If the equation x2+2x+3=0 and ax2+bx+c=0, a,b,c∈R, have a common root, then a:b:c is

Given Equation x2+2x+3=0⇒ (x+1)(x+2)=0⇒x=−1, −2∴ x=−1, −2 are roots of ax2+bx+c=0∴ −1−2=−ba, and −1(−2)=ca⇒ b= 3a,c=2a∴ a:b:c= a:3a:2a= 1:3:2

If the equation x2+2x+3=0 and ax2+bx+c=0, a,b,c∈R, have a common root, then a:b:c is

Given Equation x2+2x+3=0⇒ (x+1)(x+2)=0⇒x=−1, −2∴ x=−1, −2 are roots of ax2+bx+c=0∴ −1−2=−ba, and −1(−2)=ca⇒ b= 3a,c=2a∴ a:b:c= a:3a:2a= 1:3:2

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$I f t h e e q u a t i o n x^{2} + 2 x + 3 = 0 a n d a x^{2} + b x + c = 0, a, b, c \in R, h a v e a c o m m o n r o o t, t h e n a : b : c i s$

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$1 : 2 : 3$

$3 : 2 : 1$

$1 : 3 : 2$

$3 : 1 : 2$

answer is C.

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Detailed Solution

$\begin{array}{l} G i v e n E q u a t i o n x^{2} + 2 x + 3 = 0 \\ \Rightarrow (x + 1) (x + 2) = 0 \\ \Rightarrow x = - 1, - 2 \\ ∴ x = - 1, - 2 a r e r o o t s o f a x^{2} + b x + c = 0 \\ ∴ - 1 - 2 = \frac{- b}{a}, a n d - 1 (- 2) = \frac{c}{a} \\ \Rightarrow b = 3 a, c = 2 a \\ ∴ a : b : c = a : 3 a : 2 a \\ = 1 : 3 : 2 \end{array}$