A) \[\frac{\sqrt{m-n}+\sqrt{n}}{\sqrt{m-n}-\sqrt{n}}\]
B) \[\frac{\sqrt{n}+\sqrt{m-n}}{\sqrt{n}-\sqrt{m-n}}\]
C) \[\frac{\sqrt{m}+\sqrt{m-n}}{\sqrt{m}-\sqrt{m-n}}\]
D) None of these
Correct Answer: C
Solution :
\[\frac{\frac{a+b}{2}}{\frac{2ab}{a+b}}=\frac{m}{n}\] \[\Rightarrow \] \[\frac{{{(a+b)}^{2}}}{4ab}=\frac{m}{n}\] \[\Rightarrow \]\[\frac{{{(a+b)}^{2}}}{2ab}=\frac{2m}{n}\] Applying dividendo, we get \[\frac{{{a}^{2}}+{{b}^{2}}}{2ab}=\frac{2m-n}{n}\] Applying componendo and dividendo, We get \[\frac{{{(a+b)}^{2}}}{{{(a-b)}^{2}}}=\frac{m}{m-n}\]\[\Rightarrow \]\[\frac{a+b}{a-b}=\frac{\sqrt{m}}{\sqrt{m-n}}\] Again, applying componendo and dividendo We get\[\frac{a}{b}=\frac{\sqrt{m}+\sqrt{m-n}}{\sqrt{m}-\sqrt{m-n}}\].
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