Inverse Trigonometric Functions

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