Question

The normal to the curve given by $x=a(\cos \theta +\theta \sin \theta ),y=a(\sin \theta -\theta \cos \theta )$ at any point $\theta$ is such that it

• A. makes a constant angle with x-axis
• B. is at a constant distance from the origin
• C. touches a fixed circle
• D. passes through the origin

Answer 1

Answer: (B), (C)

The correct options are
B is at a constant distance from the origin
C touches a fixed circle

$x=a(\cos \theta +\theta \sin \theta )$ and $y=a(\sin \theta -\theta \cos \theta )$

$\frac{dx}{d\theta }=a(-\sin \theta +\sin \theta +\theta \cos \theta )=a\theta \cos \theta$

$\frac{dy}{d\theta }=a(\cos \theta -\cos \theta +\theta \sin \theta )=a\theta \sin \theta$

$\therefore \frac{dy}{dx}=\tan \theta$

So, the equation of the normal is

$y-a\sin \theta +a\theta \cos \theta =-\frac{\cos \theta }{\sin \theta }(x-a\cos \theta -a\theta \sin \theta )$

$\Rightarrow y-a\sin \theta -a{\sin }^2\theta +a\theta \cos \theta \sin \theta =-x\cos \theta +a{\cos }^2\theta +a\theta \sin \theta \cos \theta$

$\Rightarrow x\cos \theta +y\sin \theta =a$

It is always at a constant distance $a$ from the origin.