A) \[si{{n}^{-1}}x.si{{n}^{-1}}y=c\]
B) \[si{{n}^{-1}}x-si{{n}^{-1}}y=c\]
C) \[si{{n}^{-1}}x+si{{n}^{-1}}y=c\]
D) \[si{{n}^{-1}}x=c.si{{n}^{-1}}y\]
Correct Answer: C
Solution :
[c] \[\because \frac{dx}{dy}+\sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}=0\] \[\Rightarrow \sqrt{\frac{1-{{y}^{2}}}{1-{{x}^{2}}}}=-\frac{dy}{dx}\] \[\Rightarrow \frac{1}{\sqrt{1-{{x}^{2}}}}.dx=-\frac{1}{1-{{y}^{2}}}.dy\] Integrating both sides, we have \[\int{\frac{1}{\sqrt{1-{{x}^{2}}}}.dx}+\int{\frac{1}{\sqrt{1-{{y}^{2}}}}.dy=0\Rightarrow {{\sin }^{-1}}x}+{{\sin }^{-1}}y=c\]which is the required solution. Hence, option [c] is correct.
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