jmc

Dividing by $36,$ we have the standard form of the ellipse: $$\frac{(x-1{)}^2}{2^2}+\frac{y^2}{6^2}=1.$$Therefore, the distance from the center of the ellipse to $A$ is $6,$ and the distance from the center of the ellipse to $B$ is $2.$ Since the major and minor axes are perpendicular, by the Pythagorean theorem, $$AB=\sqrt{6^2+2^2}=\boxed{2\sqrt{10}}.$$