Question

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

• A. $l=8a\sqrt{3}$
• B. $l=4a\sqrt{3}$
• C. $l=16a\sqrt{3}$
• D. $l=8a\sqrt{2}$

Answer 1

The correct option is A $l=8a\sqrt{3}$

R.E.F image

let the two vertices

of the triangle be $Q+R$

In $\Delta PRT$

$tan{30}^{\circ }=\frac{RT}{K}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{RT}{K}$

$RT=\frac{K}{\sqrt{3}}$

Now R Lies on parabola $y^2=4ax$

$\Rightarrow R(K,\frac{K}{\sqrt{3}})$

$\Rightarrow {\left(\frac{K}{\sqrt{3}}\right)}^2=4QK$

$\Rightarrow \frac{K}{3}=4Q$

$\Rightarrow K=12Q$

length of side of the triangle $=2\left(RT\right)$

$=2.\frac{K}{\sqrt{3}}=2.\frac{\left(12Q\right)}{\sqrt{3}}=8\sqrt{3}Q$