Question

An equilateral triangle is inscribed in the parabola $y^2=4x$ . If a vertex of the triangle is at the vertex of the parabola, then the length of side of the triangle is

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

Answer 1

Answer: (B)

Let $OA=l,$ given, $y^2=4x$ ...(i)
In $\Delta OAM$ $\sin 30{}^{\circ }=\frac{AM}{l}=\frac{1}{2}$
$\Rightarrow$ $AM=\frac{l}{2}$ and $\cos 30{}^{\circ }=\frac{OM}{l}=\frac{\sqrt{3}}{2}\Rightarrow OM=\frac{l\sqrt{3}}{2}$ The coordinate of $A=(OM,AM)=\left(\frac{l\sqrt{3}}{2},\frac{l}{2}\right)$ which satisfy the equation of parabola Eq. (i), ${\left(\frac{l}{2}\right)}^2=4\left(\frac{l}{2}\right)\sqrt{3}$
$\Rightarrow$ $\frac{l^2}{4}=\frac{4l\sqrt{3}}{2}$
$\Rightarrow$ $l=8\sqrt{3}=OA$ Which is the required length of equilateral triangle.