A) \[{{\cos }^{-1}}\frac{x}{a}\]
B) \[-{{\cos }^{-1}}\frac{x}{a}\]
C) \[\frac{1}{2}{{\cos }^{-1}}\frac{x}{a}\]
D) None of these
Correct Answer: D
Solution :
Putting \[x=a\cos \theta \Rightarrow \theta ={{\cos }^{-1}}\frac{x}{a}\] \[y={{\tan }^{-1}}\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}=\frac{\theta }{2}=\frac{1}{2}{{\cos }^{-1}}\frac{x}{a}\] Þ \[\frac{dy}{dx}=-\frac{1}{2}\frac{1}{\sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}}}.\frac{1}{a}=-\frac{1}{2\sqrt{{{a}^{2}}-{{x}^{2}}}}\].
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