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Calculus IIIChapter 1: 3-Dimensional SpaceSection 1-4

Quadric Surfaces — Question 9

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Question 9

Classify the real point sets x2+y2+z2−2xy−2xz+2yz=cx^2+y^2+z^2-2xy-2xz+2yz=c

Tasks

  1. Factor the quadratic expression.

  2. Classify the real point set for c<0c<0, c=0c=0, and c>0c>0.

  3. Explain the degeneration geometrically.

Original worksheet page 1: question and worked solution for 1-4-009
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Question 9 – Solution

Strategy Test whether the quadratic expression is a perfect square.

Step 1: Factor Expanding (x−y−z)2=x2+y2+z2−2xy−2xz+2yz.(x-y-z)^2=x^2+y^2+z^2-2xy-2xz+2yz. Thus the equation is simply (x−y−z)2=c.(x-y-z)^2=c.

Case analysis If c<0c<0, no real square can equal cc, so the set is empty. If c=0c=0, then x−y−z=0x-y-z=0, a single plane. If c>0c>0, taking square roots gives x−y−z=±c,x-y-z=\pm\sqrt c, so the set is two distinct parallel planes.

Conclusion The classification is , , and for the three cases. Substitution into the factored form verifies each result.

Original worksheet page 2: question and worked solution for 1-4-009

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