If the AM and GM between two number 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)-nm)/(m^(2)+n^(2)+mn)`

Explanation

To find the ratio between two numbers \( a \) and \( b \) given that their Arithmetic Mean (AM) and Geometric Mean (GM) are in the ratio \( m:n \), we can follow these steps:

### Step 1: Define AM and GM
The Arithmetic Mean (AM) of two numbers \( a \) and \( b \) is given by:
\[ AM = \frac{a + b}{2} \]
The Geometric Mean (GM) of \( a \) and \( b \) is given by:
\[ GM = \sqrt{ab} \]

### Step 2: Set up the ratio
According to the problem, the ratio of AM to GM is given as:
\[ \frac{AM}{GM} = \frac{m}{n} \]
Substituting the expressions for AM and GM, we have:
\[ \frac{\frac{a + b}{2}}{\sqrt{ab}} = \frac{m}{n} \]

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

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:
\[ n^2(a + b)^2 = 4m^2ab \]

### Step 5: Expand the left-hand side
Expanding the left-hand side:
\[ n^2(a^2 + 2ab + b^2) = 4m^2ab \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ n^2a^2 + 2n^2ab + n^2b^2 - 4m^2ab = 0 \]
This simplifies to:
\[ n^2a^2 + (2n^2 - 4m^2)ab + n^2b^2 = 0 \]

### Step 7: Treat as a quadratic in \( a \)
This is a quadratic equation in terms of \( a \):
\[ n^2a^2 + (2n^2 - 4m^2)ab + n^2b^2 = 0 \]

### Step 8: Use the quadratic formula
Using the quadratic formula \( a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where:
- \( A = n^2 \)
- \( B = (2n^2 - 4m^2)b \)
- \( C = n^2b^2 \)

The discriminant \( D \) is:
\[ D = (2n^2 - 4m^2)^2b^2 - 4n^2(n^2b^2) \]

### Step 9: Find the ratio \( \frac{a}{b} \)
From the quadratic formula, we can derive the ratio \( \frac{a}{b} \):
\[ \frac{a}{b} = \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} \]

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