Let x = m n ( m , n are co-prime natural numbers) be a solution of the equation cos 2 sin - 1 x = 1 9 and let α , β ( α > β ) be the roots of the equation m x 2 - n x - m + n = 0 . Then the point ( α , β ) lies on the line

Step-by-step solution

Given: cos 2 sin - 1 x = 1 9 Let, sin - 1 x = θ ⇒ cos 2 θ = 1 9 ⇒ 1 - 2 sin 2 θ = 1 9 ⇒ 8 9 = 2 sin 2 θ ⇒ sin θ = ± 2 3 ⇒ x = ± 2 3 It is given that m and n are co-prime natural numbers, ⇒ m = 2 , n = 3 Also, m x 2 - n x - m + n = 0 . ⇒ 2 x 2 - 3 x - 2 + 3 = 0 ⇒ 2 x 2 - 3 x + 1 = 0 ⇒ 2 x 2 - 2 x - x + 1 = 0 ⇒ 2 x x - 1 - x - 1 = 0 ⇒ 2 x - 1 x - 1 = 0 ⇒ x = 1 , 1 2 ⇒ α = 1 , β = 1 2 Out of the given options, 1 , 1 2 lines on 5 x + 8 y = 9 .