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Math Help - Point on a surface

  1. #1
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    Unhappy Point on a surface

    Find a point on the surface of z = 8- 3x^2 - 2y^2 at which the tangent plane is perpendicular to the line x = 2 - 3t, y = 7 + 8t, z= 5 - t. Find the equation of the tangent line at that point.

    I don't know how to find the point. I only know how to find the equation of the tangent plane to the surface at a certain point.
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  2. #2
    Super Member malaygoel's Avatar
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    Quote Originally Posted by tdat1979 View Post
    Find a point on the surface of z = 8- 3x^2 - 2y^2 at which the tangent plane is perpendicular to the line x = 2 - 3t, y = 7 + 8t, z= 5 - t. Find the equation of the tangent line at that point.

    I don't know how to find the point. I only know how to find the equation of the tangent plane to the surface at a certain point.
    I think I can help you but first you have to tell how will you get tangent plane and what is the equation of tangent plane you got.
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  3. #3
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    Quote:
    Originally Posted by tdat1979
    Find a point on the surface of z = 8- 3x^2 - 2y^2 at which the tangent plane is perpendicular to the line x = 2 - 3t, y = 7 + 8t, z= 5 - t. Find the equation of the tangent line at that point.

    I don't know how to find the point. I only know how to find the equation of the tangent plane to the surface at a certain point.


    I think I can help you but first you have to tell how will you get tangent plane and what is the equation of tangent plane you got.
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    I don't know if if doing it right but i got the point (2,7,5) and vector
    (-3,8,-1) from the equation of the line. So the tangent plane would be
    -3(x-2) + 8(y-7) - (z-5) = 0 which would give me -3x + 8y - z = 45. If I'm wrong please tell me what I did wrong.
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  4. #4
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    Quote Originally Posted by tdat1979 View Post
    Find a point on the surface of z = 8- 3x^2 - 2y^2 at which the tangent plane is perpendicular to the line x = 2 - 3t, y = 7 + 8t, z= 5 - t. Find the equation of the tangent line at that point.

    I don't know how to find the point. I only know how to find the equation of the tangent plane to the surface at a certain point.
    So we want to know when the gradient of the surface is parallel to the tangent of the line.

    F(x,y,z)=3x^2+2y^2+z-8

    \nabla F(x,y,z)=6x \vec i +4y \vec j +\vec k

    The tangent vector to the line is -3 \vec i + 8\vec j -\vec k

    We know that 2 vectors are parallel if and only if v_1=cv_2 for some scalar (or if their cross product is 0)

    Now we need to solve the vector equation

    6x \vec i +4y \vec j +\vec k =c(-3 \vec i + 8\vec j -\vec k)

    Note that our only choice for c is c=-1 from the z component.

    so this implies that 6x=3 \iff x=\frac{1}{2} and
    4y=-8 \iff y=-2

    Now just plug these back into the function to find z.
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