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Q.

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

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a

$| \frac{r}{p} + 7 | \geq 4 \sqrt{3}$

b

$| r + 7 | \geq 4 \sqrt{3}$

c

$| r - 7 | \geq 4 \sqrt{3}$

d

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

answer is B.

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Detailed Solution

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For real roots $q^{2} - 4 p r \geq 0$

$\begin{array}{l} \Rightarrow {(\frac{p + r}{2})}^{2} - 4 p r \geq 0 (∵ p, q, r a r e i n A . P) \\ \Rightarrow p^{2} + r^{2} - 14 p r \geq 0 \Rightarrow \frac{p^{2}}{r^{2}} - 14 \frac{p}{r} + 1 \geq 0 \\ \Rightarrow (\frac{p}{r} - 7) - 48 \geq 0 \Rightarrow (\frac{p}{r} - 7) \geq 4 \sqrt{3} \end{array}$

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