Question

# $$M$$ is the foot of the perpendicular from a point $$\mathrm{P}$$ on the parabola $$y^{2}=8(x-3)$$ to its directrix and $$\mathrm{S}$$ is the focus of the parabola. If $$\triangle SPM$$ is an equilateral triangle, the length of each side of the triangle is

A
2
B
3
C
4
D
8

Solution

## The correct option is C $$8$$Since $$SPM$$ is an equilateral triangle $$\angle SPM={ 60 }^{ 0 }\Rightarrow \angle SMZ={ 30 }^{ 0 }$$FRom $$\triangle SZM$$$$\displaystyle \frac { SZ }{ SM } =\sin { { 30 }^{ 0 }\Rightarrow SM=2SZ } \\ \Rightarrow SM=2\times 4=8$$Hence length of each side of the triangle is $$8$$.Mathematics

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