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
$x^2 + 2\,ax + b = 0 \,\,\,\,...(1)$
$\therefore \, \alpha + \beta = - 2\,a$ and $\alpha \, \beta = b$
By the given condition $|\alpha - \beta| \leq \, 2m$
$\therefore \, (\alpha - \beta)^2 \leq 4m^2$
$\Rightarrow (\alpha + \beta)^2 - 4 \, \alpha \, \beta \leq 4m^2$
$ \Rightarrow \,4a^2 - 4b \leq 4m^2$
$\Rightarrow \, a^2 -b \leq m^2 \,\,\,\,...(2)$
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 $(3)$
$a^2 - m^2 \leq b < a^2$
$\therefore b \in [a^2 - m^2, a^2]$