12.6
21-28 Match the equation with its graph (labeled I-VIII). Give reasons for your choices.
21. VII. Because x2+ 9z2= 1 is an ellipse lying in the xz-plane.
22. IV. Because 9x2+ z2= 1 is an ellipse lying in the xz-plane.
23. II. Because x2 y2= 1 is a hyperbola lying in the xz-plane which passes through (1; 0; 0):
24. III. Because x2 z2= 1 is an empty set.
25. VI. Because y = 2x2 and y = z2are parabolas lying in the xy-plane and the yz-plane, respectively.
26. I. Because y2= x2is a cross lying in the xy-plane.
27. VIII. Because the graph is independant of y:
28. V. Because y = x2; y = z2 are parabolas lying in the xy-plane and the yz-plane, respectively. Moreover, they open to opposite directions.
41. See the graph in page A114.
43. When we rotate the parabola y = x2 about the y-axis with 90 , it becomes y = z2. So the equation of the revolution surface is y = x2+ z2:
46. Let P = (x; y; z). The distance from P to the x-axis is p
y2+ z2. The distance from P to the yz-plane is x. So the equation is p
y2+ z2 = 2x; i.e.
4x2 y2 z2= 0
49. It su¢ ces to check c + 2(b a)t = (b + t)2 (a + t)2 and c 2(b + a)t = (b t)2 (a + t)2: This comes immediately from the relation c = b2 a2:
50. Multiply both sides of x2+ 2y2 z2+ 3x = 1 with 2; we have 2x2+ 4y2 2z2+ 6x = 2:
Subtract 2x2+ 4y2 2z2 5y = 0 from it, we get 6x + 5y = 2; which lies in the xy-plane.
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