Question

For the curve $y = 3\sin\theta\cos\theta$, $x = e^{\theta}\sin\theta,$

$\theta \leq \theta \leq \pi$; tangent is parallel to x-axis, then $\theta$ =

• A. 0
• B. $\frac{\pi}{2}$
• C. $\frac{\pi}{4}$
• D. $\frac{\pi}{6}$

Answer 1

Answer: (A)

0

Answer 2

Given $y = 3\sin\theta\cos\theta,x = e^{\theta}\sin\theta$

For tangent parallel to $x -$axis, we should have $\frac{dy}{dx} = 0$

⇒ $\frac{dy}{d\theta} = 0$and $\frac{dx}{d\theta} \neq 0$

⇒ $\frac { d } { d \theta }$ $\left( 3\frac{\sin 2\theta}{2} \right) = 0$

⇒ $3\cos 2\theta = 0$ ⇒ 2θ = $\frac{\pi}{2}$

⇒ $\theta = \frac{\pi}{4}$

Note that for $\theta = \frac{\pi}{4}$, $\frac{dx}{d\theta} = e^{\theta}$ [$\sin\theta + \cos\theta$] ≠ 0.