Question

Find the locus of a complex number $z$ such that arg $\left(\frac{z-2}{z+2}\right)=\frac{\pi }{3}$.

Answer 1

$arg\left(\frac{z-2}{z+2}\right)=\frac{\pi }{3}$

Let $z=x+iy$

$arg(z-2)-arg(z+2)=\frac{\pi }{3}$

$\frac{y}{x-2}-\frac{y}{x+2}=\tan \frac{\pi }{3}$

$y\frac{4}{x^2-4}=\sqrt{3}$

$4y=\sqrt{3}{x}^2-4\sqrt{3}$

$y=\frac{\sqrt{3}{x}^2}{4}-\sqrt{3}$

Locus is a parabola