Circle

Let circle $C$ be the image of $x^2+y^2-2 x+4 y-4=0$ in the line $2 x-3 y+5=0$ and $A$ be the point on $C$ such that $O A$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta<4$, lies on $C$ such that the length of the arc $A B$ is $(1 / 6)^{\text {th }}$ of the perimeter of $C$, then $\beta-\sqrt{3} \alpha$ is equal to