If the sum of the roots of the equation ax2+bx+c=0 is equal to the sum of the squares of their reciprocals, then

Let α,β be the roots of the given quadratic equation. Then,  α+β=−b/a, αβ=c/aIt is given that α+β=1α2+1b2⇒α2+β2=(α+β)α2β2⇒(α+β)2−2αβ=(α+β)(αβ)2⇒b2a2−2ca=−bc2a3 ⇒2ca=b2a2+bc2a3⇒2a2c=ab2+bc2⇒c2b,a2c,b2a  are in A.P.Dividing both sides of 2a2c=ab2+bc2 by abc , we get 2ab=bc+ca⇒bc,ab,ca are in A.P.