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Problem 9 Medium Difficulty
(a) Find and identify the traces of the quadric surface $ x^2 + y^2 - z^2 = 1 $ and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1.
(b) If we change the equation in part (a) to $ x^2 - y^2 + z^2 = 1 $, how is the graph affected?
(c) What if we change the equation in part (a) to $ x^2 + y^2 + 2y - z^2 = 0 $?
Answer
a) The $z=k$ traces are circles, the $x=k$ and $y=k$ traces are hyperbolas.
b) The $y$ -axis becomes the axis of symmetry. The $y=k$ traces become the ones with the circles while the $x=k, z=k$ traces are hyperbolas.
c) The center moves from $(0,0,0)$ to $(0,-1,0)$
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Top Calculus 3 Educators
Catherine R.
Missouri State University
Caleb E.
Baylor University
Kristen K.
University of Michigan - Ann Arbor
Joseph L.
Boston College