Question

If the sum of the roots of the quadratic equation ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then show that ab², ca², bc² are in A.P.

Answer 1

Let α, β be the roots of the equation ax² + bx + c = 0.

Then α + β = –b/a, αβ = c/a

Given, 

Sum of roots = Sum of squares of reciprocals of roots

  ⇒ α + β = \(\frac{1}{α^2}\) + \(\frac{1}{β ^2}\)  ⇒ α + β = \(\frac{\alpha ^2 + \beta^2}{\alpha^2 +\beta^2}\)  ⇒ α + β = \(\frac{(\alpha + \beta)^2 - 2 \alpha \beta}{ (\alpha \beta)^2}\)

⇒ \(-b /a\) = \(\frac{(-b/a)^2 - \frac{2c}{a}}{(c/a)^2}\)

 ⇒ \(- \frac{b}{a}\) = \(\frac{b^2 -2ca}{c^2}\) 

⇒ – bc² = ab² – 2ca² 

⇒ 2ca² = ab² + bc² 

⇒ ab², ca², bc² are in A.P. (∵ a,b,c in A.P. ⇒ 2b = a + c)