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.

Answer 1

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"`.