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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
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B
3:2:1
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C
1:3:2
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D
3:1:2
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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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