Let p , q and r be real numbers p ≠ q , r ≠ 0 , such that the roots of the equation 1 x + p + 1 x + q = 1 r are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to

Step-by-step solution

Given quadratic equation is 1 x + p   + 1 x + q = 1 r . Let α and β be the roots of given equation. ⇒ 2 x + p + q r = x + p x + q ⇒ x 2 + p + q - 2 r x + p q - p r - q r = 0 Now, sum of roots α + β = - b a = - p + q - 2 r ⇒ - p + q - 2 r = 0 ( ∵ Given that roots are equal in magnitude and opposite in sign) ⇒ p + q = 2 r     . . . 1 Product of roots α β = c a = p q - p r - q r Now, α 2 + β 2 = α + β 2 - 2 α β = 0