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Q.
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.
Conic Sections
Solution:
As shown in the figure $APQ$ denotes the equilateral triangle with its equal sides of length $l$ (say).
Here $AP = l$ so $AR = l$ $cos \,30^{\circ}=l\frac{\sqrt{3}}{2}$
Also, $PR = l \,sin \, 30^{\circ}=\frac{l}{2}$
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 $y^{2}=4ax$.
