Question

If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$, $a,b,c\in R$, have a common root, then $a:b:c$ is

• A. $1:2:3$
• B. $3:2:1$
• C. $1:3:2$
• D. $3:1:2$

Answer 1

The correct option is A $1:2:3$

Let $\alpha$ be the common root.

Hence, ${\alpha }^2+2\alpha +3=0$

$a{\alpha }^2+b\alpha +c=0$

As both of equations are in $\alpha$, their coefficients should be in the same ratio.

$\Rightarrow \frac{1}{a}=\frac{2}{b}=\frac{3}{c}=\lambda$ (let)

$\Rightarrow a=\frac{1}{\lambda },b=\frac{2}{\lambda },c=\frac{3}{\lambda }$

$\Rightarrow a:b:c=\frac{1}{\lambda }:\frac{2}{\lambda }:\frac{3}{\lambda }=1:2:3$