Question

An equilatral triangle inscribed in parabola $y^2=4ax$ whose one vertex is at the vertex of parabola. Then the length of the side of the triangle is

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

Answer 1

The correct option is A $8\sqrt{3}a$

$\because$ Triangle is equilatral triangle and one of the vertex is lie on the vertex of the parbola
Let any point on the parabola is $(a{t}^2,2at)$
$\therefore$ length of the side of the triangle
$=\sqrt{(a{t}^2{)}^2+(2at{)}^2}\cdots \left(1\right)$
Coordinate of the other vertex will be $(a{t}^2,-2at)$
$\therefore$ length of the side of the trianlge
$=4at\cdots \left(2\right)$
Equating both the equation
$a^2{t}^4+4a^2{t}^2=16a^2{t}^{2}t=0,2\sqrt{3}$
For $t=0$ no triangle
For $t=2\sqrt{3}$
Length $=8\sqrt{3}a$