Question

# If the equations $$x^2+2x+3=0$$ and $$ax^2+bx+c=0$$, $$a,b,c\in R$$, 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

Solution

## 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.$$\Rightarrow \dfrac{1}{a} = \dfrac{2}{b}=\dfrac{3}{c} = \lambda$$ (let)$$\Rightarrow a=\dfrac{1}{\lambda },b= \dfrac{2}{\lambda }, c =\dfrac{3}{\lambda }$$$$\Rightarrow a:b:c = \dfrac{1}{\lambda }:\dfrac{2}{\lambda }:\dfrac{3}{\lambda } = 1:2:3$$Mathematics

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