Question

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

The correct option is A a + b + 4 = 0

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