Question

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

• A. $2a\sqrt{3}$
• B. $4a\sqrt{3}$
• C. $6a\sqrt{3}$
• D. $8a\sqrt{3}$

Answer 1

Answer: (D)

$In \Delta OCA$, $\tan\,30^{\circ}=\frac{AC}{OC}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{2at}{at^{2}}$



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

Again in $\Delta OCA$,
$OA=\sqrt{\left(OC\right)^{2}+\left(CA\right)^{2}}$

$=\sqrt{\left(at^{2}\right)^{2}+\left(2at\right)^{2}}$

$=\sqrt{\left\{a\cdot\left(2\sqrt{3}\right)^{2}\right\}^{2}+\left\{2a\cdot2\sqrt{3}\right\}^{2}}$

$=\sqrt{144\,a^{2}+48\,a^{2}}$

$=\sqrt{192\,a^{2}}$

$\Rightarrow OA=8\sqrt{3}\,a$