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 Geometric Mean (G.M.) of two positive numbers \( a \) and \( b \) is \( m:n \), then the ratio \( a:b \) can be expressed as:

\[ \frac{a}{b} = \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}} \]

### Step-by-Step Solution:

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