Explanation

(a)
Given, $x = a(\cos{\theta + \theta}\sin\theta)$
And $y = a(\sin{\theta –}\theta\cos\theta)$
$\Rightarrow \ \ \ \frac{dx}{d\theta} = a( - \sin\theta + \sin\theta + \theta\cos\theta)$
$\Rightarrow \ \ \ \ \ \frac{dx}{d\theta} = a\theta\cos\theta$
And $\frac{dy}{d\theta} = a(\cos{\theta - \cos{\theta + \theta\sin\theta}})$
$\Rightarrow \ \ \ \frac{dy}{d\theta} = a\theta\sin\theta$
$\therefore\ \ \ \ \ \ \frac{dy}{dx} = \tan\theta$
So, equation of normal is
$\Rightarrow \ \ \ \ \ \ y - {asin}{\theta + a\theta\cos{\theta = - \frac{\cos\theta}{\sin\theta}}}(x - {acos}\theta - a\theta\sin\theta)$
$y\sin\theta - a\sin^{2}\theta + a\theta\cos\theta\sin\theta$
$= - xcos\ \theta + a\cos^{2}\theta + a\theta\sin{\theta\cos\theta}$
$\Rightarrow \ \ \ \ \ \ x\cos\theta + y\sin\theta = a$
It is always at a constant distance ‘$a$’ from origin.