Question

For the curve y = 3sin $\theta cos\theta ,x=e^{\theta }sin\theta ,0\le \theta \le \pi$; tangent is parallel to x-axis, then $\theta$ =

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

Answer 1

The correct option is C $\frac{\pi }{4}$

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
$\Rightarrow \frac{dy}{dx}$ = 0 and $\frac{dx}{d\theta }\ne 0$ and $\frac{d}{d\theta }\left(3\frac{sin2\theta }{2}\right)$ = 0
$\Rightarrow 3cos2\theta =0\Rightarrow 2\theta =\frac{\pi }{2}$
$\Rightarrow \theta =\frac{\pi }{4}$
Note that for $\theta =\frac{\pi }{4},\frac{dx}{d\theta }=e^{\theta }[sin\theta +cos\theta ]\ne 0.$