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(D) The Arithmetic Mean $(AM)$ is $\frac{a+b}{2}$ and the Harmonic Mean $(HM)$ is $\frac{2ab}{a+b}$.Given $\frac{AM}{HM} = \frac{m}{n}$,so $\frac{(a+b

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$\frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}}$

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(D) The Arithmetic Mean $(AM)$ is $\frac{a+b}{2}$ and the Harmonic Mean $(HM)$ is $\frac{2ab}{a+b}$.Given $\frac{AM}{HM} = \frac{m}{n}$,so $\frac{(a+b)^2}{4ab} = \frac{m}{n}$.This implies $\frac{(a+b)^2}{ab} = \frac{4m}{n}$.Dividing by $4$,we get $\frac{(a+b)^2}{4ab} = \frac{m}{n}$.Using the property $\frac{a}{b} = \frac{AM + \sqrt{AM^2 - GM^2}}{AM - \sqrt{AM^2 - GM^2}}$,where $GM^2 = AM 	imes HM$.Since $GM^2 = \frac{a+b}{2} 	imes \frac{2ab}{a+b} = ab$,we have $\frac{a}{b} = \frac{m + \sqrt{m^2 - n^2}}{m - \sqrt{m^2 - n^2}}$.

If the ratio of the Arithmetic Mean and Harmonic Mean of two positive real numbers $a$ and $b$ is $m:n$,then find the value of $a:b$.

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