Parametrize intersection of a plane and paraboloid

  1. 1. The problem statement, all variables and given/known data

    Parametrize the intersection of
    the paraboloid z = x2 + y2
    and the plane 3x -7y + z = 4
    between 0 [tex]\leq[/tex] t [tex]\geq[/tex] 2*pi

    When t = 0, x will be greatest on the curve.


    2. Relevant equations



    3. The attempt at a solution

    I never really know how to do these kinds of problem. I am more familiar with parametrizing straight lines. Here is what I have done so far

    I substitute the z in the plane equation with the paraboloid

    3x - 7y + x2 + y2 = 4
    x2 + 3x + (3/2)2 + y2 -7y + (7/2)2 = 37/2
    (x + 3/2) 2 + (y - 7/2)2 = 37/2

    which is a circle centered at (-3/2 , 7/2) with radius 37/2

    So to parametrize x, I did

    x = [tex]\sqrt{37/2}[/tex] - (3/2) at t = 0 so
    x = ([tex]\sqrt{37/2}[/tex] - 3/2) * cos(t)

    This may be wrong, but I am not sure. Please let me know if I am on the right track and how can I continue with this problem. The y and z components seem to be more complicated.
     
    Last edited: Jul 11, 2009
  2. jcsd
  3. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    What happened to the 4?:wink:

    Close, to find x, you start at the center (-3/2) and add some fraction of the radius....that means you have x=-3/2+f(t)*radius, and so you want x=-3/2+cos(t)*radius not cos(t)*(-3/2+radius)....make sense?
     
  4. I added (3/2)2 and (7/2)2 to both side so it will be greater than 4. I think I put in the wrong numbers when I use my calculator. It should be 37/2 instead of 29/2
    And that explanation made a lot of sense heh.

    So I ended up with
    x = -3/2 + ([tex]\sqrt{37/2}[/tex]*cos(t)
    y = 7/2 + ([tex]\sqrt{37/2}[/tex]*sint(t)

    Can anyone lead me on the right track to finding z? Thanks
     
  5. gabbagabbahey

    gabbagabbahey 5,013
    Homework Helper
    Gold Member

    z=x^2+y^2 :wink:
     
  6. Love it whenever a problem that looks complicated has simple solution. heh thanks =)
     
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