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 $1:2:3$

Since, $x^2+2x+3=0$ has irrational roots, hence both the roots must be common.

Let the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0$ have common roots are $\alpha$ and $\beta$

So, $\alpha +\beta =-\frac{2}{1}=-\frac{b}{a}$
$\Rightarrow \frac{a}{b}=\frac{1}{2}...\left(1\right)$
and $\alpha \cdot \beta =\frac{3}{1}=\frac{c}{a}$
$\Rightarrow \frac{c}{a}=\frac{3}{1}...\left(2\right)$

Multiplying equation $\left(1\right)$ and $\left(2\right)$
$\Rightarrow \frac{c}{b}=\frac{3}{2}$
$\Rightarrow \frac{b}{c}=\frac{2}{3}...\left(3\right)$

From equation $\left(1\right)$ and $\left(3\right)$
$\Rightarrow a:b:c=1:2:3$