Bro points

Consider a circle of radius $r$ with centre $O$ . A point $A$ lies inside the circle. Point $B$ lies outside the circle. $B$ is called bro of $A$ if the following relation holds:

1. $O$ , $A$ , and $B$ are collinear.
2. The tangent from $B$ touches the circle at $C$ and $\angle{BAC} = 90^\circ$ .

Now if $A$ moves along a circle that passes through $O$ and has a diameter $d$ such that $d=\frac{r}{4}$ . Assuming $B$ is always bro of $A$ , what is the distance between $A$ and $B$ when $B$ is closest to $O$ ?

If the answer is $\dfrac ab r$ , where $a$ and $b$ are positive coprime integers, enter $a+b$ as your answer.