Let α and β be the roots of the equation p x 2 + q x − r = 0 , where p ≠ 0 . If p , q and r be the consecutive terms of a non-constant G.P and 1 α + 1 β = 3 4 , then the value of α − β 2 is:

Step-by-step solution

Given: α and β be the roots of equation p x 2 + q x - r = 0 So, α + β = - q p , α β = - r p It is given that, 1 α + 1 β = 3 4 ⇒ α + β α β = 3 4 ⇒ q r = 3 4 Now, p , q , r are in GP So, common ratio of this GP will be q p = r q = 4 3 ⇒ p x 2 + q x - r = 0 ⇒ x 2 + q p x - r p = 0 ⇒ x 2 + 4 3 x - 4 3 2 = 0 ⇒ 9 x 2 + 12 x - 16 = 0 ⇒ α - β 2 = α + β 2 - 4 α β ⇒ α - β 2 = - 12 9 2 - 4 - 16 9 ⇒ α - β 2 = 16 9 + 64 9 ⇒ α - β 2 = 80 9