Worksheet: Quadratic Surfaces in Three Dimensions

In this worksheet, we will practice identifying the equations and graphs of quadratic surfaces in 3D such as cylinders, cones, ellipsoids, hyperboloids, and paraboloids.


Find the trace of the hyperboloid of one sheet 𝑥𝑎+𝑦𝑏𝑧𝑐=1 in the plane 𝑥=𝑎 and in the plane 𝑦=𝑏.

  • A𝑥𝑎=𝑧𝑐 and 𝑦𝑏=𝑧𝑐
  • B𝑥𝑎+𝑦𝑏=1 and 𝑧=0
  • C𝑥𝑎=𝑧𝑐 and 𝑦𝑏=𝑧𝑐
  • D𝑥𝑎=𝑧𝑐 and 𝑦𝑏=𝑧𝑐
  • E𝑥𝑎=𝑧𝑐 and 𝑦𝑏=𝑧𝑐


Find the trace of the hyperbolic paraboloid 𝑥𝑎𝑦𝑏=𝑧𝑐 in the 𝑥𝑦-plane.

  • A𝑥𝑎=𝑦𝑏 and 𝑧=0
  • B𝑥𝑎=𝑦𝑏 and 𝑧=1
  • C𝑥𝑎=𝑦𝑏 and 𝑧=0
  • D𝑥𝑎=𝑦𝑏 and 𝑧=1
  • E𝑥𝑎=𝑦𝑏 and 𝑧=0

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