   Chapter 12.6, Problem 10E

Chapter
Section
Textbook Problem

# (a) Find and identify the traces of the quadric surface −x2 − y2 + z2 = 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 x2 − y2 − z2 = 1, what happens to the graph? Sketch the new graph.

(a)

To determine

To find: The traces of quadric surface x2y2+z2=1 and identify traces and explain the reason why graph looks like graph of hyperboloid of two sheets in Table 1.

Explanation

Given data:

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

Consider the given surface equation.

x2y2+z2=1 (1)

Case i:

Consider x=k .

Substitute k for x in equation (1),

k2y2+z2=1z2y2=1+k2

Hence, the surface equation represents a trace of hyperbola.

Case ii:

Consider y=k .

Substitute k for y in equation (1),

x2k2+z2=1x2+z2=1+k2

The trace of this expression represents a trace of hyperbola.

Case iii:

Consider z=k

(b)

To determine

To find: The affect in graph if the quadric surface is changed from equation x2y2+z2=1 to x2y2z2=1 .

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