Question

If every pair of equations $x^2+ax+bc=0,x^2+bx+ca=0,x^2+cx+ab=0$ has a common root, then product of these common roots $\alpha \beta \gamma =abc$.( Enter 1 if true or 0 otherwise)

Answer 1

Since each pair has common root, let the roots be $\alpha ,\beta$ for $x^2+ax+bc=0;\beta ,\gamma$ for $x^2+bx+ca=0$ and

$\gamma ,\alpha$ for $x^2+cx+ab=0.$

$\therefore \begin{array}{cc}\alpha +\beta =-a,& \alpha \beta =bc\\ \beta +\gamma =-b,& \beta \gamma =ca\\ \gamma +\alpha =-c,& \gamma \alpha =ab\end{array}$

Multiplying them, we get

$a^2{\beta }^2{\gamma }^2=a^2{b}^2{c}^2$

$\therefore \alpha \beta \gamma =abc=$ Product