Explanation

(a)
Given, $\left| \frac{1 - iz}{z - i} \right| = 1$
$\Rightarrow \ \ \ \ \ \left| \frac{1 - i(x + iy)}{x + iy - i} \right| = 1\ \ \ \ \ \ \Rightarrow \ \ \ \ \left| \frac{(1 + y) - ix}{x + i(y - 1)} \right| = 1$
$\Rightarrow \ \ \ \ \ \sqrt{(1 + y)^{2} + x^{2}} = \sqrt{x^{2} + (y - 1)^{2}}$
$\Rightarrow \ \ \ \ (1 + y)^{2} + x^{2} = x^{2} + (y - 1)^{2}$
$\Rightarrow \ \ \ y = 0$
$\therefore$ Locus of $z$ is $x$-axis