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Mathematics

Solve for x : a+x+a−xa+x−a−x=b\dfrac{\sqrt{a + x} + \sqrt{a - x}}{\sqrt{a + x} - \sqrt{a - x}} = b.

Ratio Proportion

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Answer

Applying componendo and dividendo, we get :

⇒a+x+a−x+a+x−a−x(a+x+a−x)−(a+x−a−x)=b+1b−1⇒2a+x2a−x=b+1b−1⇒a+xa−x=b+1b−1\Rightarrow \dfrac{\sqrt{a + x} + \sqrt{a - x} + \sqrt{a + x} - \sqrt{a - x}}{(\sqrt{a + x} + \sqrt{a - x}) - (\sqrt{a + x} - \sqrt{a - x})} = \dfrac{b + 1}{b - 1} \\[1em] \Rightarrow \dfrac{2\sqrt{a + x}}{2\sqrt{a - x}} = \dfrac{b + 1}{b - 1} \\[1em] \Rightarrow \dfrac{\sqrt{a + x}}{\sqrt{a - x}} = \dfrac{b + 1}{b - 1} \\[1em]

Squaring both sides we get :

⇒a+xa−x=(b+1)2(b−1)2\Rightarrow \dfrac{a + x}{a - x} = \dfrac{(b + 1)^2}{(b - 1)^2}

Again applying componendo and dividendo, we get :

⇒a+x+a−xa+x−(a−x)=(b+1)2+(b−1)2(b+1)2−(b−1)2⇒a+a+x−xa−a+x−(−x)=b2+1+2b+b2+1−2bb2+1+2b−(b2+1−2b)⇒2a2x=2(b2+1)4b⇒ax=b2+12b⇒x=2abb2+1.\Rightarrow \dfrac{a + x + a - x}{a + x - (a - x)} = \dfrac{(b + 1)^2 + (b - 1)^2}{(b + 1)^2 - (b - 1)^2} \\[1em] \Rightarrow \dfrac{a + a + x - x}{a - a + x - (-x)} = \dfrac{b^2 + 1 + 2b + b^2 + 1 - 2b}{b^2 + 1 + 2b - (b^2 + 1 - 2b)} \\[1em] \Rightarrow \dfrac{2a}{2x} = \dfrac{2(b^2 + 1)}{4b} \\[1em] \Rightarrow \dfrac{a}{x} = \dfrac{b^2 + 1}{2b} \\[1em] \Rightarrow x = \dfrac{2ab}{b^2 + 1}.

Hence, x = 2abb2+1\dfrac{2ab}{b^2 + 1}.

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