If p, q, r are in A.P. and are positive, the roots of the quadratic equation

Question -

If p, q, r are in A.P. and are positive, the roots of the quadratic

equation p $x^{2}$ + qx + r = 0 are all real for ___.

Options:

A . |

\frac{r}{p}

- 7| ≥ 4

\sqrt{3}

B . |

\frac{p}{r}

- 7| < 4

\sqrt{3}

C . All p and r

D . No p and r

Answer: Option A
:
A

p,q,r are positive and are in A.P.

∴ q = $\frac{p + r}{2}$ .........(i)

∵ The roots of p $x^{2}$ + qx + r = 0 are real

⇒ $q^{2}$ ≥ 4pr ⇒ ${[\frac{p + r}{2}]}^{2}$ ≥ 4pr [using (i)]

⇒ $p^{2}$ + $r^{2}$ - 14pr ≥ 0

⇒ ${(\frac{r}{p})}^{2}$ - 14 ( $\frac{r}{p}$ ) + 1 ≥ 0 ( ∵ p>0 and p ≠ 0)

${(\frac{r}{p} - 7)}^{2}$ - $({4 \sqrt{3})}^{2}$ ≥ 0

⇒| $\frac{r}{p}$ - 7| ≥ 4 $\sqrt{3}$ .

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