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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 -

    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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