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)

$[a^2 - m^2, a^2]$

Answer 2

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 $|\alpha - \beta| \leq \, 2m$ $\therefore \, (\alpha - \beta)^2 \leq 4m^2$ $\Rightarrow (\alpha + \beta)^2 - 4 \, \alpha \, \beta \leq 4m^2$