Let $x = \dfrac{m}{n}$ $(m, n$ are co-prime natural numbers$)$ be a solution of the equation $\cos(2\sin^{-1}x) = \dfrac{1}{9}$ and let $\alpha, \beta$ $(\alpha > \beta)$ be the roots of the equation $mx^2 - nx - m + n = 0$. Then the point $(\alpha, \beta)$ lies on the line: