Multivariate Calculus - Bruce E. Shapiro

LECTURE 1. CARTESIAN COORDINATES 3
Example 1.3 . Find the center and radius of the sphere given by
Completing the squares in equation (1.5),
0 = x 2 + 8x + y 2 − 4y + z 2 − 22z + 77
x 2 + y 2 + z 2 + 8x − 4y − 22z + 77 = 0 (1.5)
= x 2 + 8x + 4 2 − 4 2 + y 2 − 4y + (−2) 2 − (−2) 2 + z 2 − 22z + (−11) 2 − (−11) 2 + 77
= (x + 4) 2 − 16 + (y − 2) 2 − 4 + (z − 11) 2 − 121 + 77
= (x + 4) 2 + (y − 2) 2 + (z − 11) 2 − 64
Rearranging,
(x + 4) 2 + (y − 2) 2 + (z − 11) 2 = (8) 2 (1.6)
Comparing equations (1.6) and (1.4) we conclude that the center of the sphere is at
C 0 = (−4, 2, 11) and its radius is r = 8.
Example 1.4 Find the equation of a sphere that is tangent to the three coordinate
planes whose radius is 6 and whose center is in the first octant (see text, page
599,#27). 1
Figure 1.2: Place the ball in the corner where the three walls come together.
Now imagine placing a ball that is 6 inches in diameter right in the corner. If you
push the ball right about against both walls and the floor it will be 6 inches from
each wall. Hence the center of the ball will be at
C = (6, 6, 6)
Since the radius of the ball is 6, the equation of the sphere is
(x − 6) 2 + (y − 6) 2 + (z − 6) 2 = 36.
1 The first octant is that region of space where x > 0, y > 0, and z > 0.
Math 250, Fall 2006 Revised December 6, 2006.