Question

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

• A. it makes a constant angle with $ x $ -axis
• B. it passes through origin
• C. it is at a constant distance from origin
• D. None of the above

Answer 1

Answer: (C)

Given curves are
$x=a(\cos \theta+\theta \sin \theta)$,
$y=a(\sin \theta-\theta \cos \theta) .$
On differentiating w.r.t. $\theta$ respectively, we get
$\frac{d x}{d \theta}=a(-\sin \theta+\sin \theta+\theta \cos \theta)$
$\frac{d y}{d \theta}=a(\cos \theta-\cos \theta+\theta \sin \theta)$
$\Rightarrow \frac{d x}{d \theta}=a\, \theta \cos \theta$
$\frac{d y}{d \theta}=a \,\theta \sin \theta$
$\therefore \frac{d y}{d x}=\frac{\sin \theta}{\cos \theta}$
$\therefore $ Equation of normal is
$y-a \,\sin \theta+a \,\theta \cos \theta=-\frac{\cos \theta}{\sin \theta}$
$(x-a \,\cos \theta-a \,\theta \sin \theta)$
$\Rightarrow y \sin \theta-a \,\sin ^{2} \theta+a \,\theta \cos \theta \sin \theta$
$=-x \cos \theta+a \,\cos ^{2} \theta+a\,\theta \sin \theta \cos \theta$
$\Rightarrow x \cos \theta+y \sin \theta=a$
Which is always a constant distance ' $a$ ' from the origin.