# Thread: Find the equation of a sphere

1. ## Find the equation of a sphere

Find an equation of the sphere that passes through points (0, -2, 2), (4, 0, 0), (-2, 1, 5), and (7, 7, 9).

My first thought was to somehow find vectors from these points, but I'm not so sure that's the case.

I also think this involves a gigantic matrix with its determinant being solved for, but I'm not sure how to approach it.

On top of that, I think the midpoint rule might be involved somehow, but once again, I'm not exactly sure how to approach it without getting way off track.

Could someone help sort my ideas out and give me some direction? I just have too many thoughts about what to do, and it's frustrating me.

2. Originally Posted by wilcofan3
Find an equation of the sphere that passes through points (0, -2, 2), (4, 0, 0), (-2, 1, 5), and (7, 7, 9).

My first thought was to somehow find vectors from these points, but I'm not so sure that's the case.

I also think this involves a gigantic matrix with its determinant being solved for, but I'm not sure how to approach it.

On top of that, I think the midpoint rule might be involved somehow, but once again, I'm not exactly sure how to approach it without getting way off track.

Could someone help sort my ideas out and give me some direction? I just have too many thoughts about what to do, and it's frustrating me.
The general equation of a sphere is

$\displaystyle (x-a)^2+(y-b)^2+(z-c)^2 = r^2$

Substitute your 4 points into this and you'll get 4 equations for the 4 unknowns $\displaystyle a, b, c,\, \text{and}\,r$. If you substract the equations, say eqn 1 - eqn2, eqn2- eqn3, eqn3 - eqn4, you'll get three linear equations for $\displaystyle a, b, \, \text{and}\,c$ which you can solve. Then use any one of the original equation to find $\displaystyle r$.