jmc

We know that the point $(a,b)$ lies on an asymptote, as shown below.



Since the angle between the asymptotes is $60^{\circ },$ $a$ is the long leg of a $30^{\circ }$-$60^{\circ }$-$90^{\circ }$ triangle, and $b$ is the short leg. Thus, $\frac{a}{b}=\boxed{\sqrt{3}}.$