If , are the roots of ax^2 + bx + c = 0 and + , ^2 + ^2,^3 + ^3 are in G.P. and = b^2 - 4ac, then

aligned& (a): (^2+^2)^2=(+)(^3+^3) \\& (b^2-2 a ca^2)^2=(-ba)(-b^3+3 a b ca^3) \\& 4 a^2 c^2=a c b^2 a c(b^2-4 a c)=0aligned a 0 c =0

If \alpha,\beta are the roots of ax^{2} + bx + c = 0 and \alpha + \beta, \alpha^{2} + \beta^{2},\alpha^{3} + \beta^{3} are in G.P. and \Delta = b^{2} - 4ac, then: Sequence & Series (Mathematics)

\n\begin{aligned}\n& \text { (a): }\left(\alpha^2+\beta^2\right)^2=(\alpha+\beta)\left(\alpha^3+\beta^3\right) \\\n& \left(\frac{b^2-2 a c}{a^2}\right)^2=\left(-\frac{b}{a}\right)\left(\frac{-b^3+3 a b c}{a^3}\right) \\\n& \Rightarrow 4 a^2 c^2=a c b^2 \Rightarrow a c\left(b^2-4 a c\right)=0\n\end{aligned}\n

\n\nAs

a \neq 0 \Rightarrow c \Delta=0