The circle x^2+y^2-4x-6y+9=0 touches

The centre is $(2,3)$ and $r^2=4+9-9=4$, so $r=2$. The distance of the centre from the $y$-axis is $2$, equal to $r$, so the circle touches the $y$-axis. Its distance from the $x$-axis is $3>2$, so it does not touch the $x$-axis. The two distances must be compared with the radius separately.