Question

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

• A. 16 a $\sqrt{2}$
• B. $12a\sqrt{3}$
• C. $8a\sqrt{3}$
• D. $10a\sqrt{2}$

Answer 1

Answer: (C)

As shown in the figure $APQ$ denotes the equilateral triangle with its equal side of length / (say).
Here, $AP=I\text{ so }AR=I\cos 30^{\circ }$

Also, $PR=I\sin 30^{\circ }=\frac{1}{2}$.
Thus, $\left(\frac{1\sqrt{3}}{2},\frac{1}{2}\right)$ are the coordinates of the point $P$ lying on the parabola $y^2=4ax$.
Therefore, $\frac{I^2}{4}=4a\left(\frac{I\sqrt{3}}{2}\right)\Rightarrow I=8a\sqrt{3}$.
Thus, $8a\sqrt{3}$ is the required length of the side of the equilateral triangle inscribed in the parabola $y^2=4ax$.