Question

If the equation $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$ have a non - zero common root, then $\lambda$ is equal to:

• A. None
• B. $3$
• C. $-1$
• D. $1$

Answer 1

The correct option is C

$-1$

We have $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$
Condition for common root of the equations i.e.,
$\frac{x^2}{10\lambda -9\lambda }=\frac{-x}{5\lambda -6\lambda }=\frac{1}{3-4}$

$\Rightarrow (6\lambda -5\lambda {)}^2=(3-4\left)\right(10\lambda -9\lambda )$
$\Rightarrow \lambda =-1$
When we have $a_1{x}^2+b_{1}x+c_1=0$ and $a_2{x}^2+b_{2}x+c_2=0$ with a common root then $(c_1{a}_2-c_2{a}_1{)}^2=(a_1{b}_2-a_2{b}_1\left)\right(b_1{c}_2-b_2{c}_1)$