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sides, we have PC 2 r 2 or P(x, y, z)
r x C(h, k, l) 2 h y 2 k z 2 l r2 The result of Example 4 is worth remembering. 0 Equation of a Sphere An equation of a sphere with center C h, k, l and radius r is x x y F IGURE 10 2 h y k 2 z 2 l r2 In particular, if the center is the origin O, then an equation of the sphere is
x2 y2 EXAMPLE 5 Show that x 2 y 2 z2 4x
sphere, and ﬁnd its center and radius. z2 6y r2 2z 0 is the equation of a 6 SOLUTION We can rewrite the given equation in the form of an equation of a sphere if we
complete squares: x2 4x 4 y2
x 6y
2 y z2 9 2 2z 2 z 3 1
1 6
2 4 9 1 8 Comparing this equation with the standard form, we see that it is the equation of a
sphere with center 2, 3, 1 and radius s8 2 s2.
EXAMPLE 6 What region in 3 is represented by the following inequalities?
x2 y2 z2 4 z 1 1 x2 y2 z2 4 0 SOLUTION The inequalities
z can be rewritten as
1 0 sx 2 y2 z2 2 1
2
x FIGURE 11 y so they represent the points x, y, z whose distance from the origin is at least 1 and at
most 2. But we are also given that z 0, so the points lie on or below the xyplane.
Thus, the given inequalities represent the region that lies between (or on) the spheres
x 2 y 2 z 2 1 and x 2 y 2 z 2 4 and beneath (or on) the xyplane. It is sketched
in Figure 11. 5E13(pp 828837) 1/18/06 11:10 AM Page 833 SECTION 13.1 THREEDIMENSIONAL COORDINATE SYSTEMS  13.1 15–18 distance of 4 units in the positive direction, and then move
downward a distance of 3 units. What are the coordinates
of your position?  Show that the equation represents a sphere, and ﬁnd its
center and radius. 1 , 2, 4, 6 , and 1, 1, 2 on a single set of coordinate axes.
3. Which of the points P 6, 2, 3 , Q 5, 1, 4 , and R 0, 3, 8 is
closest to the xzplane? Which point lies in the yzplane? 4. What are the projections of the point (2, 3, 5) on the xy, yz, 15. x 2 y2 z2 6x 4y 16. x
2. Sketch the points 0, 5, 2 , 4, 0, 2 y 2 z 2 4x 2y 17. x 2 y 2 z 2 18. 4 x
■ 5. Describe and sketch the surface in y 3 7. Show...
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This note was uploaded on 02/04/2010 for the course M 56435 taught by Professor Hamrick during the Fall '09 term at University of Texas at Austin.
 Fall '09
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