Let x = (m)/(n) ( m, n are co-prime natural numbers) be a solution...

Given: $\cos \left(2\sin^{- 1}x\right) = \dfrac{1}{9}$

Let, $\sin^{- 1}x = \theta$

$⇒ \cos \left(2\theta\right) = \dfrac{1}{9}$

$⇒ 1 - 2\sin^{2}\theta = \dfrac{1}{9}$

$⇒ \dfrac{8}{9} = 2\sin^{2}\theta$

$⇒ sin\theta = \pm \dfrac{2}{3}$

$⇒ x = \pm \dfrac{2}{3}$

It is given that $m and n$ are co-prime natural numbers,

$⇒ m = 2, n = 3$

Also, $m{x}^{2} - nx - m + n = 0$.

$⇒ 2{x}^{2} - 3x - 2 + 3 = 0$

$⇒ 2{x}^{2} - 3x + 1 = 0$

$⇒ 2{x}^{2} - 2x - x + 1 = 0$

$⇒ 2x\left(x - 1\right) - \left(x - 1\right) = 0$

$⇒ \left(2x - 1\right)\left(x - 1\right) = 0$

$⇒ x = 1, \dfrac{1}{2}$

$⇒ \alpha = 1, \beta = \dfrac{1}{2}$

Out of the given options, \left(1, \dfrac{1}{2}\right)$lines on$5x + 8y = 9$.