The area (in sq.

Question

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^2=8x$, with one of its vertices on the vertex of this parabola, is:

Accepted Answer

By symmetry, the other two vertices are $(x_1,\pm y_1)$ with $y_1^2=8x_1$ and $\dfrac{y_1}{x_1}=\tan30^\circ=\dfrac1{\sqrt3}$ (each side makes $30^\circ$ with the axis).

$$\frac{x_1^2}3=8x_1\Rightarrow x_1=24,\qquad y_1=\frac{24}{\sqrt3}=8\sqrt3.$$

Side $=2y_1=16\sqrt3$; Area $=\dfrac{\sqrt3}4(16\sqrt3)^2=\dfrac{\sqrt3}4(768)=192\sqrt3.$

Solutions · 1