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.

Q1:

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

  • A 𝑥 𝑎 = 𝑧 𝑐 and 𝑦 𝑏 = 𝑧 𝑐
  • B 𝑥 𝑎 = 𝑧 𝑐 and 𝑦 𝑏 = 𝑧 𝑐
  • C 𝑥 𝑎 + 𝑦 𝑏 = 1 and 𝑧 = 0
  • D 𝑥 𝑎 = 𝑧 𝑐 and 𝑦 𝑏 = 𝑧 𝑐
  • E 𝑥 𝑎 = 𝑧 𝑐 and 𝑦 𝑏 = 𝑧 𝑐

Q2:

Find the trace of the hyperbolic paraboloid 𝑥 𝑎 𝑦 𝑏 = 𝑧 𝑐 in the 𝑥 𝑦 -plane.

  • A 𝑥 𝑎 = 𝑦 𝑏 and 𝑧 = 0
  • B 𝑥 𝑎 = 𝑦 𝑏 and 𝑧 = 0
  • C 𝑥 𝑎 = 𝑦 𝑏 and 𝑧 = 1
  • D 𝑥 𝑎 = 𝑦 𝑏 and 𝑧 = 1
  • E 𝑥 𝑎 = 𝑦 𝑏 and 𝑧 = 0

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