Question

If $A_1, \,A_2; \,G_1, \,G_2$ and $H_1,\, H_2$ be two $AM’s$, $GM’s$ and $HM’s$ between two quantities, then the value of $\frac{G_{1}\,G_{2}}{H_{1}\,H_{2}}$ is

• A. $\frac{A_{1}+A_{2}}{H_{1}+H_{2}}$
• B. $\frac{A_{1}-A_{2}}{H_{1}+H_{2}}$
• C. $\frac{A_{1}+A_{2}}{H_{1}-H_{2}}$
• D. $\frac{A_{1}-A_{2}}{H_{1}-H_{2}}$

Answer 1

Answer: (A)

Let the two quantities be $a$ and $b$. Then $a, A_{1}, A_{2}, b$ are in $A P$.
$\therefore A_{1}-a=b-A_{2}$
$\Rightarrow A_{1}+A_{2}=a+b \,\,\,\,\ldots(i)$
Again $a, G_{1}, G_{2}, b$ are in $G P$.
$\therefore \frac{G_{1}}{a}=\frac{b}{G_{2}}$
$\Rightarrow G_{1} G_{2}=a b \,\,\,\,\ldots(i i)$
Also, $a, H_{1}, H_{2}, b$ are in $H P$.
$\therefore \frac{1}{H_{1}}-\frac{1}{a}=\frac{1}{b}-\frac{1}{H_{2}}$
$\Rightarrow \frac{1}{H_{2}}+\frac{1}{H_{2}}=\frac{1}{b}+\frac{1}{a}$
$\Rightarrow \frac{H_{1}+H_{2}}{H_{1} H_{2}}=\frac{a+b}{a b}$
$\Rightarrow \frac{H_{1}+H_{2}}{H_{1}\, H_{2}}=\frac{A_{1}+A_{2}}{G_{1} \,G_{2}}$ [using Eqs. $(i)$ and $(i i)]$
$\Rightarrow \frac{G_{1} G_{2}}{H_{1} H_{2}}=\frac{A_{1}+A_{2}}{H_{1}+H_{2}}$