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Question 89
If $$x=\frac{4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}$$, then what is the value of $$\frac{x+2\sqrt{a}}{x-2\sqrt{a}}+\frac{x+2\sqrt{b}}{x-2\sqrt{b}}$$ (When a$$\neq$$b) ?
Solution
We have :
$$x=\frac{4\sqrt{\ ab}}{\sqrt{\ a}+\sqrt{\ b}}$$
Now $$\frac{x+2\sqrt{\ a}}{x-2\sqrt{\ a}}+\frac{x+2\sqrt{\ b}}{x-2\sqrt{\ b}}\ =\frac{2a+6\sqrt{\ ab}}{2\sqrt{\ ab}-2a}+\frac{2b+6\sqrt{\ ab}}{2\sqrt{\ ab}-2b}$$
Cross multiplying we get $$\frac{16ab-8b\sqrt{\ ab}-8a\sqrt{\ ab}}{8ab-4b\sqrt{\ ab}-4a\sqrt{\ ab}}$$=2
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