Let   $S_1:x^2-10x+y^2+16=0$ and  $S_2$ be image of $S_1$  w.r.t line $x-y=0$  and  $S_3$ be image of  $S_2$ w. r. t.  $y=0.$ If S’ be the circle which cuts all three circles orthogonally, then :

$\therefore$  Centre of $S_1=(5,0)$  and radius  $r_1=3$
$\therefore$  Centre of  $S_2=(0,5)$ and radius  $r_2=3$
And  Centre of  $S_3=(0,-5)$ and radius  $r_2=3$
$\therefore$  Radical centre of  $S_1,S_2$ and $S_3$  will be $(0,0)$  Length of tangent from (0, 0) upon  $S_1$ or $S_2$  or   $S_3=4$
$\therefore$ Equation of S’ will be $\Rightarrow x^2+y^2=16$  and radius =4