An Equilateral triangle is inscribed in the parabola if one vertex of triangle is at the vertex of parabola then
Answers
Answer:
The vertex of the parabola y
2
=8x is (0,0)
Since the triangle is equilateral, the axis of the parabola bisects the angle and so we get 30
o
as the angle above.
If the side intersects the parabola at (2t
2
,4t),tan(30
o
)=
2t
2
4t
⇒t=2
3
The point thus becomes (24,8
3
)
Length of the side thus becomes
576+192
=
768
=16
3
Step-by-step explanation:
Given : An Equilateral triangle is inscribed in the parabola one vertex of triangle is at the vertex of parabola
To find : length of the side of the triangle.
Solution:
y² = 4ax parabola
Vertex of triangle
(0 , 0) , ( k , √4ak) , ( k , -√4ak)
Side = √4ak +√4ak = 2√4ak = 4√ak
Side = √(k -0)² + (√4ak - 0)² = √(k² + 4ak )
Equating side
=> √(k² + 4ak ) = 4√ak
Squaring both sides
k² + 4ak = 16ak
=> k² = 12ak
=> k =12a
Side = 4√ak
= 4√a*12a
= 8a√3
length of the side of the triangle. 8a√3
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