Linear Algebra / Gaussian Elimination

[page13]

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?

 

[Dick]

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.

 

[page13]

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.

 

[Dick]

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.