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. $3:1:2$
• C. $3:2:1$
• D. $1:3:2$

Answer 1

The correct option is A

$1:2:3$

$x^2+2x+3=0$ . . . $\left(i\right)$

$a{x}^2+bx+c=0$ . . . $\left(ii\right)$

Since equation $\left(i\right)$ has imaginary roots, so, equation $\left(ii\right)$ will also have both roots same as $\left(i\right)$ , thus

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

$\Rightarrow a=\lambda ,b=2\lambda ,c=3\lambda$

Hence, $a:b:c=1:2:3$.