Find an equation in rectangular coordinates for the sphere consisting of points whose distance to $(3,3,3)$ is twice their distance to $(0,0,0)$.

Question

For example, $(1,1,1)$ is such a point. The sphere must contain all points that satisfy the condition.

So, I've been milling over this question on and off for the past few days and just can't seem to figure it out. I think I would have to use the distance formula from some point to (0,0,0) or (3,3,3) but can not wrap my head around how to set it up, or what point to use. Any help would be greatly appreciated.

Answer 1

Following your lead, here is how to get started:

$$\left(\text{distance to }(3,3,3)\right)^2=\left(2\cdot{}\left(\text{distance to }(0,0,0)\right)\right)^2$$ $$(x-3)^2+(y-3)^2+(z-3)^2=4\left(x^2+y^2+z^2\right)$$

A few more steps of algebra and you can expand this, regroup terms, normalize, complete the square, and end up with a sphere equation in standard form.

Answer 2

The standard way to do this is

• let $(x, y, z)$ be an arbitrary point on your surface,

• using your geometry toolkit write the formulas for the various distances you're interested in,

• set the appropriate quantities equal to each other,

• and finally clean up the algebra by possibly squaring thing and collecting like terms.