If two circles touching both the axes are passing through (2,3) then length of their common chord is

The centres of  circles touching both axes lies on the line y=x

and PQ is common chord of two circles touching  coordinate axes where P=(2,3)

Other point of intersection is the image of the given point w.r.t. the lie y=x

∴Q is the image of P on the line   y=x

i.e Q=(3,2)

$Now PQ=\sqrt{{(3-2)}^2+{(2-3)}^2}=\sqrt{2}$