Question

An equilateral triangle is inscribed in the parabola $y^2=4ax$ whose one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

• A. $8a\sqrt{3}$
• B. $a\sqrt{3}/8$
• C. $2a/\sqrt{3}$
• D. $4a\sqrt{3}$

Answer 1

Answer: (A)

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

Here $AP=l$ so $AR=l$ $cos30^{\circ }=l\frac{\sqrt{3}}{2}$
Also, $PR=lsin30^{\circ }=\frac{l}{2}$
Thus, $\left(\frac{l\sqrt{3}}{2},\frac{l}{2}\right)$ are the coordinates of the point $P$ lying
on the parabola $y^2=4ax$.
Therefore, $\frac{l^2}{4}=4a\left(\frac{l\sqrt{3}}{2}\right)$
$\Rightarrow l=8a\sqrt{3}$
Thus, $8a\sqrt{3}$ is tlie required length of the side of the equilateral triangle inscribed in the parabola $y^2=4ax$.