Constructing Equations for a Circle with 3 Points

[IniquiTrance]

Homework Statement

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

$$ax^{2} + ay^{2} + bx + cy + d = 0$$

Points:

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

Homework Equations

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!

 

[Dick]

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.

 

[IniquiTrance]

What if I used Gauss Jordan elimination to find a,b and c in terms of parameter d, would that sufficiently answer the question?

 

[Dick]

Sure, I suppose. The 'fourth parameter' is really that you can divide your whole equation by anyone 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.