   Chapter 12.6, Problem 33E

Chapter
Section
Textbook Problem

# Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 + 2y − 2z2 = 0

To determine

To classify: The given surface equation and sketch the surface equation x2+2y2z2=0.

Explanation

Formula used:

Consider the standard equation of a hyperbolic paraboloid along the y axis.

yb=z2c2x2a2 (1)

Calculation:

Consider the given surface equation.

x2+2y2z2=0

Rearrange the equation.

2y=2z2x2y=2z2x22y=2z22x22

y=z2x22 (2)

Rearrange the equation.

y1=z21x22 (3)

By comparing equation (3) with (1), the computed expression satisfies the equation of circular paraboloid along the y-axis.

Thus, the surface equation x2+2y2z2=0 is a hyperbolic paraboloid.

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

a2=2a=2a=±1.414

The value of a is ±1.414.

b=1

The value of b is 1.

c2=1c=1c=±1

The value of c is ±1.

Let x=k.

Substitute k for x in equation (2),

y=z2k22

The trace of this expression for x=k represents a family of parabola.

Let y=k

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