Question

The coordinates of the point P 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 x-axis, are

• A. $(a(\frac{\pi }{2}-1),a)$
• B. $(a(\frac{\pi }{2}+1),a)$
• C. $(a\frac{\pi }{2},a)$
• D. $(a,a)$

Answer 1

The correct option is B $(a(\frac{\pi }{2}+1),a)$

$x=a(\theta +\sin \theta ),y=a(1-\cos \theta )$
$\frac{dy}{d\theta }=a\sin \theta ,\frac{dx}{d\theta }=a(1+\cos \theta )$
$\frac{dy}{dx}=\frac{\sin \theta }{1+\cos \theta }=\tan \frac{\theta }{2}$
$\Rightarrow \tan \frac{\theta }{2}=1$
$\Rightarrow \frac{\theta }{2}=\frac{\pi }{4}\Rightarrow \theta =\frac{\pi }{2}$
$x=a(\theta +\sin \theta )$
$y=a(1-\cos \theta )$
$x=a(\theta +\sin \theta )$
$y=a(1-\cos \theta )$
$(x,y)\equiv (a(\frac{\pi }{2}+1),a)$.