Question

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

Answer 1

REF.image

slope of $OA=tan30$ $=\frac{1}{\sqrt{3}}$

$\Rightarrow$ equation of $OA:$ $y=\frac{1}{\sqrt{3}}x$,

putting in parabola

$\frac{x^2}{3}=4ax\Rightarrow x=12a\Rightarrow y=4\sqrt{3}a$

$\Rightarrow AB=2\times 4\sqrt{3}a=8\sqrt{3}a$