If A.M. and G.M. between two numbers is in the ratio `m : n` then prove that the numbers are in the ratio `(m+sqrt(m^2-n^2)):(m-sqrt(m^2-n^2))`

Explanation

To prove that if the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) between two numbers are in the ratio \( m:n \), then the numbers are in the ratio \( \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} \), we will follow these steps:

### Step 1: Define the Numbers
Let the two numbers be \( a \) and \( b \).

### Step 2: Write the Formulas for A.M. and G.M.
The Arithmetic Mean (A.M.) of \( a \) and \( b \) is given by:
\[
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