• # question_answer If  $a,\ b,\ c,\ d$ be in H.P., then A) ${{a}^{2}}+{{c}^{2}}>{{b}^{2}}+{{d}^{2}}$ B) ${{a}^{2}}+{{d}^{2}}>{{b}^{2}}+{{c}^{2}}$ C) $ac+bd>{{b}^{2}}+{{c}^{2}}$ D) $ac+bd>{{b}^{2}}+{{d}^{2}}$

Correct Answer: C

Solution :

$a,\ b,\ c,\ d$ be in H.P., then $\frac{1}{a},\ \frac{1}{b},\ \frac{1}{c},\ \frac{1}{d}$ will be in A.P. Therefore $\frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}=\frac{1}{d}-\frac{1}{c}\Rightarrow b=\frac{2ac}{a+c}$ G.M. between $a$ and $c$=$\sqrt{ac}$. Now as $G.M>H.M$., so here $\sqrt{ac}>b$ or $ac>{{b}^{2}}$. Similarly $\sqrt{bd}>c$ or $bd>{{c}^{2}}$ Adding, $ac+bd>{{b}^{2}}+{{c}^{2}}$

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