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An equilateral triangle is inscribed in the parabola y2=4 ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.
- a)8√3a
- b)2√3a
- c)3√2a
- d)√3a
Correct answer is option 'A'. Can you explain this answer?
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An equilateral triangle is inscribed in the parabolay2=4ax, where one ...
Let the vertices of the equilateral triangle be A, B, and C, with A at the vertex of the parabola.
Since the triangle is equilateral, all sides are equal in length. Let the side length be s.
The coordinates of A are (0, 0) and the equation of the parabola is y^2 = 4ax. Therefore, the x-coordinate of B and C is s/2.
The y-coordinate of B and C can be found by substituting s/2 for x in the equation of the parabola:
y^2 = 4a(s/2)
y^2 = 2as
y = ±√(2as)
Since the triangle is equilateral, the height of the triangle is also s√3/2.
We can use the distance formula to find the distance between A and B (or A and C):
AB^2 = (s/2 - 0)^2 + (√(2as) - 0)^2
AB^2 = s^2/4 + 2as
Since all sides are equal in length, we can set AB^2 = BC^2 = AC^2 = s^2:
s^2/4 + 2as = s^2
s^2/4 = 2as
s = 4a/√3
Therefore, the side length of the equilateral triangle is 4a/√3, and the coordinates of B and C are (4a/√3, ±2a√2/√3).
Note: The condition that one vertex is (8a, 8a√3) is not necessary to solve the problem.
Since the triangle is equilateral, all sides are equal in length. Let the side length be s.
The coordinates of A are (0, 0) and the equation of the parabola is y^2 = 4ax. Therefore, the x-coordinate of B and C is s/2.
The y-coordinate of B and C can be found by substituting s/2 for x in the equation of the parabola:
y^2 = 4a(s/2)
y^2 = 2as
y = ±√(2as)
Since the triangle is equilateral, the height of the triangle is also s√3/2.
We can use the distance formula to find the distance between A and B (or A and C):
AB^2 = (s/2 - 0)^2 + (√(2as) - 0)^2
AB^2 = s^2/4 + 2as
Since all sides are equal in length, we can set AB^2 = BC^2 = AC^2 = s^2:
s^2/4 + 2as = s^2
s^2/4 = 2as
s = 4a/√3
Therefore, the side length of the equilateral triangle is 4a/√3, and the coordinates of B and C are (4a/√3, ±2a√2/√3).
Note: The condition that one vertex is (8a, 8a√3) is not necessary to solve the problem.
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An equilateral triangle is inscribed in the parabolay2=4ax, where one ...
The given parabola is y2=4ax ...(i)
Let OA(=l) be the side of equilateral triangle.
Then OL=lcos30°= √3l/2
and LA=lsin 30°= l/2
∴ The co-ordinates of A are
⇒ l=8√3a
Hence the length of the side of the triangle = 8√3a units.
Let OA(=l) be the side of equilateral triangle.
Then OL=lcos30°= √3l/2
and LA=lsin 30°= l/2
∴ The co-ordinates of A are
⇒ l=8√3a
Hence the length of the side of the triangle = 8√3a units.
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