Posted by **sonu** on Monday, August 8, 2011 at 10:18am.

An equitorial triangles is inscribed in the parabola y^2=4ax, where one vertex is vertex of parabola. what will be the length of the side of triangle?

- math -
**Reiny**, Monday, August 8, 2011 at 2:55pm
Never heard of an "equitorial" triangle.

Did you mean equilateral ?

I will assume you did.

let P(x,y) be the point of contact , the the other is Q(x,-y)

but y^2 = 4ax, then x = y^2/(4a)

(0,0) is the vertex of the parabola, so (0,0) must be the third point of the equilateral triangle.

From P to the origin is

√( y^2 + y^4/(16a^2))

and PQ = 2y

so

√( y^2 + y^4/(16a^2)) = 2y

( y^2 + y^4/(16a^2) = 4y^2

y^4/(16a^2) = 3y^2

y^4= 48a^2y^2

y^2= 48a^2

y = ± a√48

= ±4a√3

since PQ = 2a

each of the sides is 8a√3

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