Question

If $A_1,A_2,G_1,G_2$ and $H_1,H_2$ are two AM's, GM's and HM's between two quantities, then the value of $\frac{G_1{G}_2}{H_1{H}_2}$?

• 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

The correct option is A

$\frac{(A_1+A_2)}{(H_1+H_2)}$

Explanation for the correct option:

Step 1. Given data in the question :

Let $a,A_1,A_2,b$ are in AP.

Then$\frac{(a+b)}{2}=\frac{(A_1+A_2)}{2}$

$\Rightarrow (a+b)=(A_1+A_2)..........\left(i\right)$

Let $a,G_1,G_2,b$ are in GP.

$$\begin{gathered} \Rightarrow \sqrt{ab}=\sqrt{G_1{G}_2} \\ \Rightarrow ab=G_1{G}_2.........\left(ii\right) \end{gathered}$$

Let $a,H_1,H_2,b$ are in HP.

So ,$\frac{1}{a},\frac{1}{H_1},\frac{1}{H_2},\frac{1}{b}$ are in AP.

Step2. Find the value of $\frac{{\mathbf{G}}_{\mathbf{1}}{\mathbf{G}}_{\mathbf{2}}}{{\mathbf{H}}_{\mathbf{1}}{\mathbf{H}}_{\mathbf{2}}}$:

$$\begin{gathered} \frac{\left[\left({\displaystyle \frac{1}{a}}\right)+\left({\displaystyle \frac{1}{b}}\right)\right]}{2}=\frac{\left[\left({\displaystyle \frac{1}{H_1}}\right)+\left({\displaystyle \frac{1}{H_2}}\right)\right]}{2} \\ \Rightarrow \frac{(a+b)}{ab}=\frac{(H_1+H_2)}{H_1{H}_2} \end{gathered}$$

$\Rightarrow \frac{(A_1+A_2)}{G_1{G}_2}=\frac{(H_1+H_2)}{H_1{H}_2}$ ( from equation (i) and (ii) )

$\Rightarrow \frac{G_1{G}_2}{H_1{H}_2}=\frac{(A_1+A_2)}{(H_1+H_2)}$

Hence, the correct option is (A).