$\text{ From a point }(\sin \theta ,\cos \theta )\text{ , if three normals can be drawn to the parabola }{y}^2=4ax,\text{ then the value of }a\text{ is }$

$\text{ Point }(\sin \theta ,\cos \theta )\text{ lies on the circle }{x}^2+y^2=1$$\mathrm{\forall }\theta \in R$

$\text{ Now, three normals can be drawn to the parabola }{y}^2=4ax\text{ if }$

$x=\left|2a\right|\text{ meets this circle. }$

$\text{ Hence, we must have }$

$\begin{array}{l}\cos \theta >\left|2a\right|\\ \text{ or }0<\left|2a\right|<1\\ \text{ or }0<\left|a\right|<\frac{1}{2}\\ \text{ or }a\in \left(-\frac{1}{2},0\right)\cup \left(0,\frac{1}{2}\right)\end{array}$