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# 12.6-sm.dvi - S E C T I O N 13.6 A Survey of Quadric...

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S E C T I O N 13.6 A Survey of Quadric Surfaces (ET Section 12.6) 397 or 1 6 4 , - 2 , 4 · x , y , z = 4 or 2 3 , - 1 3 , 2 3 · x , y , z = 4 , in normal form. 71. Let n = -→ O P , where P = ( x 0 , y 0 , z 0 ) is a point on the sphere x 2 + y 2 + z 2 = r 2 , and let P be the plane with equation n · x , y , z = r 2 . Show that the point on P nearest the origin is P itself and conclude that P is tangent to the sphere at P (Figure 13). FIGURE 13 The terminal point of n lies on the sphere of radius r . SOLUTION First notice that the terminal point P of n lies on the plane P , since substituting x , y , z = -→ O P = n in the equation of the plane gives n · x , y , z = n · -→ O P = n · n = n 2 = r 2 Since the vector -→ O P = n is orthogonal to the plane, that is, the radius O P is perpendicular to the plane, we conclude that P is the point on P closest to the origin and that P is tangent to the sphere of radius r centered at the origin. Of course, P is the tangency point. 72. Use Exercise 71 to find the equation of the plane tangent to the unit sphere at P = 1 3 , 1 3 , 1 3 . SOLUTION We substitute n = -→ O P = 1 3 , 1 3 , 1 3 and R = -→ O P = 1 in the equation n · x , y , z = R 2 of the plane tangent to the unit sphere at P . This gives 1 3 , 1 3 , 1 3 · x , y , z = 1 1 3 x + 1 3 y + 1 3 z = 1 x + y + z = 3 13.6 A Survey of Quadric Surfaces (ET Section 12.6) Preliminary Questions 1. True or false: All traces of an ellipsoid are ellipses. SOLUTION This statement is true, mostly. All traces of an ellipsoid ( x a ) 2 + ( y b ) 2 + ( z c ) 2 = 1 are ellipses, except for the traces obtained by intersecting the ellipsoid with the planes x = ± a , y = ± b and z = ± c . These traces reduce to the single points ( ± a , 0 , 0 ) , ( 0 , ± b , 0 ) and ( 0 , 0 , ± c ) respectively. 2. True or false: All traces of a hyperboloid are hyperbolas.

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398 C H A P T E R 13 VECTOR GEOMETRY (ET CHAPTER 12) SOLUTION The statement is false. For a hyperbola in the standard orientation, the horizontal traces are ellipses (or perhaps empty for a hyperbola of two sheets), and the vertical traces are hyperbolas. 3. Which quadric surfaces have both hyperbolas and parabolas as traces? SOLUTION The hyperbolic paraboloid z = ( x a ) 2 - ( y b ) 2 has vertical trace curves which are parabolas. If we set x = x 0 or y = y 0 we get z = x 0 a 2 - y b 2 z = - y b 2 + C z = x a 2 - y 0 b 2 z = x a 2 + C The hyperbolic paraboloid has vertical traces which are hyperbolas, since for z = z 0 , ( z 0 > 0), we get z 0 = x a 2 - y b 2 4. Is there any quadric surface whose traces are all parabolas? SOLUTION There is no quadric surface whose traces are all parabolas. 5. A surface is called bounded if there exists M > 0 such that every point on the surfaces lies at a distance of at most M from the origin. Which of the quadric surfaces are bounded? SOLUTION The only quadric surface that is bounded is the ellipsoid x a 2 + y b 2 + z c 2 = 1 . All other quadric surfaces are not bounded, since at least one of the coordinates can increase or decrease without bound.
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12.6-sm.dvi - S E C T I O N 13.6 A Survey of Quadric...

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