Q79 / 200 · Manipal (MET) 2010 · Mathematics
Let $a, b, c$ be real numbers with $a \neq 0$ and let $\alpha, \beta$ be the roots of the equation $a x^2+b x+c=0$, then $a^3 x^2+a b c x+c^3=0$ has roots
- Correct answerA$\alpha^2 \beta, \beta^2 \alpha$
- B$\alpha, \beta^2$
- C$\alpha^2 \beta, \beta \alpha$
- D$\alpha^3 \beta, \beta^3 \alpha$
Solution
Dividing the equation, $a^3 x^2+a b c x+c^3=0$ by $c^2$, we get $a\left(\frac{a x}{c}\right)^2+b\left(\frac{a x}{c}\right)+c=0$
$\Rightarrow \frac{a x}{c}=\alpha, \beta$ are roots.
$\Rightarrow x=\frac{c}{a} \alpha$ and $x=\frac{c}{a} \beta$
$\Rightarrow x=\alpha^2 \beta$ and $x=\alpha \beta^2$
$\left[\because \alpha \beta=\frac{c}{a}\right.$ as $\alpha, \beta$ are the roots of $\left.a x^2+b x+c=0\right]$ are roots of above equation.
Hence, the correct answer is option: 'A'.