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System of linear equations

  1. Oct 23, 2009 #1
    1. The problem statement, all variables and given/known data

    Find coefficients a, b, c, d so that the circle with the following 3 points satisfies the equation:

    [tex]ax^{2} + ay^{2} + bx + cy + d = 0[/tex]

    Points:

    (-4, 5)
    (4, -3)
    (-2, 7)



    2. Relevant equations



    3. The attempt at a solution
    I'm wondering if since I can only construct 3 equations from the 3 points, if I will have to make one unknown a parameter - probably d.

    Is there a way to construct a 4 th equation which I'm missing?

    Thanks!
     
  2. jcsd
  3. Oct 23, 2009 #2

    Dick

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    The parameters a,b,c and d are not independent if you are given that it's a circle. Write the equation of a circle in the form (x-a)^2+(y-b)^2=r^2. Now you only have three parameters. And you have three points.
     
    Last edited: Oct 23, 2009
  4. Oct 23, 2009 #3
    What if I used Gauss Jordan elimination to find a,b and c in terms of parameter d, would that sufficiently answer the question?
     
  5. Oct 23, 2009 #4

    Dick

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    Sure, I suppose. The 'fourth parameter' is really that you can divide your whole equation by any one of the four parameters that is nonzero and eliminate it. It was never really there to begin with. I.e. x^2+y^2+bx+cy+d=0 is also just as good.
     
    Last edited: Oct 23, 2009
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