If the geometric mea is `(1)/(n)` times the harmonic mean between two numbers, then show that the ratio of the two numbers is `1+sqrt(1-n^(2)):1-sqrt(1-n^(2))`.

Explanation

To solve the problem, we need to show that if the geometric mean (G) of two numbers \(a\) and \(b\) is \(\frac{1}{n}\) times the harmonic mean (H) of the same two numbers, then the ratio of the two numbers is given by:

\[ \frac{a}{b} = \frac{1 + \sqrt{1 - n^2}}{1 - \sqrt{1 - n^2}} \]

Let's break this down step by step.

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