Discussion Forum

 The coordinates of a a point on the curve  $x=a(\theta +\sin \theta), y =a(1-\cos \theta)$   where the tangent is inclined at an angle $\frac{\pi}{4}$  to the positive X-axis , are

Answer:

Explanation:

 According  to the given information,

    $\frac{dy}{dx}=1  $            ..............(i)

 $\because$   $\frac{dy}{d \theta}= a\sin \theta$ and $\frac{dx}{d \theta}= a(1+\cos \theta)$ 

$\therefore$    $\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{a \sin\theta}{a(1+\cos \theta)}=1$

 $\Rightarrow\frac{ \sin\theta}{1+\cos \theta}=1\Rightarrow\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2}\frac{\theta}{2}}=1\Rightarrow \tan \frac{\theta}{2}=1$

  $\Rightarrow \frac{\theta}{2}=\frac{\pi}{4}\Rightarrow \theta = \frac{\pi}{2}$

 So, the required point on the curve is    $\left(a\left(\frac{\pi}{2}+1\right)a\right)$

 hence  , option (b) is correct