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Arithmetic Progression

If p,q,r in...

Question

If $p, q, r$ in harmonic progression and $p$ and $r$ be different having same sign then the roots of the equation $p x^{2} + q x + r = 0$ are

Real and equal

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Real and distinct

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Irrational

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Imaginary

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Solution

The correct option is D Imaginary
given $p, q, r$ are in harmonic progression.

ie, $q = \frac{2 p r}{p + r}$

So, $\sqrt{q^{2} - 4 p r} = 2 \sqrt{p r} \sqrt{\frac{p r}{(p + r)^{2}} - 1}$

As $p r < (p + r)^{2}$ , (when $p r > 0$ )
$\frac{p r}{(p + r)^{2} - 1} < 0$

ie, roots of the equation, $p x^{2} q x + r = 0$ are imaginary.
So, Option D is the correct answer.

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