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Math Help - find the points of the surface where tangent line is parallel to xy-plane?

  1. #1
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    find the points of the surface where tangent plane is parallel to xy-plane?

    hi

    Question:
    find the points on the surfaces xy + yz + zx - x- z^2 = 0
    where the tangent plane is parallel to the xy-plane?

    my solution:
    tangent line is parallel to xy-plane = normal line to surface orthognal to xy-plane

    since normal line is orthognal to xy-plane that mean its intersect xy-plane
    and the normal is gradient f
    all what i will do is finding the parametric equation of the normal line and putting z=0

    equation of normal line:
    x = x0 + (y+z-1) t
    y = y0 + (x+z) t
    z = z0 + (y+x-2z) t
    putting z=0
    -z0 = (y+x-2z)t
    and then ??!!

    Edit:
    i saw question like this .. but it was given a point at which the normal line starts from
    i.e. x0,y0 and z0 are known ..
    Last edited by TWiX; December 22nd 2009 at 07:28 AM. Reason: edit the title
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  2. #2
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    f(x,y,z)=xy+yz+zx-x-z^{2}=0

    Compute the partials:

    {\nabla}F(x,y,z)=(y+z-1)i+(x+z)j+(-2z+x+y)k

    Now, try to find those points where the tangent plane is horizontal.

    The points where the derivatives equal 0.

    Solve for x,y, and z.
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  3. #3
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    y+z-1 = 0 ... (1)
    x+z = 0 ... (2)
    x + y - 2z = 0 ... (3)

    from (1) ---> y = -z + 1
    from (2) ---> x = -z
    substitute this values in (3)
    -z - z + 1 - 2z = 0
    -4z + 1 = 0
    z = 1/4
    x = -1/4
    y = 1 - (1/4) = 3/4
    point is ( -1/4 , 3/4 , 1/4 )

    is this right?
    Last edited by TWiX; December 22nd 2009 at 07:11 AM. Reason: changing z-coordinate from 3/4 to 1/4 ..
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  4. #4
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    but there is a problem
    in the exam he said
    find the points
    it will be more than one point??
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  5. #5
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    Oh Sorry
    its tangent plane
    not tangent line @@
    in the title of the thread
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