   Chapter 12.6, Problem 18E

Chapter
Section
Textbook Problem

# Use traces to sketch and identify the surface.3x2 − y2 + 3z2 = 0

To determine

To identify: The given surface equation and sketch it.

Explanation

Given data:

Surface equation is 3x2y2+3z2=0 .

Formula used:

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

y2b2=x2a2+z2c2 (1)

Consider the given surface equation.

3x2y2+3z2=0 (2)

Rearrange the equation.

y2=3x23z2y2=3x2+3z2

y2=x213+z213 (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 3x2y2+3z2=0 is a circular cone along the y-axis.

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

a2=13a=±13

The value of a is ±13 .

b2=1b=1b=±1

The value of b is ±1 .

c2=13c=±13

The value of c is ±13 .

Case i:

Let x=k .

Substitute k for x in equation (2),

3k2y2+3z2=0y2+3z2=3k2

3z2y2=3k2 (4)

Modify equation (4) for k=0 ,

3z2y2=3(0)23z2y2=0(3z)2y2=0(3z+y)(3zy)=0

(3z+y)=0,(3zy)=0y=3z,y=3z

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

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