Consider circles $C_1$ & $C_2$ touching both the axes and passing through $\left(\mathrm{4,4}\right)$, then the product of radii of these circles is

Since circle touches both the axes and passes through $\left(\mathrm{4,4}\right)$, it lies in the first quadrant, so its equation is
${\left(x-r\right)}^2+{\left(y-r\right)}^2=r^2$
$\left(\mathrm{4,4}\right)$ lies on it hence
$r^2-16r+32=0$
Hence, the product of roots (radii) $=32$