If the length of the side of an equilateral triangle inscribed in the parabola, y2 = 4x so that one of its angular point is at the vertex is then l equals to....
Question
If the length of the side of an equilateral triangle inscribed in the parabola, y2 = 4x so that one of its angular point is at the vertex is then l equals to....
Solution
To find the length of the side of the equilateral triangle inscribed in the parabola y^2 = 4x, we need to follow these steps:
Step 1: Determine the coordinates of the vertex of the parabola. The vertex of the parabola y^2 = 4x is at the point (0, 0).
Step 2: Find the coordinates of one of the angular points of the equilateral triangle. Since one of the angular points is at the vertex of the parabola, we already know that it is at (0, 0).
Step 3: Find the coordinates of the other two angular points of the equilateral triangle. To find the other two angular points, we need to consider the symmetry of the equilateral triangle. Since the triangle is equilateral, the other two angular points will be equidistant from the vertex and lie on the parabola.
Step 4: Use the distance formula to find the distance between the vertex and one of the other angular points. Let's assume one of the other angular points is at (x, y). The distance between the vertex (0, 0) and (x, y) can be calculated using the distance formula: d = sqrt((x - 0)^2 + (y - 0)^2).
Step 5: Set up an equation using the equation of the parabola to find the coordinates of the other angular points. Since the other two angular points lie on the parabola y^2 = 4x, we can substitute the coordinates (x, y) into the equation to get y^2 = 4x. This will give us an equation that we can solve to find the values of x and y.
Step 6: Solve the equation to find the values of x and y. By substituting (x, y) into the equation y^2 = 4x, we can solve for x and y.
Step 7: Calculate the distance between the vertex and one of the other angular points. Using the values of x and y obtained from the previous step, substitute them into the distance formula to calculate the distance between the vertex and one of the other angular points.
Step 8: Multiply the distance calculated in Step 7 by 3 to find the length of one side of the equilateral triangle. Since the equilateral triangle has three equal sides, we can multiply the distance calculated in Step 7 by 3 to find the length of one side of the equilateral triangle.
Step 9: The length of one side of the equilateral triangle is equal to l. Therefore, l is equal to the length calculated in Step 8.
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