac{\pi }{3}$. The points representing this complex number will lie on

• A. An ellipse
• B. A parabola
• C. A circle
• D. A straight line

Answer 1

The correct option is C A circle

$arg\left(\frac{z-2}{z+2}\right)=\frac{\pi }{3}$
$\Rightarrow arg\left(\frac{x-2+iy}{x+2+iy}\right)=\frac{\pi }{3}$
$\Rightarrow arg(x-2+iy)-arg(x+2+iy)=\frac{\pi }{3}$
$\Rightarrow {\tan }^{-1}\left(\frac{y}{x-2}\right)-{\tan }^{-1}\left(\frac{y}{x+2}\right)=\frac{\pi }{3}$
$\Rightarrow \frac{4y}{x^2+y^2-4}=\sqrt{3}$
$\Rightarrow \sqrt{3}(x^2+y^2)-4y-4\sqrt{3}=0$
which is an equation of a circle.