An equilateral triangle is inscribed in the parabola `y^2=4a x ,` such that one vertex of this triangle coincides with the vertex of the parabola. Then find the side length of this triangle.

Explanation

As shown in the figure, equilateral triangle OPQ is symmetrical about x-axis.

Let OP=l.
In triangle OMP,
`OM=OPcos30^(@)=(sqrt(3l))/(2)`
`and Mp=Opsin 3^(@)=(1)/(2)`
`:.p-=((sqrt(3l))/(2),(l)/(2))`
P lies on parabola.
`:." "((l)/(2))^(2)=4a((sqrt(3l))/(2))`
`:." "l=8sqrt(3)a" "rArr" "OP=8sqrt(3)a`