Linear Algebra / Gaussian Elimination

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Homework Help Overview

The problem involves finding coefficients a, b, c, and d for the equation of a circle in the xy-plane, given three specific points. The context is within linear algebra, specifically focusing on Gaussian elimination to solve a system of equations derived from substituting the points into the circle equation.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster expresses uncertainty about how to begin the problem. One participant suggests using Gaussian elimination to derive linear equations from the given points, noting the potential for expressing some variables in terms of others due to insufficient equations for a unique solution.

Discussion Status

Participants are actively discussing the setup of the problem and the implications of the resulting equations. There is acknowledgment of the complexity involved, particularly regarding the potential for complicated fractions in the Row Echelon Form. Guidance has been provided regarding the nature of the solution space.

Contextual Notes

There is an indication that the problem may not yield a unique solution due to the number of equations being less than the number of variables, which is a point of consideration in the discussion.

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Homework Statement



Find coefficients a,b,c and d so that the curve of a circle in an xy plane, with points (-4,5), (-2,7) and (4,-3), is given by the equation ax2 + ay2 + bx + cy + d = 0.

Not even sure where to start. Can anyone help me with this?
 
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Use Gaussian elimination. Substitute the x and y values for each point into the equation and get three linear equations for a, b, c and d. But you don't have enough equations to get a unique solution. You'll have to express three of those variables in terms of another. You've got to expect this. (ax^2 + ay^2 + bx + cy + d)/k=0 is the same circle for any nonzero k.
 
Thanks Dick. I guess I'll get some big ugly fractions in my Row Echelon Form, correct? So far I've got numbers over 53 in the 1st row, over 65 in the 2nd row, and the 3rd row looks like it'll be a 4 digit denominator.
 
page13 said:
Thanks Dick. I guess I'll get some big ugly fractions in my Row Echelon Form, correct? So far I've got numbers over 53 in the 1st row, over 65 in the 2nd row, and the 3rd row looks like it'll be a 4 digit denominator.

Probably. I didn't actually work it out, but it doesn't look like it was set up to come out nice. Sounds like you are the right track though.
 

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