An equilateral triangle is inscribed in parabola y^2=4axsuch that the vertex of the triangle is at vertex of parabola find area of triangle
To solve this problem, let's break it down step by step:
Step 1: Understand the problem
We are given that an equilateral triangle is inscribed in the parabola y^2 = 4ax, and the vertex of the triangle coincides with the vertex of the parabola. We need to find the area of the triangle.
Step 2: Understand the properties of an equilateral triangle
An equilateral triangle is a triangle in which all three sides are equal in length, and all three angles are equal to 60 degrees.
Step 3: Determine the coordinates of the vertex of the parabola
The vertex of the parabola y^2 = 4ax is (0, 0) since the equation represents a parabola opening to the right and its vertex lies on the y-axis.
Step 4: Determine the coordinates of the other two vertices of the triangle
Since the triangle is equilateral, we can determine the coordinates of the other two vertices by rotating the vertex of the parabola by 60 degrees in both clockwise and counterclockwise directions.
To rotate a point (x, y) by 60 degrees counterclockwise, we can use the following formulas:
x' = x * cos(60) - y * sin(60)
y' = x * sin(60) + y * cos(60)
Using these formulas, let's find the coordinates of the other two vertices of the triangle.
Step 5: Calculate the coordinates of the other two vertices
Let the coordinates of the vertex of the triangle be (0, 0).
Using the formulas from Step 4:
For the first vertex:
x' = 0 * cos(60) - 0 * sin(60) = 0
y' = 0 * sin(60) + 0 * cos(60) = 0
So, the coordinates of the first vertex are (0, 0).
For the second vertex, we substitute the coordinates of the vertex of the parabola (0,0) into the rotation formulas:
x' = 0 * cos(60) - 0 * sin(60) = 0
y' = 0 * sin(60) + 0 * cos(60) = 0
So, the coordinates of the second vertex are also (0, 0).
Therefore, all three vertices of the triangle are (0, 0).
Step 6: Calculate the area of the triangle
Since all three vertices of the triangle coincide, the triangle formed is a degenerate triangle (a triangle with area zero). Therefore, the area of the triangle is zero.
So, the area of the triangle inscribed in the parabola is 0.