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Question
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.
Solution
Parabola y2 = 4ax, an equilateral triangle is formed.
Let the length of its side be p.
ΔOLP in OL2 = OP2 + LP2
p2 = `"OP"^2 + ("p"/2)^2`
∴ OP2 = `"p"^2 - "p"^2/4 = 3/4"p"`
∴ The coordinates of L are `(sqrt3/2, "p"/2)`.
This parabola is situated at y2 = 4ax.
∴ `("p"/2)^2 = 4"a". (sqrt3/2"p")`
or `"p"^2/4 = 4"a" . sqrt3/2 "p"`
p = `8sqrt3"a"`
Hence, the length of the side of an equilateral triangle is `8sqrt3"a"`.
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