Question

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

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

Answer 1

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

Given equations are
$x^2+2x+3=0...\left(i\right)$
$a{x}^2+bx+c=0...\left(ii\right)$
roots of equation $\left(i\right)$ are imaginary.

According to the question $\left(ii\right)$ will also have both roots same as $\left(i\right)$.
Thus,
$\frac{a}{1}=\frac{b}{2}=\frac{c}{3}=\lambda \left(say\right)\Rightarrow a=\lambda ,b=2\lambda ,c=3\lambda$

Hence, the required ratio is $1:2:3$