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Tangent plane.

  1. Feb 5, 2008 #1
    Please,help me with this problem.
    1. The problem statement, all variables and given/known data
    Prove that the two families of parabolas
    [tex]y^2=4a(a-x),a>0[/tex] and [tex]y^2=4b(b+x),b>0[/tex]
    form an orthogonal net. Specifically, check that for any a, b > 0 these two parabolas
    are perpendicular to each other at the points where they intersect.

    3. The attempt at a solution

    Their tangent spaces at point [tex](x_0,y_0)[/tex] are

    [tex]2y_0(y-y_0)+4a(x-x_0)=0[/tex]

    [tex]2y_0(y-y_0)-4b(x-x_0)=0[/tex]

    If they are perpendicular then we have

    [tex]4y_0^2-16ab=0\Rightarrow y_0^2=4ab[/tex]

    from the equations of parabolas we have

    [tex]y_0^2=4a(a-x_0)[/tex]

    [tex]y_0^2=4b(b+x_0)[/tex]

    if we substitute [tex]x_0[/tex]

    [tex]y_0^2=4ab[/tex]
    So they are perpendicular.
     
  2. jcsd
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