Question

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

• A. $\left(a\left(\frac{\pi}{2}-1\right), a\right)$
• 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)

According to the given information, $\frac{d y}{d x}=1\,\,\,...(i)$
$\because \frac{d y}{d \theta}=a \sin \theta$ and $\frac{d x}{d \theta}=a(1+\cos \theta)$
$\therefore \frac{d y}{d x}=\frac{\frac{d y}{d \theta}}{\frac{d x}{d \theta}}=\frac{a \sin \theta}{a(1+\cos \theta)}=1$
$\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 $\left(a\left(\frac{\pi}{2}+1\right), a\right)$