Question

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

• A. $\Delta \neq 0$
• B. $b \Delta=0$
• C. $c \Delta=0$
• D. $\Delta=0$

Answer 1

Answer: (C)

$\left(a^2+b^2\right)^2=(a+b)\left(a^3+b^3\right)$
$\alpha^4+\beta^4+2 \alpha^2 \beta^2=\alpha^4+\beta^4+\alpha \beta\left(\beta^2+\alpha^2\right) $
$\alpha \beta\left(\alpha^2+\beta^2-2 \alpha \beta\right)=0$
$\alpha \beta\left[(\alpha+\beta)^2-4 \alpha \beta\right]=0 $
$\frac{c}{a}\left(\frac{b^2}{a^2}-\frac{4 c}{a}\right)=0$
$c\left(b^2-4 a c\right)=0 $
$c \Delta=0$