   Chapter 12.6, Problem 23E

Chapter
Section
Textbook Problem

# Match the equation with its graph (labeled I-VIII). Give reasons for your choice.x2 − y2 + z2 = 1

To determine

To match: The given surface equation with their corresponding graph shown in Figure I-VIII.

Explanation

Given data:

Surface equation is x2y2+z2=1 .

Formula used:

Consider the standard equation of hyperboloid of one sheet along the y axis.

x2a2+z2c2y2b2=1 (1)

Consider the given surface equation.

x2y2+z2=1 (2)

Rearrange the equation.

x2+z2y2=1

x21+z21y21=1 (3)

By comparing equation (3) with (1), the computed expression satisfies the equation of hyperboloid of one sheet.

Case i:

Let x=k .

Substitute k for x in equation (2),

k2y2+z2=1y2+z2=1k2

z2y2=1k2 (4)

Modify equation (4) for k=1 ,

z2y2=(112)=0

The integer solution of this equation is,

(y,z)=(0,0)

Hence, the surface equation has a origin point for k=0 .

Substitute 0 for y and z in equation (4),

0202=1k21k2=0k2=1k=±1

Similarly, the surface equation (4) have two different oriented hyperbolas trace for |k|<1 and |k|>1

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