The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b = `(m+sqrt(m^2-n^2)):(m-sqrt(m^2-n^2))`.

Explanation

To solve the problem, we need to show that if the ratio of the Arithmetic Mean (A.M.) and the Geometric Mean (G.M.) of two positive numbers \( a \) and \( b \) is \( m : n \), then the ratio \( a : b \) can be expressed as \( (m + \sqrt{m^2 - n^2}) : (m - \sqrt{m^2 - n^2}) \).

### 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}
...