14.2 A BRIEF CATALOG OF THE QUADRIC SURFACES; PROJECTIONS 829
z
Example 1 The paraboloid of revolution z = x2 + y2 and the plane
z = 2y + 3
intersect in a curve C. See Figure 14.2.11. The projection of this curve onto the xy-plane
is the set of all points (x, y, 0) with
x2 + y2 = 2y + 3.
y
(0, 1, 0)
This equation can be written
x
x2 + (y − 1)2 = 4.
Figure 14.2.11
The projection of C onto the xy-plane is the circle of radius 2 centered at (0, 1, 0).
EXERCISES 14.2
33. 4x2 + 9z 2 − 36y = 0.
Identify the surface.
1. x + 4y − 16z = 0.
34. 9x2 + 4z 2 − 36y2 = 0.
2. x2 + 4y2 + 16z 2 − 12 = 0.
35. 9y2 − 4x2 − 36z 2 − 36 = 0.
3. x − 4y2 = 0.
36. 9y2 + 4z 2 − 36x = 0.
4. x − 4y − 2z = 0.
38. 36x + 9y + 4z − 36 = 0.
39. Identify all possibilities for the surface
2
2
2
2
2
2
5. 5x + 2y − 6z − 10 = 0.
2
2
2
6. 2x2 + 4y2 − 1 = 0.
2
2
8. 5x2 + 2y2 − 6z 2 + 10 = 0.
9. x2 + 2y2 − 4z = 0.
11. x − y2 + 2z 2 = 0.
10. 2x2 − 3y2 − 6 = 0.
12. x − y2 − 6z 2 = 0.
13. 25y2 + 4z 2 − 100 = 0.
14. 25x2 + 4y2 − 100 = 0.
15. y − z = 0.
16. x2 − y + 1 = 0.
17. y2 + z = 0.
18. 25x2 − 9y2 − 225 = 0.
y2
x2
20.
+
= 1.
4
9
22. z = x2 .
19. x2 + y2 = 9.
21. y2 − 4x2 = 4.
23. y = x2 + 1.
24. (x − 1)2 + (y − 1)2 = 1.
Identify the surface and find the traces. Then sketch the surface.
25. 9x2 + 4y2 − 36z = 0.
26. 9x2 + 4y2 + 36z 2 − 36 = 0.
27. 9x2 + 4y2 − 36z 2 = 0.
28. 9x2 + 4y2 − 36z 2 − 36 = 0.
29. 9x2 + 4y2 − 36z 2 + 36 = 0.
30. 9x2 − 4y2 − 36z = 0.
31. 9x − 4y − 36z = 36.
2
2
40.
41.
42.
Sketch the cylinder.
2
37. x2 + y2 − 4z = 0.
2
z = Ax2 + By2
7. x + y + z − 4 = 0.
2
2
2
32. 9x2 + 4z 2 − 36y2 − 36 = 0.
43.
taking (a) AB > 0.
(b) AB < 0.
(c) AB = 0.
Find the planes of symmetry for the cylinder x − 4y2 = 0.
Write an equation for the surface obtained by revolving the
parabola 4z − y2 = 0 about the z-axis.
The hyperbola c2 y2 − b2 z 2 − b2 c2 = 0 is revolved about the
z-axis. Find an equation for the resulting surface.
(a) The equation
x2 + y2 = kz
with k > 0
represents the upper nappe of a cone, with vertex at the
origin and the positive z-axis as the axis of symmetry.
Describe the section in the plane z = z0 , z0 > 0.
(b) Let S be one nappe of a cone, with vertex at the origin.
Write an equation for S given that
(i) the negative z-axis is the axis of symmetry and the
section in the plane z = −2 is a circle of radius 6,
(ii) the positive y-axis is the axis of symmetry and the
section in the plane y = 3 is a circle of radius 1.
44. Form the elliptic paraboloid
x2 +
y2
= z.
b2
(a) Describe the section in the plane z = 1.
(b) What happens to this section as b tends to infinity?
(c) What happens to the paraboloid as b tends to infinity?
830 CHAPTER 14 FUNCTIONS OF SEVERAL VARIABLES
The surfaces intersect in a space curve C. Determine the
projection of C onto the xy-plane.
45. The planes x + 2y + 3z = 6 and x + y − 2z = 6.
46. The planes x − 2y + z = 4 and 3x + y − 2z = 1.
47. The sphere x2 + y2 + (z − 1)2 = 32 and the hyperboloid
x2 + y2 − z 2 = 1.
48. The sphere x2 + y2 + (z − 2)2 = 2 and the cone x2 + y2 = z 2 .
49. The paraboloids x2 + y2 + z = 4 and x2 + 3y2 = z.
50. The cylinder y2 +z −4 = 0 and the paraboloid x2 +3y2 = z.
51. The cone x2 + y2 = z 2 and the plane y + z = 2.
52. The cone x2 + y2 = z 2 and the plane y + 2z = 2.
x2
y2
z2
c 53. The ellipsoid 2 + 2 + 2 = 1 can be parametrized by
a
b
c
the vector function of two variables
r (u, v) = a cos u cos v i + b cos u sin v j + c sin u k.
(a) Verify that r parametrizes an ellipsoid.
(b) Use a graphing utility to draw the ellipsoid with a = 3,
b = 4, c = 2.
(c) Experiment with other values of a, b, c to see how the
ellipsoid changes shape. How would you choose a, b, c
to obtain a sphere?
x
c 54. The hyperboloid of one sheet
a
2
y2
z2
− 2 = 1 can be
2
b
c
parametrized by the vector function of two variables
2
+
r (u, v) = a sec u cos v i + b sec u sin v j + c tan u k.
(a) Verify that r parametrizes a hyperboloid.
(b) Use a graphing utility to draw the hyperboloid with
a = 2, b = 3, c = 4.
(c) Experiment with other values of a, b, c to see how the
hyperboloid changes shape.
x2 y2
z2
c 55. The elliptic cone 2 + 2 = 2 can be parametrized by the
a
b
c
vector function of two variables
r (u, v) = a v cos u i + b v sin u j + c v k
(a) Verify that r parametrizes an elliptic cone.
(b) Use a graphing utility to draw the elliptic cone with
a = 1, b = 2, c = 3.
(c) Experiment with other values of a, b, c to see how the
cone changes shape. In particular, what effect does c
have on the cone?
14.3 GRAPHS; LEVEL CURVES AND LEVEL SURFACES
We begin with a function f of two variables defined on a subset D of the xy-plane. By
the graph of f we mean the graph of the equation
z
z = f (x, y)
(x, y) ∈ D.
In the case of f (x, y) = x2 + y2 , the domain is the entire plane. The
graph of f is a paraboloid of revolution:
Example 1
z = x2 + y2 .
This surface can be generated by revolving the parabola
y
x
z = x2 + y 2
Figure 14.3.1
z = x2
about the z-axis. See Figure 14.3.1.
Example 2
(in the xz-plane)
Let a, b, and c be positive constants. The domain of the function
g(x, y) = c − ax − by
is also the entire xy-plane. The graph of g is the plane
z = c − ax − by
with intercepts x = c/a, y = c/b, z = c.
(Figure 14.3.2)