Equidistant from 2 points

In summary, the set of all points equidistant from the points A(-1,5,3) and B(6,2,-2) can be described as a plane perpendicular to the line that connects A and B, passing through the midpoint of this line. This plane is the set of all points that are equidistant from A and B.
  • #1
fk378
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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?
 
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  • #2
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?
 
  • #3
fk378 said:

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?
 

1. What does it mean to be equidistant from two points?

Being equidistant from two points means that the distance from a specific point to each of the two given points is the same.

2. How do you find a point that is equidistant from two given points?

To find a point that is equidistant from two given points, you can use the midpoint formula or the perpendicular bisector method.

3. Can a point be equidistant from two points on a straight line?

Yes, a point can be equidistant from two points on a straight line. This point would be the midpoint of the line segment connecting the two given points.

4. Is the point equidistant from the two points always on the same plane?

Yes, the point equidistant from two points is always on the same plane as the two given points.

5. How many points can be equidistant from two given points?

There are infinite points that can be equidistant from two given points. These points lie on a perpendicular bisector of the line segment connecting the two given points.

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