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
$\frac{d x}{d \theta}=a(1+\cos \theta), \frac{d y}{d \theta}=a(0+\sin \theta)$
$\therefore \frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{a \sin \theta}{a(1+\cos \theta)}$
$=\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{2 \cos ^{2} \frac{\theta}{2}}=\tan \frac{\theta}{2}$
$ \tan \frac{\pi}{4}=\tan \frac{\theta}{2}\left(\because \frac{d y}{d x}=\tan x\right) $
$ \Rightarrow \frac{\pi}{4}=\frac{\theta}{2} $
$ \Rightarrow \theta=\frac{\pi}{2} $
$\therefore $ Coordinate of
$P\left[a\left(\frac{\pi}{2}+\sin \frac{\pi}{2}\right), a\left(1-\cos \frac{\pi}{2}\right)\right]$
$=P\left[a\left(\frac{\pi}{2}+1\right), a\right]$