Question

If the equations $x^2+ax+bc=0$ and $x^2+bx+ca=0$ have a common root, then their other roots satisfy the equation

• A. $x^2+(a+b+c)x+ab=0$
• B. $x^2+cx+ab=0$
• C. $x^2-cx+ab=0$
• D. $x^2+(a+b)x+ab=0$

Answer 1

Answer: (B)

$x^2+cx+ab=0$

Let $\alpha$ be the common root; $\beta ,\gamma$ be other
roots resp.
$\Rightarrow \alpha +\beta =-a;\alpha \beta =bc$

$\alpha +\gamma =-b;\alpha \gamma =ca$

$\frac{\beta }{y}=\frac{b}{a}\Rightarrow \beta =\frac{\gamma b}{a}$

$\beta -\gamma =b-a$

$\Rightarrow \gamma (\frac{b}{a}-1)=b-a$

$\to \gamma =a;\beta =b$

$\therefore \gamma \beta =ab;\gamma +\beta =(a+b)=-c$

$\therefore$ They satisfy $x^2+cx+ab=0.$