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Math Help - spherical

  1. #1
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    spherical

    Doubts with the response

    To determine the equation of the sphere that passes in A and B, and has straight center in s
    A(0,4,3) B(1,1,-5)
    (s):
    x=z+2
    y=z-3

    My solution:
    For the straight, center: C(1,1,1)

    R=d(C,A)=\sqrt{14}

    The equation is:
    (x-1)^2+(y-1)^2+(z-1)^2=14

    BUT THE ANSWER IS:
    (x-3)^2+(y+2)^2+(z-1)^2=49
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  2. #2
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    How did you find the center ? It is supposed to lie on the line S, so it must satisfy both equations, which it does'nt.

    The center must be equidistant from A and B, and satisfy the two equations. This makes 3 equations, allowing you to find the three components of the center.

    Laurent.
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  3. #3
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    Each of the points is the same distance from the center.
    So solve \left( {z + 2} \right)^2  + \left( {z - 7} \right)^2  + \left( {z - 8} \right)^2  = \left( {z + 1} \right)^2  + \left( {z - 4} \right)^2  + \left( {z + 5} \right)^2 .
    If correct, you will find z=1.
    Then you can find x&y on s.
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  4. #4
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    How do you get this equation, Plato ?

    A and B are at the same distance from the center O, so the coordinates (x,y,z) of the center satisfy:

    x^2+(y-4)^2+(z-3)^2=(x-1)^2+(y-1)^2+(z+5)^2.

    Expand this. The squares simplify, so you get a linear equation. The equations of S provide two others, and you have a linear system of three equations in three unknowns.
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  5. #5
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    Quote Originally Posted by Laurent View Post
    How do you get this equation, Plato ?
    It is easy to see. The center is on s. On s, x & y are defined in terms of z.
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  6. #6
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    Well done, Plato ! This is indeed quicker to write down this way.
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  7. #7
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    thanks
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