What Is the Equation for Points Equidistant from Two Given Points in 3D Space?

[fk378]

Homework Statement

Find an equation of the set of all points equidistant from the set points A(-1,5,3) and B(6,2,-2). Describe the set.

Homework Equations

d= sqrt [(x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2]

The Attempt at a Solution

I solved the distance between A and B and got d=sqrt 83. If a sphere is constructed that passes through these 2 points, then the center will be equidistant from both. Therefore that sphere will have radius=(sqrt 83)/2.

I don't know where to take it from here. Any thoughts?

 

[symbolipoint]

Find the midpoint of AB.
Determine slope of AB. Determine negative reciprocal of this slope (Do you understand why?)
One or two more steps... can you do this?

 

[dynamicsolo]

Homework Statement

Find an equation of the set of all points equidistant from the set points A(-1,5,3) and B(6,2,-2). Describe the set.

...

If a sphere is constructed that passes through these 2 points, then the center will be equidistant from both.

This is not the set the problem is asking for. It is true that the midpoint of the segment AB is equidistant from both A and B. But plainly points A and B would each have to be on the surface of that sphere you describe, yet each of those can hardly be equidistant from both.

But consider the "equator" of that sphere, where A and B are the poles. Would all of those points be equidistant from both A and B? Fill in the circle enclosed by the equator -- are all of those points equidistant from A and B? Could there be any other points in this "equidistant set"? What must the complete set be then?