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 [JEE Main 2024, 29 Jan (Shift 2)]

Question

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 [JEE Main 2024, 29 Jan (Shift 2)]

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Let \(x=\frac{m}{n}\) ( \(m, n\) are co-prime natural numbers) be a solution of the equation \(\cos \left(2 \sin ^{-1} x…

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