Prerequisites : Coordinate Geometry I and Coordinate Geometry II

Spheres
A sphere is an idealized entity, defined as the locus of a point in 3D space, such that its distance from a fixed point remains constant. If is the fixed point and is the fixed distance, then by distance formula,
is the radius and is the center of the sphere.

General Form of Equation of Sphere
Radius is and center is .
The terms above can be rearranged to get the equation similar to

Diameter Form
If and are endpoints of diameter of a sphere, then its equation is given by

Externally Touching Spheres
The segment joining and is divided internally in the ratio

Internally Touching Spheres
The segment joining and gets divided externally in the ratio

Tangent Plane to a Sphere
Let be a point on the sphere . The equation of plane which touches the sphere at is given by
It goes without saying that the length of perpendicular to the plane from the center of the circle is equal to the radius of the sphere.

Section of Sphere by Plane
When a sphere is sliced by a plane, the section obtained is a circle. If the plane passes through the center of the sphere, the circle is known as great circle. If the equation of sphere is and that of plane is , then the equations together represent the circle.

Section of Sphere by Another Sphere
When 2 spheres intersect each other, the section obtained is a circle. If the equations of spheres are and , the equations together represent the circle.
The equation represents the plane in which the circle lies. The plane is known as radical plane. Radical plane is the locus of all those points, from which the lengths of tangents to the spheres are equal.

Family of Spheres
I) If and represent a circle, then, represents a family of spheres passing through the circle.
II) If and represent a circle, then represents a family of spheres passing through the circle.

Orthogonal Spheres
If and intersect such that the tangent planes and at the point of contact are perpendicular to each other, then the spheres are said to be orthogonal.
Let the equations be
and
Then, the condition for orthogonality is