Question

In the quadratic equation $a{x}^2+$ bx $+c=0$, if $\Delta =b^2-4ac$ and $\alpha +\beta ,{\alpha }^2+{{\beta }^2,\alpha }^3+{\beta }^3$ are in G.P. where $\alpha ,\beta$ are the roots of $a{x}^2+$ bx $+c=0$, then

• A. $a=0$
• B. $b=0$
• C. $c=0$
• D. $c=1$
  

Answer 1

Answer: (C)

$c=0$

Given,
$\alpha +\beta ,{\alpha }^2+{\beta }^2,{\alpha }^3+{\beta }^3$ are in G.P.
$\Rightarrow ({\alpha }^2+{\beta }^2{)}^2=(\alpha +\beta \left)\right({\alpha }^3+{\beta }^3)$
$\Rightarrow {\alpha }^4+{\beta }^4+2{\alpha }^2{\beta }^2={\alpha }^4+{\beta }^4+\alpha \beta ({\alpha }^2+{\beta }^2)$
$\Rightarrow \alpha \beta ({\alpha }^2+{\beta }^2-2\alpha \beta )=0$
$\Rightarrow \alpha \beta (\alpha -\beta {)}^2=0$
$\Rightarrow \alpha \beta =0$ or $\alpha =\beta$
$\Rightarrow c/a=0\Rightarrow c=0$ or $\Delta =b^2-4ac=0$