Problem with polynomials and spheres

[markcholden]

Hi all,

I've got this problem:

Consider the points P such that the distance from P to A(-1,5,3) is twice the distance from P to B(6,2,-2). Show that the set of all such points is a sphere, and find its center and radius.

I think the setup should be this:

sqrt[(x+1)^2 + (y-5)^2 + (z-3)^2] = 2*sqrt[(x-6)^2 + (y-2)^2 + (z+2)^2]

But when I try to work it out, I just end up with a mess and no equation for a sphere. Any help appreciated.

 

[AKG]

Don't use "code" tags like that - it makes people have to scroll to read your problem/solution. Anyways, square both sides, move everything to the left side, complete the square in each of x, y, and z, and take the constant numbers left over from completing the square over to the right side, and express the right side as a square. You'll end up, then, with something of the form:

(x-a)² + (y-b)² + (z-c)² = r²

from which you can easily read off the center and radius.