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

Option 1)
$\frac{3\pi }{4}$

Option 2)
$\frac{\pi }{2}$

Option 3)
$\frac{\pi }{4}$

Option 4)
$\frac{\pi }{6}$

Answers (1)

As we learnt in

Slope of a line -

If $\Theta$ is the angle at which a straight line is inclined to a positive direction of x-axis, then the slope is defined by

$m= \tan \Theta$ .

- wherein

y = $y=3sin\theta cos\theta =\frac{3}{2}sin2\theta$

$xe^{\theta }sin\theta$

$\frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$ $=\frac{\frac{3}{2}cos2\theta }{e^{\theta}(sin\theta +cos\theta )}$

For tangent to be parellel to x-axis, slope = 0

$\frac{3cos2\theta }{e^{0}(sin\theta +cos\theta )}=0$

$cos2\theta =0$

$\Rightarrow 2\theta =\frac{\pi}{2}$

$\Rightarrow \theta =\frac{\pi}{4}$

Option 1)

$\frac{3\pi }{4}$

This is incorrect

Option 2)

$\frac{\pi }{2}$

This is incorrect

Option 3)

$\frac{\pi }{4}$

This is correct

Option 4)

$\frac{\pi }{6}$

This is incorrect

Posted by

Sabhrant Ambastha

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