Question

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

• A. an ellipse
• B. a parabola
• C. a circle
• D. a straight line

Answer 1

Answer: (C)

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