Question

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

Find the length of the side of the triangle.

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

Answer 1

The correct option is A

$8a\sqrt{3}$

The explanation for the correct option:

Step 1. Draw the diagram :

Step 2. Find the length of the side of the triangle :

Let the two vertices of the triangle be $Q$ and $R$.

The points $Q$ and $R$ will have the same $x-$coordinate $=k$(say)

In the right-angled triangle $PRT$

$\begin{array}{rcl}\tan 30°& =& RT/k\\ \frac{1}{\sqrt{3}}& =& RT/k\\ RT& =& \frac{k}{\sqrt{3}}\end{array}$

The coordinate of $R=(k,k/\sqrt{3})$

$R$lies on the parabola $y^2=4ax$

$$\begin{gathered} \Rightarrow {\left(\frac{k}{\sqrt{3}}\right)}^2=4ak \\ \Rightarrow \frac{k^2}{3}=4ak \\ \Rightarrow k=12a \end{gathered}$$

Therefore the length of the side of the triangle $\Rightarrow 2RT=2\left(\frac{k}{\sqrt{3}}\right)=2\left(\frac{12a}{\sqrt{3}}\right)=\frac{24a\sqrt{3}}{3}=8a\sqrt{3}$

Hence, the correct option is (A).