If the A.M. and G.M. between two numbers are in the ratio m.n., then what is the ratio between the two numbers ?

Options

A. `(m+sqrt(m^(2)-n^(2)))/(m-sqrt(m^(2)-n^(2)))`
B. `(m+n)/(m-n)`
C. `(m^(2)-n^(2))/(m^(2)+n^(2))`
D. `(m^(2)+n^(2)-mn)/(m^(2)+n^(2)+mn)`

Answer

To solve the problem, we need to find the ratio between two numbers \( a \) and \( b \) given that the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of these two numbers are in the ratio \( m:n \).

### Step-by-Step Solution:

1. **Define A.M. and G.M.**:
- The Arithmetic Mean (A.M.) of two numbers \( a \) and \( b \) is given by:
\[ A.M. = \frac{a + b}{2} \]
- The Geometric Mean (G.M.) of two numbers \( a \) and \( b \) is given by:
\[ G.M. = \sqrt{ab} \]

2. **Set up the ratio**:
- According to the problem, the ratio of A.M. to G.M. is given as:
\[ \frac{A.M.}{G.M.} = \frac{m}{n} \]
- Substituting the expressions for A.M. and G.M. into the ratio gives:
\[ \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{m}{n} \]

3. **Cross-multiply to eliminate the fraction**:
- Cross-multiplying yields:
\[ n(a + b) = 2m\sqrt{ab} \]

4. **Rearranging the equation**:
- Rearranging the equation gives:
\[ n(a + b) = 2m\sqrt{ab} \]

5. **Using the method of components and dividends**:
- We can apply the method of components and dividends to express \( a \) and \( b \) in terms of \( m \) and \( n \). We can rewrite the equation as:
\[ \frac{a + b}{2\sqrt{ab}} = \frac{m}{n} \]
- This can be manipulated into a form suitable for applying the method.

6. **Expressing the ratio**:
- By applying the component and dividend method, we can express the ratio of \( a \) to \( b \):
\[ \frac{a}{b} = \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} \]

### Final Result:
Thus, the ratio of the two numbers \( a \) and \( b \) is:
\[ \frac{a}{b} = \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} \]