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Find the mean proportion between a−ba+b and a2b2a2−b2\dfrac{a - b}{a + b} \text{ and } \dfrac{a^2b^2}{a^2 - b^2}.

Ratio Proportion

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Let mean proportion between a−ba+b and a2b2a2−b2\dfrac{a - b}{a + b} \text{ and } \dfrac{a^2b^2}{a^2 - b^2} be x.

∴a−ba+bx=xa2b2a2−b2⇒x2=a−ba+b×a2b2a2−b2⇒x2=(a2b2)(a−b)(a+b)(a2−b2)⇒x2=(a2b2)(a−b)(a+b)(a−b)(a+b)⇒x2=a2b2(a+b)2⇒x=a2b2(a+b)2⇒x=aba+b.\therefore \dfrac{\dfrac{a - b}{a + b}}{x} = \dfrac{x}{\dfrac{a^2b^2}{a^2 - b^2}} \\[1em] \Rightarrow x^2 = \dfrac{a - b}{a + b} \times \dfrac{a^2b^2}{a^2 - b^2} \\[1em] \Rightarrow x^2 = \dfrac{(a^2b^2)(a - b)}{(a + b)(a^2 - b^2)} \\[1em] \Rightarrow x^2 = \dfrac{(a^2b^2)(a - b)}{(a + b)(a - b)(a + b)} \\[1em] \Rightarrow x^2 = \dfrac{a^2b^2}{(a + b)^2} \\[1em] \Rightarrow x = \sqrt{\dfrac{a^2b^2}{(a + b)^2}} \\[1em] \Rightarrow x = \dfrac{ab}{a + b}.

Hence, mean proportion between a−ba+b and a2b2a2−b2 is aba+b\dfrac{a - b}{a + b} \text{ and } \dfrac{a^2b^2}{a^2 - b^2} \text{ is } \dfrac{ab}{a + b}.

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