Find the length of $AB$

Let the radius of the largest circle be $r$ and the distance between the feet of perpendiculars from $O_1$ and $O_2$ to the line $\overline {AB}$ be $p$ . Then $(r-6)^2+p^2=(r-4)^2$ or $p^2=4r-20$ . Also, $(4+5)^2=p^2+(5-4)^2$ or $p^2=80$ . Therefore $4r-20=80$ or $r=25$ . Hence $|\overline {AB}|=2\sqrt {25^2-15^2}=\boxed {40}$ .