Explanation

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