Explanation

(a)
Given, $\left| \frac{z - 2i}{z + 2i} \right| = 1$
$\Rightarrow \ \ \ \ \ \ \ \left| \frac{x + iy - 2i}{x + iy + 2i} \right| = 1$
$\Rightarrow \ \ \ \ \ \ \ \ \ \sqrt{x^{2} + (y - 2)^{2}} = \sqrt{x^{2} + (y + 2)^{2}}$
$\Rightarrow \ \ \ \ \ \ \ \ \ x^{2} + y^{2} + 4 - 4y = x^{2} + y^{2} + 4 + 4y$
$\Rightarrow \ \ \ \ \ \ y = 0$
Thus, the locus of $z$ is $x$-axis