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 the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15x$ subtends an angle $\theta$ at the vertex of the parabola,then $\sin \frac{\theta}{3}+\cos \frac{2\theta}{3}-\sec \frac{4\theta}{3}=$

The parametric equations of the parabola $x^2-8 x+12 y+15=0$ are

The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$,if

The equation of the latus rectum of a parabola is $x + y = 8$ and the equation of the tangent at the vertex is $x + y = 12$. The length of the latus rectum is:

If all the vertices of an equilateral triangle lie on the parabola $y^2=16x$ and one of them coincides with the vertex of that parabola,then the length of the side of that triangle is

Vedclass Test Series

Exam Paper Generator

Online Exam Module