question_answer

A curve passes through the point\[\left( 1,\,\,\frac{\pi }{6} \right)\]. If the slope of the curve at each point (x, y) is \[\frac{y}{x}+\sec \,\,\left( \frac{y}{x} \right),\]\[x>0\]. Then, the equation of the curve is ________.

A)  \[\sin \,\,\left( \frac{y}{x} \right)=\log \,\,x+\frac{1}{2}\]

B)  \[\operatorname{cosec}\,\,\left( \frac{y}{x} \right)=\log \,\,x+2\]

C)  \[sec\,\,\left( \frac{2y}{x} \right)=\log \,\,x+2\]

D)  \[\cos \,\,\left( \frac{2y}{x} \right)=\log \,\,x+\frac{1}{2}\]

E)  None of these