A) \[\frac{ay}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}\]
B) \[\frac{ay}{\sqrt{{{a}^{2}}-{{x}^{2}}}}\]
C) \[\frac{ay}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}\]
D) None of these
Correct Answer: A
Solution :
\[y=\frac{\sqrt{a+x}-\sqrt{a-x}}{\sqrt{a+x}+\sqrt{a-x}}\Rightarrow y=\frac{{{(\sqrt{a+x}-\sqrt{a-x})}^{2}}}{(a+x)-(a-x)}\] \[\Rightarrow y=\frac{(a+x)+(a-x)-2(\sqrt{{{a}^{2}}-{{x}^{2}}})}{2x}\] \[=\frac{2a-2\sqrt{{{a}^{2}}-{{x}^{2}}}}{2x}\] or \[y=\frac{a-\sqrt{{{a}^{2}}-{{x}^{2}}}}{x}\] ?..(i) Differentiating w.r.t. x of y, we get \[\frac{dy}{dx}=\frac{x\left[ -\frac{1}{2\sqrt{{{a}^{2}}-{{x}^{2}}}}(-2x) \right]-(a-\sqrt{{{a}^{2}}-{{x}^{2}}})}{{{x}^{2}}}\] \[=\frac{{{x}^{2}}-a\sqrt{{{a}^{2}}-{{x}^{2}}}+{{a}^{2}}-{{x}^{2}}}{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}}=\frac{a(a-\sqrt{{{a}^{2}}-{{x}^{2}}})}{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}}\] \[=\frac{a}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}\left[ \frac{a-\sqrt{{{a}^{2}}-{{x}^{2}}}}{x} \right]=\frac{ay}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}\] [By (i)]
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