The ratio of the A.M. and G.M. of two positive numbers \( a \) and ...

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