The equation of normal at any point $\mathrm{\theta }$ to the curve $\mathrm{x}=\mathrm{acos}\mathrm{\theta }+a \mathrm{\theta sin}\mathrm{\theta }, \mathrm{y}=\mathrm{asin}\mathrm{\theta }-a \mathrm{\theta cos}\mathrm{\theta }$ is always at a distance of

$\begin{array}{l}\mathrm{x}=\mathrm{acos}\mathrm{\theta }+\mathrm{a\theta sin}\mathrm{\theta },\mathrm{y}=\mathrm{asin}\mathrm{\theta }-\mathrm{a\theta cos}\mathrm{\theta }\\ \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{a\theta sin}\mathrm{\theta }}{\mathrm{a\theta cos}\mathrm{\theta }}=\frac{\sin \mathrm{\theta }}{\cos \mathrm{\theta }}\end{array}$

Slope of normal $=\frac{-\cos \mathrm{\theta }}{\sin \mathrm{\theta }}$

Equation of normal at $\mathrm{\theta }$

$\begin{array}{l}\mathrm{y}-[\mathrm{asin}\mathrm{\theta }-\mathrm{a\theta cos}\mathrm{\theta }]=\frac{-\cos \mathrm{\theta }}{\sin \mathrm{\theta }}[\mathrm{x}-(\mathrm{acos}\mathrm{\theta }+\mathrm{a\theta sin}\mathrm{\theta }\left)\right]\\ \mathrm{ysin}\mathrm{\theta }-{\mathrm{asin}}^2\mathrm{\theta }+\mathrm{a\theta sin}\mathrm{\theta cos}\mathrm{\theta }=-\mathrm{xcos}\mathrm{\theta }+{\mathrm{acos}}^2\mathrm{\theta }+\mathrm{a\theta sin}\mathrm{\theta cos}\mathrm{\theta }\\ \mathrm{xcos}\mathrm{\theta }+\mathrm{ysin}\mathrm{\theta }-\mathrm{a}=0\end{array}$

The distance from origin to the normal

$=\frac{|-\mathrm{a}|}{\sqrt{{\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }}}=\mathrm{a}$