If $M$ is the foot of the perpendicular from a point $P$ on a parabola to its directrix and $SPM$ is an equilateral triangle, where $S$ is the focus , then $SP=$

$\mathrm{△}SPM$ is equilateral $\Rightarrow \mathrm{\angle }PSM={60}^{\circ }\Rightarrow \mathrm{\angle }PSX={60}^{\circ }$ 

$\therefore \text{ Slope of }\overleftrightarrow{SP}\text{ is }\tan {60}^{\circ }=\sqrt{3}\cdot S(a,0)$

Equation to $\overleftrightarrow{SP}\text{ is }y=\sqrt{3}(x-a).$

Parabola equation is $y^2=4ax$

$\begin{array}{l}\Rightarrow \left[\sqrt{3}\right(x-a){]}^2=4ax\Rightarrow 3x^2-10ax+3a^2=0\\ \Rightarrow (3x-a)(x-3a)=0\Rightarrow x=a/3\text{ or }3a\\ \therefore P(3a,2\sqrt{3}a).SP=\sqrt{4a^2+(2\sqrt{3}a{)}^2}=4a\end{array}$