Prove the normal will be at constant distance form origin in this parametric function?

Question

Given a function,

$x = a(cos \theta + \theta \sin\theta])$, $y = a(sin\theta - \theta\ cos\theta)$, $a \in R$

Prove that the normal drawn on each point is at constant distance form the origin? If possible? I tried to plot it but found that it is not the case?

Answer 1

$$r(t):=\left(a(\cos t+t\sin t)\,,\,a(\sin t-t\cos t)\right)\implies$$

$$T(t):=r'(t)=(at\cos t\,,\,at\sin t)\implies \frac{T(t)}{||T(t)||}=(\cos t, \sin t)$$

$$T'(t)=(a\cos t-at\sin t\,,\,a\sin t+at\cos t)\implies $$

$$\implies N(t):=\frac{T'(t)}{||T'(t)}||=\left(\frac{\cos t}{a(1+t^2)}-\frac{t\sin t}{a(1+t^2)}\;,\;\frac{\sin t}{a(1+t^2)}+\frac{t\cos t}{a(1+t^2)}\right)$$

and now check that $\;||N(t)||\;$ doesn't depend on $\;t\;$ ...