Question

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

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

Answer 1

Answer: (C)

Let $A$ be the vertex and $ABC$ be the equilateral triangle inscribed in the hyperbola $y^2=4ax.AM$ is $\perp$ on $BC$.

Then if $AB=l,AM=lcos30^{\circ }=\frac{\sqrt{3}l}{2}$
and $BM=lsin30^{\circ }=\frac{1}{2}$
$\therefore$ the co-ordinates of the point $B$ are $\left(\frac{\sqrt{3}l}{2},\frac{l}{2}\right)$
Since $B$ lies on the parabola $y^2=4x$
$\therefore {\left(\frac{l}{2}\right)}^2=4l\cdot \frac{\sqrt{3}}{2}$
$\Rightarrow l=8\sqrt{3}$
Thus length of each side of the triangle $=-8\sqrt{3}$