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. $(a^2,a^2+m^2)$
• B. $(a^2-m^2,a^2)$
• C. $[a^2-m^2,a^2)$
• D. None of these

Answer 1

The correct option is C $[a^2-m^2,a^2)$

Let $\alpha ,\beta$ be the roots of $x^2+2ax+b=\mathrm{0......}\left(1\right)$

$\Rightarrow \alpha +\beta =-2a$ and $\alpha \beta =b$

By hypothesis

$|\alpha -\beta |\le 2m$

$\Rightarrow (\alpha -\beta {)}^2\le 4m^2$

$\Rightarrow (\alpha +\beta {)}^2-4\alpha \beta \le 4m^2$

$\Rightarrow 4a^2-4b\le 4m^2$

$\Rightarrow a^2-b\le m^2......\left(2\right)$

And discriminant of (1) is >0

$\Rightarrow 4a^2-4b>0$

$\Rightarrow b<a^2......\left(3\right)$

From (2) and (3)

$a^2>b\ge (a^2-m^2)$

$b\in [a^2-m^2,a^2)$