If the AM and GM of two positive numbers a and b are in the ratio m:n show that
$a : b = (m + \sqrt{m^{2} - n^{2}}) : (m - \sqrt{m^{2} - n^{2}}) .$

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Solution

Let A and G be respectively the AM and GM of a and b. Then,
$A = \frac{a + b}{2} and G = \sqrt{a b} \Rightarrow a + b = 2 A and G^{2} = a b . Now, the equation having roots a and b is x^{2} - (a + b) x + a b = 0 \Rightarrow x^{2} - (a + b) x + a b = 0 \Rightarrow x^{2} - 2 L x + G^{2} = 0 \Rightarrow x = \frac{2 A \pm \sqrt{4 A^{2} - 4 G^{2}}}{2} = A \pm \sqrt{A^{2} - G^{2}} \Rightarrow a = A + \sqrt{A^{2} - G^{2}} and b = A - \sqrt{A^{2} - G^{2}} . Now m, A : G = m : n Let A = km and G = kn for some constant k. Then, \frac{a}{b} = \frac{A + \sqrt{A^{2} - G^{2}}}{A - \sqrt{A^{2} - G^{2}}} = \frac{k m + \sqrt{k^{2} m^{2} - k^{2} n^{2}}}{k m - \sqrt{k^{2} m^{2} - k^{2} n^{2}}} = \frac{m + \sqrt{m^{2} - n^{2}}}{m - \sqrt{m^{2} - n^{2}}} Hence, a : b = (m + \sqrt{m^{2} - n^{2}}) : (m - \sqrt{m^{2} - n^{2}}) .$

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If the AM and GM of two positive numbers a and b are in the ratio m:n show that a:b=(m+√m2−n2):(m−√m2−n2).

If the AM and GM of two positive numbers a and b are in the ratio m:n show that
$a : b = (m + \sqrt{m^{2} - n^{2}}) : (m - \sqrt{m^{2} - n^{2}}) .$