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 $x$ -axis, are

• A. $\left[a\left(\frac{\pi }{4}-1\right),a\right]$
• B. $\left[a\left(\frac{\pi }{2}+1\right),a\right]$
• C. $\left(a\frac{\pi }{2},a\right)$
• D. $(a,a)$

Answer 1

Answer: (B)

Given coordinate is
$x=a(\theta +\sin \theta ),y=a(1-\cos \theta )$
On differentiating w.r.t. $x$, we get
$\frac{dx}{d\theta }=a(1+\cos \theta ),\frac{dy}{d\theta }=a(0+\sin \theta )$
$\therefore \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=\frac{a\sin \theta }{a(1+\cos \theta )}$
$=\frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{2{\cos }^2\frac{\theta }{2}}=\tan \frac{\theta }{2}$
$\tan \frac{\pi }{4}=\tan \frac{\theta }{2}\left(\because \frac{dy}{dx}=\tan x\right)$
$\Rightarrow \frac{\pi }{4}=\frac{\theta }{2}$
$\Rightarrow \theta =\frac{\pi }{2}$
$\therefore$ Coordinate of
$P\left[a\left(\frac{\pi }{2}+\sin \frac{\pi }{2}\right),a\left(1-\cos \frac{\pi }{2}\right)\right]$
$=P\left[a\left(\frac{\pi }{2}+1\right),a\right]$