Three-Dimensional Coordinate Systems

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  1. Which of the points P623 , Q514 , and R038 is closest to the xz-plane ? Which point lies in the yz-plane ?
    Q ; R .
  2. Describe and sketch a surface in 3 represented by the equation x+y=2 .
    A vertical plane that intersects the xy-plane in the line y=2x , z=0 .
  3. Find an equation of the sphere with center 264 and radius 5 . Describe its intersection with each of the coordinate planes.
    x22+y+62+z42=25
    circle of radius 3 for z=0 , circle of radius 21 for x=0 , and empty set for y=0 .
  4. Show that the following equation is that of a sphere, and find its center and radius.

    4x2+4y2+4z28x+16y=1

  5. Find an equation of the sphere if one of its diameters has endpoints 214 and 4310 .
  6. Describe in words the region of 3 represented by the equation y=4 .
    A plane parallel to the xz-plane and 4 units to the left of it.
  7. Describe in words the region of 3 represented by the inequality x2+y2+z2>2z .
  8. Write inequalities describing the solid upper hemisphere of the sphere of radius 2 centered at the origin.