If the equations $\mathrm{x}^{2}+2 \mathrm{x}+3=0$ and $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0, \mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathrm{R}$, have a common root, then $\mathrm{a}: \mathrm{b}: \mathrm{c}$ is

JEE Main · 2013 · 07 Apr 2013 · Offline · Quadratic Equations and Inequalities

If the equations

x^{2} + 2 x + 3 = 0

and

ax^{2} + bx + c = 0, a, b, c \in R

, have a common root, then

a : b : c

(A) $3 : 2 : 1$
(B) $1 : 3 : 2$
(C) $3 : 1 : 2$
(D) $1 : 2 : 3$ ✓ Correct

Answer: (D)

Solution

The equation

x^{2} + 2 x + 3 = 0

has discriminant

4 - 12 = - 8 < 0

, so both its roots are complex. Since

a, b, c

are real, the second equation's complex roots must come in conjugate pairs, so sharing one complex root forces it to share both. The two quadratics are then proportional,

\frac{a}{1} = \frac{b}{2} = \frac{c}{3}

, giving

a : b : c = 1 : 2 : 3

JEE Main · 2013 · 07 Apr 2013 · Offline · Single correct option

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