   Chapter 12, Problem 34RE

Chapter
Section
Textbook Problem

# Identify and sketch the graph of each surface.34. y2 + z2 = 1 + x2

To determine

To identify: The given surface equation and sketch it.

Explanation

Given data:

Surface equation is y2+z2=1+x2 .

Formula used:

Consider the standard equation of hyperboloid of one sheet along the x axis.

y2b2+z2c2x2a2=1 (1)

Consider the given surface equation.

y2+z2=1+x2 (2)

Rearrange the equation.

y2+z2x2=1 (3)

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

Thus, the surface equation y2+z2=1+x2 is a hyperboloid of one sheet.

Case i:

Let x=k .

Substitute k for x in equation (2),

y2+z2=1+k2

The trace of this expression in x=k represents family of ellipse.

Case ii:

Let y=k .

Substitute k for y in equation (2),

k2+z2=1+x2

z2x2=1k2 (4)

Modify equation (4) for k=1 ,

z2x2=112=11=0

The integer solution of this equation is,

(x,z)=(0,0)

Hence, the surface equation has a origin point for k=0 .

Substitute 0 for x and z in equation (4),

(0)2(0)2=1k200=1k21k2=0k2=1

k=1k=±1

Similarly, the surface equation (4) have two different oriented hyperbolas trace for |k|<1 and |k|>1

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