A) \[sec\,\,x\,\,sec\,\,y=\sqrt{2}\]
B) \[cos\,\,x\,\,cos\,\,y=\sqrt{2}\]
C) \[\sec \,\,x=\sqrt{2}\,\,\cos \,\,y\]
D) \[cos\,\,y=\sqrt{2}\,\,\sec \,\,y\]
Correct Answer: A
Solution :
[a] The given differential equation is \[\sin x\cos ydx+\cos x\sin ydy=0\] dividing by \[\cos x\cos y\Rightarrow \frac{\sin x}{\cos x}dx+\frac{\sin y}{\cos y}dy=0\] Integrating, \[\int{\tan xdx+\int{\tan ydy=\log c}}\] Or \[\log \sec x\sec y=\log cor\sec x\sec y=c\] curve passes through the point \[\left( 0,\frac{\pi }{4} \right)\] \[\sec 0\sec \frac{\pi }{4}=c=\sqrt{2}\] Hence. The required equation of the curve is \[\sec x\sec y=\sqrt{2}\]
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