Equation of a Sphere

1. A sphere is defined as the set of points in space that are at a fixed distance from a point . This point is called the center of the sphere, and the line segment joining the center and any point on the sphere is the radius.
2. The equation of a sphere in coordinate space with center and radius is:
   
   
3. If the center is at the origin and the radius is , the equation of the sphere simplifies to:

For a sphere with center that is tangent to the coordinate planes:

1. The equation of a sphere tangent to the -plane is:
2. The equation of a sphere tangent to the -plane is:
3. The equation of a sphere tangent to the -plane is:

Equation of the Form

The general quadratic equation in , , and of the form: represents a sphere if the condition holds. The center of this sphere is: and the radius is: