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Finding The Distance From A Paraboloid To A Plane.

  1. Feb 24, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the distance from the paraboloid z = X2 + 2Y2 to the plane
    2X + 8Y + Z = -8.


    2. Relevant equations

    The partial derivatives with respect to X, And Y for the paraboloid.



    3. The attempt at a solution

    My professor said we need to find the point where the tangent plane of the paraboloid is parallel to the plane. I can take the X, and Y partial derivatives, but then I do not know what to do.
     
  2. jcsd
  3. Feb 24, 2013 #2

    Dick

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    You should know what the normal vector to the plane is by looking at it. You want to find a point on the paraboloid whose normal vector is parallel to that. How would you find a normal vector to the surface at a point (x,y,z)?
     
  4. Feb 24, 2013 #3
    Oh I just need to move the Z over then take the gradient.
     
  5. Feb 24, 2013 #4

    Dick

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    Yes, and the simple way to check if two vectors are parallel is to see if one is a multiple of the other.
     
  6. Feb 24, 2013 #5
    Ok, so now I am stuck at the finding the vectors that are parallel. I know that they can be a multiple of each other. I got the gradient of < 2X, 4y, -1>. I know the normal vector is
    <2, 8, 1>.

    I set the equations to make:

    2X = 2K 4Y = 8K -1 = 1K

    K is just a constant. How do I solve these?
     
  7. Feb 24, 2013 #6

    Dick

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    Good job setting the equations up, but I'm having a hard time figuring out why you can't solve -1=1*K. Take another look at them.
     
    Last edited: Feb 24, 2013
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