question_answer

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)  \[2\sqrt{3}\]

Correct Answer: B

Solution :

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.