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

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