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

Explanation

To find the length of the side of an equilateral triangle inscribed in the parabola \( y^2 = 4ax \) with one vertex at the vertex of the parabola, we can follow these steps:

### Step 1: Understand the Geometry
The parabola \( y^2 = 4ax \) opens to the right, and its vertex is at the origin \( O(0, 0) \). Let the vertices of the equilateral triangle be \( A(0, 0) \) (the vertex of the parabola), \( B(k, 2\sqrt{ak}) \), and \( C(k, -2\sqrt{ak}) \).

### Step 2: Determine the Coordinates of Points B and C
Since the points \( B \) and \( C \) lie on the parabola, we can find their coordinates:
- For point \( B \):
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