Question

An equilateral triangle is inscribed in the parabola $y^2=8x$, with one of its vertices is the vertex of the parabola. Then, the length of the side of that triangle is

• A. $24\sqrt{3}$
• B. $16\sqrt{3}$
• C. $8\sqrt{3}$
• D. $4\sqrt{3}$

Answer 1

The correct option is B $16\sqrt{3}$

The vertex of the parabola $y^2=8x$ is $(0,0)$

Since the triangle is equilateral, the axis of the parabola bisects the angle and so we get ${30}^o$ as the angle above.

If the side intersects the parabola at $(2t^2,4t),tan\left({30}^o\right)=\frac{4t}{2t^2}$

$\Rightarrow t=2\sqrt{3}$

The point thus becomes $(24,8\sqrt{3})$

Length of the side thus becomes $\sqrt{576+192}=\sqrt{768}=16\sqrt{3}$