Practicing Success
A curve passes through the point $(1, \pi / 6)$. Let the slope of the curve at each point $(x, y)$ be $\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0$. Then, the equation of the curve is |
$\sin \left(\frac{y}{x}\right)=\log x+\frac{1}{2}$ $cosec\left(\frac{y}{x}\right)=\log x+2$ $\sec \left(\frac{2 y}{x}\right)=\log x+2$ $\cos \left(\frac{2 y}{x}\right)=\log x+\frac{1}{2}$ |
$\sin \left(\frac{y}{x}\right)=\log x+\frac{1}{2}$ |
It is given that $\frac{d y}{d x}=\frac{y}{x}+\sec \left(\frac{y}{x}\right), x>0$ ....(i) Let $y=v x$. Then, $\frac{d y}{d x}=v+x \frac{d v}{x}$ Substituting these values in (i), we obtain $v+x \frac{d v}{d x}=v+\sec v$ $\Rightarrow x \frac{d v}{d x}=\sec v$ $\Rightarrow \cos v d v=\frac{1}{x} d x$ Integrating both sides, we get $\sin v=\log x+C$ $\Rightarrow \sin \left(\frac{y}{x}\right)=\log x+C$ ....(ii) It passes through the point $\left(1, \frac{\pi}{6}\right)$. ∴ $\sin \frac{\pi}{6}=\log 1+C \Rightarrow C=\frac{1}{2}$ Putting $C=\frac{1}{2}$ in (ii), we obtain the equation of the curve as $\sin \left(\frac{y}{x}\right)=\log x+\frac{1}{2}$ |