   Chapter 12, Problem 33RE

Chapter
Section
Textbook Problem

# Identify and sketch the graph of each surface.33. −4x2 + y2 −4z2 = 4

To determine

To identify: The given surface equation and sketch it.

Explanation

Given data:

Surface equation is 4x2+y24z2=4 .

Formula used:

Consider the standard equation of hyperboloid of two sheets.

x2a2+y2b2z2c2=1 (1)

Consider the given surface equation.

4x2+y24z2=4 (2)

Rearrange the equation.

4x2+y24z24=444x24+y244z24=1

x21+y24z21=1 (3)

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

Thus, the surface equation 4x2+y24z2=4 is a hyperboloid of two sheets.

Case i:

Let x=k .

Substitute k for x in equation (2),

4k2+y24z2=4y24z2=4+4k2y2=4z2+4+4k2

The trace of this expression in x=k represents hyperbolas.

Case ii:

Let z=k .

Substitute k for z in equation (2),

4x2+y24k2=44x2+y2=4+4k2y24x2=4+4k2

The trace of this expression in z=k represents hyperbolas

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