To find the area of the equilateral triangle inscribed in the parabola \( y^2 = 8x \) with one vertex at the vertex of the parabola, we can follow these steps:
### Step 1: Understand the Parabola
The given parabola is \( y^2 = 8x \). This can be rewritten in the standard form \( y^2 = 4ax \) where \( a = 2 \). The vertex of the parabola is at the origin (0, 0).
### Step 2: Set Up the Triangle
Let the vertices of the equilateral triangle be \( A(0, 0) \) (the vertex of the parabola), \( B(x_1, y_1) \), and \( C(x_2, y_2) \). Since the triangle is symmetric about the x-axis, we can assume that \( B \) is at \( (x_1, y_1) \) and \( C \) is at \( (x_1, -y_1) \).
### Step 3: Use the Parabola Equation
Since points \( B \) and \( C \) lie on the parabola, they must satisfy the equation \( y^2 = 8x \). Therefore, we have:
\[
y_1^2 = 8x_1
\]
\[
y_2^2 = 8x_2
\]
Since \( y_2 = -y_1 \), we can write:
\[
(-y_1)^2 = 8x_2 \implies y_1^2 = 8x_2
\]
Thus, we have:
\[
8x_1 = 8x_2 \implies x_1 = x_2
\]
### Step 4: Find Coordinates of B and C
Let \( x_1 = x \). Then from the parabola equation:
\[
y_1^2 = 8x \implies y_1 = \sqrt{8x}
\]
So, the coordinates of points \( B \) and \( C \) are \( B(x, \sqrt{8x}) \) and \( C(x, -\sqrt{8x}) \).
### Step 5: Calculate the Length of the Sides
The distance \( AB \) is:
\[
AB = \sqrt{(x - 0)^2 + (\sqrt{8x} - 0)^2} = \sqrt{x^2 + 8x} = \sqrt{x(x + 8)}
\]
The distance \( AC \) is the same due to symmetry:
\[
AC = \sqrt{x^2 + 8x} = \sqrt{x(x + 8)}
\]
The distance \( BC \) is:
\[
BC = \sqrt{(x - x)^2 + (\sqrt{8x} - (-\sqrt{8x}))^2} = \sqrt{(2\sqrt{8x})^2} = 2\sqrt{8x}
\]
### Step 6: Set the Lengths Equal
Since the triangle is equilateral, we have:
\[
AB = AC = BC
\]
Thus:
\[
\sqrt{x(x + 8)} = 2\sqrt{8x}
\]
Squaring both sides:
\[
x(x + 8) = 4 \cdot 8x
\]
\[
x^2 + 8x = 32x
\]
\[
x^2 - 24x = 0
\]
Factoring gives:
\[
x(x - 24) = 0
\]
Thus, \( x = 0 \) or \( x = 24 \). Since \( x = 0 \) corresponds to the vertex, we take \( x = 24 \).
### Step 7: Find the Height
Substituting \( x = 24 \) into the parabola equation:
\[
y_1 = \sqrt{8 \cdot 24} = \sqrt{192} = 8\sqrt{3}
\]
### Step 8: Calculate the Area of the Triangle
The area \( A \) of the triangle can be calculated using the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base \( BC = 2\sqrt{8 \cdot 24} = 16\sqrt{3} \) and the height from \( A \) to \( BC \) is \( 8\sqrt{3} \):
\[
A = \frac{1}{2} \times 16\sqrt{3} \times 8\sqrt{3} = \frac{1}{2} \times 16 \times 8 \times 3 = 192
\]
### Final Area
Thus, the area of the equilateral triangle is:
\[
\boxed{192}
\]
