If `A_(1),A_(2)` be two A.M.'s and `G_(1),G_(2)` be two G.M.,s between a and b, then `(A_(1)+A_(2))/(G_(1)G_(2))` is equal to

Options

A. `(a+b)/(2ab)`
B. `(2ab)/(a+b)`
C. `(a+b)/(ab)`
D. `(a+b)/(sqrt(ab))`

Explanation

To solve the problem, we need to understand the definitions of Arithmetic Means (A.M.) and Geometric Means (G.M.) between two numbers \(a\) and \(b\).

### Step-by-Step Solution:

1. **Identify the Arithmetic Means (A.M.)**:
- The two A.M.s between \(a\) and \(b\) can be expressed as:
\[ A_1 = \frac{a + b}{2} \quad \text{and} \quad A_2 = \frac{a + b}{2} \]
- However, since we are looking for two distinct A.M.s, we can express them as:
\[ A_1 = a + d \quad \text{and} \quad A_2 = b - d \]
- Here, \(d\) is a common difference.

2. **Calculate \(A_1 + A_2\)**:
- Adding the two A.M.s:
\[ A_1 + A_2 = (a + d) + (b - d) = a + b \]

3. **Identify the Geometric Means (G.M.)**:
- The two G.M.s between \(a\) and \(b\) can be expressed as:
\[ G_1 = \sqrt{ab} \quad \text{and} \quad G_2 = \sqrt{ab} \]
- However, since we are looking for two distinct G.M.s, we can express them as:
\[ G_1 = \sqrt{a \cdot b} \quad \text{and} \quad G_2 = \sqrt{a \cdot b} \]

4. **Calculate \(G_1 \cdot G_2\)**:
- Multiplying the two G.M.s:
\[ G_1 \cdot G_2 = \sqrt{ab} \cdot \sqrt{ab} = ab \]

5. **Formulate the Expression**:
- Now, we need to find the value of:
\[ \frac{A_1 + A_2}{G_1 \cdot G_2} \]
- Substituting the values we calculated:
\[ \frac{A_1 + A_2}{G_1 \cdot G_2} = \frac{a + b}{ab} \]

6. **Final Result**:
- Thus, the final result is:
\[ \frac{A_1 + A_2}{G_1 \cdot G_2} = \frac{a + b}{ab} \]