Question

An equilateral triangle is inscribed in the parabola y² = 4ax whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

Answer 1

As shown in the figure APQ denotes the equilateral triangle with its equal sides of length l (say).

Here AP = l

So AR = l cos30°

= `l  sqrt(3)/2`

Also, PR = `l  sin 30^circ = l/2`.

Thus `(lsqrt(3))/2, l/2` are the coordinates of the point P lying on the parabola y² = 4ax.

Therefore, `l^2/4 = 4a  (lsqrt(3))/2`

⇒ `l = 8 asqrt(3)`.

THus, 8 `asqrt(3)` is the required length of the side of the equilateral triangle inscribed in the parabola y² = 4ax.