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.

 

[tacman]

Hello,

Thank you for sharing your problem with us. Polynomials and spheres can definitely be tricky to work with, but with a systematic approach, we can solve this problem together.

First, let's simplify the equation to make it easier to work with. We can square both sides of the equation to eliminate the square roots, giving us:

(x+1)^2 + (y-5)^2 + (z-3)^2 = 4((x-6)^2 + (y-2)^2 + (z+2)^2)

Expanding the parentheses and simplifying, we get:

x^2 + y^2 + z^2 - 2x - 10y - 6z + 35 = 4x^2 - 48x + 144 + 4y^2 - 16y + 16 + 4z^2 + 16z + 16

Now, let's collect like terms and get all the variables on one side, giving us:

3x^2 - 2y^2 - 2z^2 + 50x + 26y + 22z - 205 = 0

We can now see that this equation is in the form of a general equation for a sphere, which is:

(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2

Where (h,k,l) is the center of the sphere and r is the radius. We can use this information to find the center and radius of the sphere in our problem.

By comparing our equation to the general equation, we can see that:

h = -50/6 = -25/3

k = -26/4 = -13/2

l = -22/4 = -11/2

And the radius can be found by taking the square root of the constant term, giving us:

r = sqrt(205/3)

Therefore, the set of all points that satisfy the given condition is a sphere with center (-25/3, -13/2, -11/2) and radius sqrt(205/3).

I hope this helps you solve the problem. Don't get discouraged by the messiness of polynomial equations, with practice and patience, you'll become more confident in working with them. Good luck!