Question

If p, q, r are positive and are in A.P., the roots of quadratic equation px² + qx + r = 0 are all real for:

• A. $\left| \frac{r}{p} - 7 \right|$³ 4$\sqrt{3}$
• B. $\left| \frac{p}{r} - 7 \right|$³ 4$\sqrt{3}$
• C. All p and r
• D. No p and r

Answer 1

Answer: (B)

$\left| \frac{p}{r} - 7 \right|$³ 4$\sqrt{3}$

Answer 2

Roots are real, so D ³ 0 p, q, r are in A.P.

q² ³ 4pr So, q =$\frac{p + r}{2}$

Ž $\left( \frac{p + r}{2} \right)^{2}$ ³ 4pr

Ž p² + r² + 2pr ³ 16 pr

Ž p² + r² –14 pr ³ 0

Ž $\left( \frac{p}{r} \right)^{2}$+ $\left( - \frac{p}{r} \right)$ (14) + 1 ³ 0

Ž $\left( \frac{p}{r} \right)^{2}$– $\frac{14p}{r}$+ 1 ³ 0

$\left( \frac{p}{r} \right)$= $\frac{14 \pm \sqrt{(14)^{2} - 4}}{2}$= 7 ± 4$\sqrt{3}$

Ž $\frac{P}{r}$£ 7 + 4$\sqrt{3}$, $\frac{P}{r}$ Ī 7 – 4$\sqrt{3}$

Ž $\left| \frac{P}{r} - 7 \right|$³ 4$\sqrt{3}$

Hence choice (2) is correct.