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M is the foot of the perpendicular from ...

`M` is the foot of the perpendicular from a point `P` on a parabola `y^2=4a x` to its directrix and `S P M` is an equilateral triangle, where S is the focus. Then find `S P` .

A

a

B

2a

C

3a

D

4a

Text Solution

Verified by Experts

The correct Answer is:
D
From the definition of the parabola, we have
`SP=PM`

It is given that SPM is an equilateal.
`:." "SP=PM=SMrArr/_PMS=60^(@)rArr/_SMZ=30^(@)`
In `Delta SMZ`, we have
`sin30^(@)=(SZ)/(SM)rArr1/2=(2a)/(SM)rArrSM=4a`
Hence `SP=SM=4a.`
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