An equilateral triangle is inscribed in the parabola `y^2=4ax` where are at the vertex of the parabola. find the length of the side of the triangle.

Options

A. `8sqrt3a`
B. `8sqrt2a`
C. `4sqrt3a`
D. `4sqrt2a`

Answer

A

Explanation

First, we draw the parabola in the positive side of X-axis and inside that, draw an equilaaterl `Delta oab.` let OB =1=OA=AB `angle BOA=60^(@)rArrangleBOP=30^(@)` in `DeltaBOP`
`sin30^(@)=(pb)/(ob)` `rArr1/2=(OP)/(1)rArrPB=1/2`
`and cos30^(@)=(OP)/(OB)`
`rArr sqrt3/2=(op)/(1)rArrop=1sqrt(3)/(2)`
`therefore` coordinates of B=(OP,PB)=( `(sqrt((1))/(2),1/2) will satisfy `y^2=4ax`
`(1/2)^(2)=(4axx1)/sqrt3(2)`
`rArr (1^2)/(1)=(4al)sqrt3/(2) `rArr 1=8sqrt3a`