If α, β are the nonzero roots of ax2+bx+c=0 and α2, β2 are the roots of a2x2+b2x+c2=0, then a, b, c are in

α, β are roots of ax2+bx+c=0. Hence,α+β=−ba                 (1)αβ=ca                        (2)α2, β2are roots of a2x2+b2x+c2=0. Hence,α2+β2=−b2a2                  (3)α2β2=c2a2                         (4)Now, from (3), we have(α+β)2−2αβ=−b2a2⇒ −ba2−2ca=−b2a2or 2b2a2=2caor b2=ac⇒a,b,c are in G.P.