The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola, is :

To find the area of an equilateral triangle inscribed in the parabola y^2 = 8x, with one of its vertices on the vertex of the parabola, we can follow these steps:

Step 1: Determine the coordinates of the vertex of the parabola. The given equation y^2 = 8x represents a parabola with its vertex at the origin (0, 0).

Step 2: Find the coordinates of one of the vertices of the equilateral triangle. Since one of the vertices of the equilateral triangle is on the vertex of the parabola, we can take the vertex of the parabola as one of the vertices of the triangle. Therefore, one of the vertices of the equilateral triangle is (0, 0).

Step 3: Find the coordinates of the other two vertices of the equilateral triangle. To find the other two vertices, we need to consider the properties of an equilateral triangle. Since all sides of an equilateral triangle are equal, we can determine the length of one side by finding the distance between the vertex of the parabola and any other point on the parabola.

Let's consider a point (x, y) on the parabola. The distance between this point and the vertex of the parabola is given by the formula: distance = sqrt((x - 0)^2 + (y - 0)^2) = sqrt(x^2 + y^2)

Since the triangle is equilateral, the distance between any two vertices is equal to the length of one side. Therefore, the distance between the vertex of the parabola and any other point on the parabola is equal to the length of one side of the equilateral triangle.

Step 4: Find the length of one side of the equilateral triangle. To find the length of one side, we need to find the distance between the vertex of the parabola and any other point on the parabola. Let's consider a point (x, y) on the parabola.

Substituting y^2 = 8x into the distance formula, we have: distance = sqrt(x^2 + (8x)^2) = sqrt(x^2 + 64x^2) = sqrt(65x^2) = sqrt(65)x

Since the distance between the vertex of the parabola and any other point on the parabola is equal to the length of one side of the equilateral triangle, we have: length of one side = sqrt(65)x

Step 5: Find the area of the equilateral triangle. The area of an equilateral triangle can be calculated using the formula: area = (sqrt(3)/4) * (length of one side)^2

Substituting the length of one side from Step 4 into the formula, we have: area = (sqrt(3)/4) * (sqrt(65)x)^2 = (sqrt(3)/4) * 65x = (65sqrt(3)/4) * x

Therefore, the area of the equilateral triangle inscribed in the parabola y^2 = 8x, with one of its vertices on the vertex of the parabola, is (65sqrt(3)/4) * x square units.

To find the area of an equilateral triangle inscribed in the parabola y^2 = 8x, we need to follow these steps:

Step 1: Determine the coordinates of the vertex of the parabola. The given equation y^2 = 8x represents a parabola with its vertex at the origin (0, 0).

Step 2: Find the coordinates of one of the vertices of the equilateral triangle. Since one of the vertices of the equilateral triangle is on the vertex of the parabola, we can take the vertex of the parabola as one of the vertices of the equilateral triangle. Therefore, one of the vertices of the equilateral triangle is (0, 0).

Step 3: Find the coordinates of the other two vertices of the equilateral triangle. To find the other two vertices, we need to consider the properties of an equilateral triangle. Since all sides of an equilateral triangle are equal, we can determine the length of one side by finding the distance between the vertex of the parabola and any other point on the parabola.

Let's consider a point (x, y) on the parabola. The distance between the vertex (0, 0) and the point (x, y) can be found using the distance formula:

Distance = sqrt((x - 0)^2 + (y - 0)^2) = sqrt(x^2 + y^2)

Since the triangle is equilateral, the distance between the vertex (0, 0) and any other vertex is equal to the distance between the other two vertices. Let's denote this distance as "d".

Therefore, we have: d = sqrt(x^2 + y^2)

To form an equilateral triangle, the other two vertices must be symmetric with respect to the y-axis. So, the coordinates of the other two vertices are (-x, y) and (-x, -y).

Step 4: Find the length of one side of the equilateral triangle. To find the length of one side of the equilateral triangle, we need to find the distance between any two vertices. Let's consider the distance between (0, 0) and (-x, y):

Distance = sqrt((-x - 0)^2 + (y - 0)^2) = sqrt(x^2 + y^2)

Therefore, the length of one side of the equilateral triangle is sqrt(x^2 + y^2).

Step 5: Find the area of the equilateral triangle. The area of an equilateral triangle can be found using the formula:

Area = (sqrt(3) / 4) * (side length)^2

Substituting the length of one side of the equilateral triangle, we have:

Area = (sqrt(3) / 4) * (sqrt(x^2 + y^2))^2 = (sqrt(3) / 4) * (x^2 + y^2)

So, the area of the equilateral triangle inscribed in the parabola y^2 = 8x, with one of its vertices on the vertex of this parabola, is (sqrt(3) / 4) * (x^2 + y^2) square units.