Given that
$\left|\frac{z-25}{z-1}\right|=5 \Rightarrow \left|z-25\right|=5\left|z-1\right|$
Let Z = x + iy, then
$\left|x+iy-25\right|=5\left|x+iy-1\right|$
$\Rightarrow \left|\left(x-25\right)+iy\right|=5\left|x-1+iy\right|$
Squaring both sides, we get
$\left(x-25\right)^{2}+y^{2}=25 \left\{\left(x-1\right)^{2}+y^{2}\right\}$
$\Rightarrow x^{2}-50x+625+y^{2} $
$\quad\quad\quad\quad\quad\quad=25x^{2} -50x+25+25y^{2}$
$\Rightarrow 24x^{2}+24 y^{2}-600=0$
$\Rightarrow x^{2}+y^{2}-25=0$
$\Rightarrow \left|x+iy\right|^{2}=25 \Rightarrow \left|Z\right|^{2}=5^{2}$
$\Rightarrow \left|Z\right|=5$