Let a ,   b and c satisfy the system of equations a + 2 b + 3 c = 6 ,   4 a + 5 b + 6 c = 12 and 6 a + 9 b = 4 . If the roots of the equation a + b + c x 2 - a b c x + a - 1 + b - 1 + c - 1 = 0 are α and β , then 1 α + 1 β is equal to

Step-by-step solution

Using cramer’s rule ∆ = 1 2 3 4 5 6 6 9 0 = 1 - 54 - 2 - 36 + 3 6 = - 54 + 72 + 18 = 36 ∆ 1 = 6 2 3 12 5 6 4 9 0 = 6 - 54 - 2 - 24 + 3 88 = - 324 + 48 + 264 = - 12 ∆ 2 = 1 6 3 4 12 6 6 4 0 = 1 - 24 - 6 - 36 + 3 - 56 = - 24 + 216 - 168 = 24 ∆ 3 = 1 2 6 4 5 12 6 9 4 = 1 - 88 - 2 - 56 + 6 6 = - 88 + 112 + 36 = 60 ⇒ a = ∆ 1 ∆ = - 1 3 , b = ∆ 2 ∆ = 2 3 , c = ∆ 3 ∆ = 5 3 ⇒ equation is 2 x 2 + 10 27 x - 9 10 = 0 ⇒ 540 x 2 + 100 x - 243 = 0 α + β = - 5 27 , α β = - 9 20 1 α + 1 β = α + β α β = - 5 27 - 9 20 = 100 243