Question

An equitorial triangles is inscribed in the parabola y^2=4ax, where one vertex is vertex of parabola. what will be the length of the side of triangle?

Answer 1

Never heard of an "equitorial" triangle.
Did you mean equilateral ?

I will assume you did.

let P(x,y) be the point of contact , the the other is Q(x,-y)
but y^2 = 4ax, then x = y^2/(4a)
(0,0) is the vertex of the parabola, so (0,0) must be the third point of the equilateral triangle.
From P to the origin is
√( y^2 + y^4/(16a^2))
and PQ = 2y
so
√( y^2 + y^4/(16a^2)) = 2y
( y^2 + y^4/(16a^2) = 4y^2
y^4/(16a^2) = 3y^2
y^4= 48a^2y^2
y^2= 48a^2
y = ± a√48
= ±4a√3

since PQ = 2a
each of the sides is 8a√3