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Math Help - Help with equation of plane tangent to surface

  1. #1
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    Help with equation of plane tangent to surface

    x^2 + y^2 +z^2 = 1
    Let (a,b,c) be a point on the surface, find the equation of plane tangent to this surface at (a, b, c)

    Do I put it in terms of z?
    f(x,y) = z = sqroot( 1-x^2-y^2)
    Solve for fx(x,y) and fy(x,y) and plugging in 'a' and 'b' for x and y

    z= (fx(a,b)(x-a) + (fy(a,b)(y-b) + c

    Is this correct? Thanks in advance.
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  2. #2
    MHF Contributor
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    Hi

    Here is a simple way but I do not know if it is in the spirit of the exercise
    x + y + z = 1 is the sphere (center O, radius 1)
    A plane tangent to the sphere at point M(a,b,c) is perpendicular to vector OM whose coordinates are (a,b,c)
    Therefore an equation of the plane is ax + by + cz + k = 0

    To find k you just have to substitute (a,b,c) to (x,y,z), M being on the plane
    a + b + c + k = 0 therefore k = - a - b - c

    An equation of the plane is ax + by + cz - a - b - c = 0
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