An equitorial triangles is inscribed in the parabola y^2=4ax, where one vertex is vertex of parabola. what will be the length of the side of triangle?
Never heard of an "equitorial" triangle.
Did you mean equilateral ?
I will assume you did.
let P(x,y) be the point of contact , the the other is Q(x,-y)
but y^2 = 4ax, then x = y^2/(4a)
(0,0) is the vertex of the parabola, so (0,0) must be the third point of the equilateral triangle.
From P to the origin is
√( y^2 + y^4/(16a^2))
and PQ = 2y
so
√( y^2 + y^4/(16a^2)) = 2y
( y^2 + y^4/(16a^2) = 4y^2
y^4/(16a^2) = 3y^2
y^4= 48a^2y^2
y^2= 48a^2
y = ± a√48
= ±4a√3
since PQ = 2a
each of the sides is 8a√3
To find the length of the side of the equilateral triangle inscribed in the parabola y^2 = 4ax, we need to find the coordinates of the three vertices of the triangle.
Let's start by finding the coordinates of the vertex of the parabola. The vertex of a parabola in the form y^2 = 4ax is located at (h, 0), where h is the x-coordinate of the vertex. In this case, the vertex lies on the y-axis, so its x-coordinate is 0.
The vertex of the parabola is (0, 0).
Next, we need to find the coordinates of the other two vertices of the equilateral triangle. Since the triangle is equilateral, the other two vertices will lie on the parabola and will have the same y-coordinate as the vertex of the parabola (0, 0). Let's call these vertices A and B.
To find the coordinates of A and B, we can substitute y = 0 into the equation y^2 = 4ax:
0^2 = 4a(0)
0 = 0
Thus, A and B will have the same x-coordinate as the vertex of the parabola, which is 0.
The coordinates of the three vertices of the equilateral triangle are:
Vertex A: (0, 0)
Vertex B: (0, 0)
Vertex C: (h, 0), where h is the x-coordinate of the vertex of the parabola.
Since the triangle is equilateral, all sides of the triangle will have the same length. To find the length of one side, we simply need to find the distance between any two vertices of the triangle.
Let's calculate the distance between vertices A and C:
Distance AC = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((h - 0)^2 + (0 - 0)^2)
= sqrt(h^2)
= |h|
Therefore, the length of one side of the equilateral triangle inscribed in the parabola y^2 = 4ax is |h|.