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 $\bot$ on $BC$.

Then if $AB =l, AM = l\, cos\, 30^{\circ} = \frac{\sqrt{3}l}{2} $
and $BM= l \,sin \,30^{\circ} = \frac{1}{2}$
$\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 $=- 8\sqrt{3}$