The locus of points in space...?

Question

The locus of points in space at a given distance from point P and 3 cm from a plane Q may be:

1. no points

2. one point

3. two points

4. a circle

5. two circles

6. a circle and a point

A. 1, 2, and 3

B. 1, 3, and 5

C. 4

D. All of these

Answer 1

D.

We are looking at the intersection of a sphere centered at P with two planes 6 cm apart.

If the plane is far away from P, the sphere and plane will not intersect.

If the sphere is exactly 3 cm away from the plane, the intersection is one point.

If the plane is exactly in the middle of the sphere and the sphere has radius 3 cm, then the answer is two points.

If the sphere is less than 3 cm away from the plane but the plane isn't far enough inside the sphere, you have just one circle.

If the plane is in the middle of the sphere and the radius is greater than 3 cm, the answer is two circles.

Finally, if you put the plane inside the sphere and the sphere has a radius bigger than 3 cm, but you slide the plane so that it's 3 cm from one side of the sphere, the answer is a circle and a point.

So the final answer is D, all of these.