Squaring both sides we have pc 2 r 2 or x h 2 y k 2 z

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Unformatted text preview: both 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 xy-plane. 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 xy-plane. It is sketched in Figure 11. 5E-13(pp 828-837) 1/18/06 11:10 AM Page 833 SECTION 13.1 THREE-DIMENSIONAL 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 xz-plane? Which point lies in the yz-plane? 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.

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