   Chapter 12.6, Problem 26E

Chapter
Section
Textbook Problem

# Match the equation with its graph (labeled I-VIII). Give reasons for your choice.y2 = x2 + 2z2

To determine

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

Explanation

Given data:

Surface equation is y2=x2+2z2 .

Formula used:

Consider the standard equation of circular cone along the y axis.

y2b2=x2a2+z2c2 (1)

Consider the given surface equation.

y2=x2+2z2 (2)

Rearrange the equation.

y21=x21+z212 (3)

By comparing equation (3) with (1), the given surface equation satisfies the equation of a circular cone along the y-axis.

Find a, b, and c by comparing equation (3) with equation (1).

a2=1a=1a=±1

The value of a is ±1 .

b2=1b=1b=±1

The value of b is ±1 .

c2=12c=±12

The value of c is ±12 .

Case i:

Let x=k .

Substitute k for x in equation (2),

y2=k2+2z2

y22z2=k2 (4)

Modify equation (4) for k=0 ,

y22z2=02y22z2=0y2(2z)2=0(y+2z)(y2z)=0

(y+2z)=0,(y2z)=0y=2z,y=2z

So, the surface equation (4) has two intersecting line equations for k=0

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