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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
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B
3
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C
4
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D
8
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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 $$8$$.

390482_37143_ans_fc26797908fa4d0bb531bc08c6af1719.png

Mathematics

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