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Problem 10 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 two sheets in Table 1.
(b) If the equation in part (a) is changed to $ x^2 - y^2 - z^2 = 1 $, what happens to the graph? Sketch the new graph.

Answer

a) The traces parallel to the $y z$ -plane are hyperbolas. $z^{2}-y^{2}=1+k^{2}$
The traces parallel to the $x z$ -plane are hyperbolas. $z^{2}-x^{2}=1+k^{2}$
b) See Graph

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Video Transcript

So here we want to find and identify the traces of the quadratic surface or aquatic surface. So it's going to be negative acts squared minus y squared minus z squared one. Um And we want to explain why the graph looks like the graph of a hyperba Lloyd of two sheets. And that's because the traces parallel to the y z plane are going to be hyperbole as and if we change the equation uh so this is actually plus one R plus t squared. And if we change it to minus the squared, then we end up getting that the traces parallel to xz the xz plane will be hyper bolos. Um So that would be the resulting graph that we end up getting. And it's going to be able to form Z squared minus x squared equals one plus k squared or z squared minus y squared equals one plus k squared.