Question
If the equations $${x^2} + 2x + 3 = 0\,\,{\text{and }}\,a{x^2} + bx + c = 0,a,b,c \in R,$$ 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 :
$$1 : 2 : 3$$
Solution :
Given equation are
$$\eqalign{
& {x^2} + 2x + 3 = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\text{i}} \right) \cr
& a{x^2} + bx + c = 0\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{ii}}} \right) \cr} $$
Roots of equation (i) are imaginary roots.
According to the question (ii) will also have both roots same as (i). Thus
$$\eqalign{
& \frac{a}{1} = \frac{b}{2} = \frac{c}{3} = \lambda \,\,\,\,\,\left( {{\text{say}}} \right) \cr
& \Rightarrow \,\,a = \lambda ,b = 2\lambda ,c = 3\lambda \cr} $$
Hence, required ratio is $$1 : 2 : 3$$