   Chapter 12.6, Problem 31E

Chapter
Section
Textbook Problem

# Reduce the equation to one of the standard forms, classify the surface, and sketch it. y 2  =  x 2  +  1 9 z 2

To determine

To classify: The given surface equation and sketch the surface equation y2=x2+19z2 .

Explanation

Given data:

Surface equation is y2=x2+19z2 .

Formula used:

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

y2b2=x2a2+z2c2 (1)

Consider the given surface equation.

y2=x2+19z2 (2)

Rearrange the equation.

y21=x21+z29 (3)

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

Thus, the surface equation y2=x2+19z2 is 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=9c=9c=±3

The value of c is ±3 .

Case i:

Let x=k .

Substitute k for x in equation (2),

y2=k2+19z2

y219z2=k2 (4)

Modify equation (4) for k=0 ,

y219z2=02y219z2=0y2(13z)2=0(y+13z)(y13z)=0

(y+13z)=0,(y13z)=0y=13z,y=13z

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

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