Question

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. $ac+bd>b^2+d^2$

Answer 1

The correct option is C $ac+bd>b^2+c^2$

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 b = $\sqrt{ac}.$
Now as G.M > H.M., so here $\sqrt{ac}>b$ or $ac>b^2.$
Similarly $\sqrt{bd}>c$ or $bd>c^2$
Adding, $ac+bd>b^2+c^2$