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, $\frac{dy}{dx}=1...(i)$
$\because \frac{dy}{d\theta }=a\sin \theta$ 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 $\left(a\left(\frac{\pi }{2}+1\right),a\right)$