Question

If the roots of the equation $x^2-ax+b=0$ are real and differ by a quantity which is less than $c(c>0)$, then $b$ lies between

• A. $\frac{a^2-c^2}{4}$ and $\frac{a^2}{4}$
• B. $\frac{a^2+c^2}{4}$ and $\frac{a^2}{4}$
• C. $\frac{a^2-c^2}{2}$ and $\frac{a^2}{4}$
• D. $Noneofthese$

Answer 1

Answer: (A)

$\frac{a^2-c^2}{4}$ and $\frac{a^2}{4}$

Let roots be $\alpha ,\beta$

$|\alpha -\beta |<c\sqrt{{(\alpha +\beta )}^2-4\alpha \beta }<c$

So, $a^2-4b<c^2$

$b>\frac{a^2-c^2}{4}$

and since $b<\frac{a^2}{4}$ {max value of $x^2-ax+b=0$}

So, $b$ lies between $\frac{a^2-c^2}{4}$ and $\frac{a^2}{4}$