JEE Main 2014 (Online) 11th AprilQuestion 1
For the curve $y=3\sin\theta\cos\theta$, $x=e^{\theta}\sin\theta$, $0\le\theta\le\pi$, the tangent is parallel to $x$-axis when $\theta$ is :
Differentiation and its Application
Solution
Concept. A horizontal tangent needs $\frac{dy}{d\theta}=0$ with $\frac{dx}{d\theta}\ne0$. Formulas used. (1) $\dfrac{dy}{dx}=\dfrac{dy/d\theta}{dx/d\theta}$Step 1 - $\frac{dy}{d\theta}$. $y=\frac32\sin2\theta$, so $\frac{dy}{d\theta}=3\cos2\theta=0$ at $\theta=\frac\pi4,\frac{3\pi}{4}$. Step 2 - $\frac{dx}{d\theta}$. $e^\theta(\sin\theta+\cos\theta)$: non-zero at $\frac\pi4$, zero at $\frac{3\pi}{4}$. Step 3 - Conclude. At $\theta=\frac{3\pi}{4}$ both vanish, and the limit of $\frac{dy}{dx}$ there is not $0$. The tangent is horizontal at $\theta=\frac\pi4$. Answer: (C)