Explanation

(c)
Let $z = x + iy$
$\therefore\ \ \ \ \left| \frac{z - 25}{z - 1} \right| = 5$
$\Rightarrow \ \ \left| \frac{(x - 25) + iy}{(x - 1) + iy} \right| = 5$
$\Rightarrow \ \ \ \left| (x - 25) + iy \right| = 5|(x - 1) + iy|$
$\Rightarrow \ \ \ \ \ \sqrt{(x - 25)^{2} + y^{2}} = 5\sqrt{(x - 1)^{2} + y^{2}}$
On squaring both sides, we get
$(x - 25)^{2} + y^{2} = 25\{(x - 1)^{2} + y^{2}\}$
$\Rightarrow \ \ \ \ \ \ x^{2} - 50x + 625 + y^{2} = 25x^{2} - 50x + 25 + 25y^{2}$
$\Rightarrow \ \ \ \ 24x^{2} + 24y^{2} = 600$
$\ \ \Rightarrow \ \ \ \ x^{2} + y^{2} = 25$
$\Rightarrow \ \ \ \ \sqrt{x^{2} + y^{2}} = 5\ \ \ \ \ \ \ \ \ \lbrack\because|z| = \sqrt{(x^{2} + y^{2})}\rbrack$
$\Rightarrow \ \ \ \ \ |z| = 5$