Analytical Geometry

Chapter 14

Sphere

14.1  DEFINITION OF SPHERE

The locus of a moving point in space such that its distance from a fixed point is constant is called a sphere. The fixed point is called the centre of the sphere. The constant distance is called the radius.

14.2  THE EQUATION OF A SPHERE WITH CENTRE AT (a, b, c) AND RADIUS R

Let P(x, y, z) be any point on the sphere. Let C(a, b, c) be the centre.

Then,

This is the equation of the required sphere.

Show that the equation x² + y² + z² + 2ux + 2vy + 2wz + d = 0 always represents a sphere. Find its centre ...