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

The correct option is D $$1 : 2 : 3$$
For equation $$x^{2}+2x+3=0$$ 

$$\Delta=2^2-4(1)(3)=4-12=-8<0$$ so both roots are imaginary .

Hence, the roots are non-real. They will exist in complex conjugate pairs. 

As one of the roots is common to $$ax^2+bx+c = 0 $$ , the other root will also be the complex conjugate of it. 

Hence, the roots of the two equations will be the same. 

Since $$a,\ b,\ c \in R$$.

If one root is common, then both roots are common .

Hence, $$\dfrac{a}{1}=\dfrac{b}{2}=\dfrac{c}{3}$$

 $$a:b:c= 1:2:3$$

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