Question

If the roots of the equation $x^2+2ax+b=0$ are real and distinct and they differ by at most $2m$, then $b$ lies in the interval

• A. $(a^2-m^{2-},a^2)$
• B. $[a^2-m^2,a^2]$
• C. $[a^2,a^2+m^2]$
• D. none of these

Answer 1

Answer: (B)

Let $\alpha ,\beta$ be the roots of
$x^2+2ax+b=0...(1)$
$\therefore \alpha +\beta =-2a$ and $\alpha \beta =b$
By the given condition $|\alpha -\beta |\le 2m$
$\therefore (\alpha -\beta {)}^2\le 4m^2$
$\Rightarrow (\alpha +\beta {)}^2-4\alpha \beta \le 4m^2$
$\Rightarrow 4a^2-4b\le 4m^2$
$\Rightarrow a^2-b\le m^2...(2)$
Since roots of (1) are real and distinct.
$\therefore$ Disc $>0$.
$\therefore 4a^2-4b>0$
$\Rightarrow a^2>b$
$\Rightarrow b<a^2...(3)$
From $(2)$ and $(3)$
$a^2-m^2\le b<a^2$
$\therefore b\in [a^2-m^2,a^2]$