An equilateral triangle is inscribed in the parabola y² = 4ax whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

APQ denotes the equilateral triangle with its equal sides of length l (say).

Here, AP = l, so AR = l cos 30° = (l√3)/2

Also, PR = l sin 30° = l/2

Thus, (l√3/2, l/2) are the coordinates of the point P lying on the parabola y² = 4ax.

Therefore, (l²/4) = 4a(l√3/2) ⇒ l = 8a√3

Hence, 8a√3 is the required length of the side of the equilateral triangle inscribed in the parabola y² = 4ax.