If one of the vertices of an equilateral tria | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the side length of the equilateral triangle be $a$. Since one vertex is at the origin $(0,0)$ and the triangle is symmetric about the $x$-axis

Vedclass · Accepted Answer

$432$

Vedclass · Answer

(D) Let the side length of the equilateral triangle be $a$. Since one vertex is at the origin $(0,0)$ and the triangle is symmetric about the $x$-axis, the other two vertices lie on the parabola $y^2=12x$ at angles of $30^{\circ}$ and $-30^{\circ}$ with the $x$-axis. The coordinates of the vertex $A$ are $(a \cos 30^{\circ}, a \sin 30^{\circ})$. Since $A$ lies on the parabola $y^2=12x$, we have: $(a \sin 30^{\circ})^2 = 12(a \cos 30^{\circ})$ $a^2 \sin^2 30^{\circ} = 12a \cos 30^{\circ}$ $a^2 (1/4) = 12a (\sqrt{3}/2)$ $a/4 = 6\sqrt{3}$ $a = 24\sqrt{3}$. The area of an equilateral triangle with side $a$ is $\frac{\sqrt{3}}{4} a^2$. Area $= \frac{\sqrt{3}}{4} (24\sqrt{3})^2 = \frac{\sqrt{3}}{4} (576 	imes 3) = \frac{\sqrt{3}}{4} (1728) = 432\sqrt{3}$ sq. units.

If one of the vertices of an equilateral triangle inscribed in the parabola $y^2=12x$ coincides with the vertex of the parabola, then the area (in sq. units) of that triangle is (in $\sqrt{3}$)

Similar Questions

If $x-2=t^2$ and $y=2t$ are the parametric equations of the parabola $y^2=a(x-b)$,then the value of $a+b$ equals

If $PSQ$ is a focal chord of the parabola $y^2 = 8x$ such that $SP = 6$,then find the length of $SQ$.

The area of the triangle formed by the lines joining the vertex of the parabola $x^{2}=12y$ to the ends of the latus rectum is

If two tangents drawn from a point $P$ to the parabola $y^2 = 4x$ are such that the slope of one tangent is double the other,then $P$ lies on the curve:

Two parabolas have the same focus. If their directrices are the $x$-axis and the $y$-axis respectively,then the slope of their common chord is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module