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 OA is parallel to x -axis and A lies on…

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