Question

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

• A. $6a\sqrt{2}$
• B. $8a\sqrt{3}$
• C. $4a\sqrt{3}$
• D. None of these

Answer 1

The correct option is B $8a\sqrt{3}$

Let $AB=l$. Then,

$AM=l\cos {30}^o=\frac{l\sqrt{3}}{2}$ and, $BM=l\sin {30}^o=\frac{l}{2}$

So, the coordinates of B are $(\frac{\sqrt{3}l}{2},\frac{l}{2})$. Since, B lies on $y^2=4ax$. So, coordinates of B satisfy $y^2=4ax$.

Therefore, $\frac{l^2}{4}=4a\left(\frac{l\sqrt{3}}{2}\right)$

$l=8a\sqrt{3}$