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