Question

If $p,q,r$ are positive and are in A.P., the roots of quadratic equation $p{x}^2+qx+r=0$ are all real for

• A. $â\frac{r}{p}−7ââ‰\yen 4\sqrt{3}$
• B. $â\frac{p}{r}−7ââ‰\yen 4\sqrt{3}$
• C. all $p$ and $r$
• D. no $p$ and $r$

Answer 1

Answer: (B)

Since $p,q,r$ are in A.P. so $2q=p+r$.
The roots of the equation $p{x}^2+qx+r=0$ are real if and only if
$q^2−4prâ‰\yen 0$
$⇒{\left(\frac{p+r}{2}\right)}^2−4prâ‰\yen 0$
$⇒p^2+r^2−14prâ‰\yen 0$
$⇒\frac{p^2}{r^2}−14\frac{p}{r}+1â‰\yen 0$
$⇒{\left(\frac{p}{r}−7\right)}^2−48â‰\yen 0$
$⇒â\frac{p}{r}−7ââ‰\yen 4\sqrt{3}$