Explanation

(a)
Given, $\frac{x + iy - 5i}{x + iy + 5i} = 1$
$\Rightarrow \ \ \ \ \ |x + iy - 5i| = |x + iy + 5i|$
$\Rightarrow \ \ \ \ x^{2} + (y - 5)^{2} = x^{2} + (5 + y)^{2}\ \ \ \ \ \ \ \ \ \ \ \ \left\lbrack \because\ \ \ \ \left| \frac{z_{1}}{z_{2}} \right| = \frac{{|z}_{1}|}{|z_{2}|} \right\rbrack$
$\Rightarrow \ \ \ \ y = 0$
$\Rightarrow \ \ \ \ z = x\ ie$, then $z$ lies on the axis of $x$.