# Show that the normal at any point $$\theta$$ to the curve $x = a \cos\theta + a \: \theta \sin\: \theta, y = a\sin\theta - a\theta \cos\theta$ is at a constant distance from the origin.

Toolbox:
• $\large\frac{d}{dx}$$(\sin\theta)=\cos\theta • \large\frac{d}{dx}$$(\cos\theta)=-\sin\theta$
• $\large\frac{d}{dx}$$(x^n)=nx^{n-1} Step 1: We have x=a\cos\theta+a\theta\sin\theta \qquad\quad\; y=a\sin\theta-a\theta\sin\theta Differentiating with respect to x \large\frac{dx}{d\theta}$$=-a\sin\theta+a\theta\cos\theta+a\sin\theta$
$\quad\;=a\theta\cos\theta$