JEE MAIN - Mathematics (2013 (Offline) - No. 15)
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
$$1:2:3$$
$$3:2:1$$
$$1:3:2$$
$$3:1:2$$
Explanation
Given equations are
$$\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 2x + 3 = 0\,\,\,\,\,...\left( i \right)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,a{x^2} + bx + c = 0\,\,\,...\left( {ii} \right)$$
Roots of equation $$(i)$$ are imaginary roots.
According to the question $$(ii)$$ will also have both roots same as $$(i).$$
Thus $${a \over 1} = {b \over 2} = {c \over 3} = \lambda \left( {say} \right)$$
$$ \Rightarrow a = \lambda ,b = 2\lambda ,c = 3\lambda $$
Hence, required ratio is $$1:2:3$$
$$\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 2x + 3 = 0\,\,\,\,\,...\left( i \right)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,a{x^2} + bx + c = 0\,\,\,...\left( {ii} \right)$$
Roots of equation $$(i)$$ are imaginary roots.
According to the question $$(ii)$$ will also have both roots same as $$(i).$$
Thus $${a \over 1} = {b \over 2} = {c \over 3} = \lambda \left( {say} \right)$$
$$ \Rightarrow a = \lambda ,b = 2\lambda ,c = 3\lambda $$
Hence, required ratio is $$1:2:3$$
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