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 \(AR = l \cos30°\)

\(= l\, \frac{\sqrt 3}2\)

Also,  \(PR = l \cos30° =\frac l2\)

Thus \( l\, \frac{\sqrt 3}2, \frac l2\) are the coordinates of the point P lying on the parabola y² = 4ax.

Therefore, \(\frac {l^2}4 = 4a\frac {l\sqrt 3}2\)

⇒ \(l = 8a\sqrt 3\)

Thus, 8√3 is the required length of the side of the equilateral triangle inscribed in the parabola y² = 4ax.

Answer 2

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

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

Therefore,

Thus, 8a√3 is the required length of the side of the equilateral triangle inscribed in the parabola y² = 4ax.