Question

The difference between the corresponding roots of $x^2+ax+b=0$ and $x^2+bx+a=0$ is same and $a\ne b,$ then

• A. $a+b+4=0$
• B. $a+b-4=0$
• C. $a-b-4=0$
• D. $a-b+4=0$

Answer 1

Answer: (A)

Let $\alpha ,\beta$ and $\gamma ,\delta$ be the roots of the equations
$x^2+ax+b=0$ and $x^2+bx+a=0,$ respectively
Therefore,
$\alpha +\beta =-a,\alpha \beta =b$
and $\delta +\gamma =-b,\gamma \delta =a.$
Given $\left|\alpha -\beta \right|=\left|\gamma -\delta \right|$
$\Rightarrow {\left(\alpha +\beta \right)}^2-4\alpha \beta ={\left(\gamma +\delta \right)}^2-4\gamma \delta$
$\Rightarrow a^2-4b=b^2=4a$
$\Rightarrow \left(a^2=b^2\right)+4\left(a-b\right)=0$
$\Rightarrow a+b+4=0\left(\because a\ne b\right)$