13.6 EXERCISES SECTION 13.6 A Survey of Quadric Surfaces 731 Preliminary Questions True or false? All traces of an ellipsoid are ellipses. 2 . True or false? All traces of a hyperboloid are hyperbolas . 3. Which quadric surfaces have both hyperbolas and parabolas as 5. A surface is called bounded if there exists M > 0 such that every point on the surface lies at a distance of at most M from the origin. Which of the quadric surfaces are bounded? traces ? 4. Is there any quadric surface whose traces are all parabolas? 6. What is the definition of a parabolic cylinder? Exercises In Exercises 1-8, state whether the given equation defines an ellipsoid or hyperboloid, and if a hyperboloid, whether it is of one or two sheets. 1 . ( # ) ? + ( 3 ) 2 + ( 5 ) 2 - 1 2 . ( 3 ) ? + ( 3 ) 2 - ( 3 ) ? = 1 3 . 1 2 + 3y2 + 92 2 = 1 4 . - (2 ) - (3)2 + (3 ) 2 = 1 5. x2 - 3y2 + 922 = 1 6. x2 - 3y2-972 =1 (B) (C) 7 . 12 + 12 =4-422 8. x2 + 3y2 =9+22w FIGURE 14 In Exercises 9-14, state whether the given equation defines an elliptic paraboloid, a hyperbolic paraboloid, or an elliptic cone. 26. Describe the surface that is obtained when, in the equation 18x2 +3y +z =1, we choose (a) all plus signs, (b) one minus sign, and (c) two minus signs. 9 . 2 = ( # ) + (3) () ()10. 82 - (#)2 + (3)2 27. What is the equation of the surface obtained when the elliptic nega) wist paraboloid z = () + ( is rotated about the x-axis by 90? Refer 1 . 2 = ( * ) - ( 1 2 ) ? 12. 42 = 9x2 + 5y2 to Figure 15. 13 . 3x 2 - 7y2 = z 14. 3x2 + 7y2 = 1472 15. y2 = 5x2 - 4z 16. y = 3x2 - 4z2 gienl 1arthur In Exercises 17-24, state the type of the quadric surface and describe the trace obtained by intersecting with the given plane. 17 . x 2 + ( 2 ) + 2 2 = 1 , y = 0 18 . x 2 + ( 2 ) + 2 2 = 1 , y = 5 19 . x 2 + ( 2 ) + 2 2 = 1 , 2 = 7 FIGURE 15 20 . ( # ) + ( 5) 2 - 52 2 = 1, x= 0 28. Describe the intersection of the horizontal plane z = h and the hy- perboloid -x2 - 4y + 4z = 1. For which values of h is the intersection 21 . ( 3 ) + ( 2 ) " - 52 2 = 1 , y = 1 empty? In Exercises 29-42, sketch the given surface. 22. 4x 2 + ( 2 ) ' - 22 2 = -1, = =1 29. x2 + y2 - 22 = 1 23. y = 3x2, z=27 24. y = 3x2, y = 27 30 . ( # ) + ( 8 ) ? + ( 12 ) ? = 1 25. Match each of the ellipsoids in Figure 14 with the correct equation: 31. z = (* ) + (8)2 32. 2 = ( ) - (3 ) 2 (a) x2 + 4y2 + 422 = 16 (b) 4x2 + y2 + 422 = 16 (c) 4x2 + 4y2 + 22 = 16 33 . 2 2 = (* ) 2 + ( 8 ) 2 34. y = -x2