Let $a,\ p,\ q,\ r,\ s \in R\ \sim\ \left\{ 0 \right\}.$ If $3a^{2} + 2\left( \frac{1}{p} - \frac{1}{s} \right)a + \frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}} - 2\left( \frac{1}{pq} + \frac{1}{qr} + \frac{1}{rs} \right) \leq 0$ for some real $a,$ then $p,\ q,\ r,\ s$ are in

Question

Dr. Abubacker · Accepted Answer

(c)
$\left( a + \frac{1}{p} - \frac{1}{q} \right)^{2} + \left( a + \frac{1}{q} - \frac{1}{r} \right)^{2} + \left( a + \frac{1}{r} - \frac{1}{s} \right)^{2} \leq 0$
$\Rightarrow \frac{1}{p} - \frac{1}{q} = \frac{1}{q} - \frac{1}{r} = \frac{1}{r} - \frac{1}{s}$
$\Rightarrow \ p,\ q,\ r,\ s$ are in HP.

Q. Let $a,\ p,\ q,\ r,\ s \in R\ \sim\ \left\{ 0 \right\}.$ If $3a^{2} + 2\left( \frac{1}{p} - \frac{1}{s} \right)a + \frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}} - 2\left( \frac{1}{pq} + \frac{1}{qr} + \frac{1}{rs} \right) \leq 0$ for some real $a,$ then $p,\ q,\ r,\ s$ are in

Explanation

Reviewed by Dr. Abubacker

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