Chapter 12.6, Problem 10E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

(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.

(a)

To determine

**To find:** The traces of quadric surface

Explanation

**Given data:**

Quadric surface is

Refer to Table 1 in the textbook for hyperboloid of two sheets.

Consider the given surface equation.

**Case i:**

Consider

Substitute *k* for *x* in equation (1),

Hence, the surface equation represents a trace of hyperbola.

**Case ii:**

Consider

Substitute *k* for *y* in equation (1),

The trace of this expression represents a trace of hyperbola.

**Case iii:**

Consider

(b)

To determine

**To find:** The affect in graph if the quadric surface is changed from equation

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