QJEEM13Ma95cc: JEE Main 2013 Mathematics - Class 11 JEEM13Ma95cc Solution: If the equations and have a common root, then is

Question

Verified Solution • 2025–26 Session moderate

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

Accepted Answer

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

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