Question

$M$ is the foot of perpendicular from a point $P$ on a parabola $y^2=4ax$ to its directrix and $SPM$ is an equilateral triangle, where $S$ is the focus of the parabola. Then the value of $\frac{SP}{a}$ is

Answer 1

From the definition of parabola, we have
$SP=PM$
$SPM$ is an equilateral triangle.
Therefore, $SP=PM=SM$
$\Rightarrow \angle PMS={60}^{\circ }$
$\Rightarrow \angle SMZ={30}^{\circ }$

In $△SMZ$, we have
$\sin {30}^{\circ }=\frac{SZ}{SM}$
$\Rightarrow \frac{1}{2}=\frac{2a}{SM}$
$\Rightarrow SM=4a$
$\therefore SP=SM=4a$
$\Rightarrow \frac{SP}{a}=\frac{4a}{a}=4$