Question

If every pair of equations $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $x^2+cx+ab=0$ has a common root, then product of these common roots is

• A. $\frac{ab}{c}$
• B. $\frac{c}{ab}$
• C. $abc$
• D. $a^2{b}^2{c}^2$

Answer 1

The correct option is C $abc$

Each pair has a common root

$\Rightarrow$ Let $\alpha ,\beta$ be the roots of $x^2+ax+bc=0\Rightarrow \alpha \cdot \beta =bc$
Let $\beta ,\gamma$ be roots of $x^2+bx+ac=0\Rightarrow \beta \cdot \gamma =ac$
Let $\alpha ,\gamma$ be roots of $x^2+cx+ab=0\Rightarrow \alpha \cdot \gamma =ab$

$\therefore (\alpha \cdot \beta )(\beta \cdot \gamma )(\alpha \cdot \gamma )=\left(bc\right)\left(ac\right)\left(ab\right)$
$\Rightarrow (\alpha \beta \gamma {)}^2=(abc{)}^2$
$\Rightarrow \alpha \beta \gamma =abc$