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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Reduce the equation to one of the standard forms, classify the surface, and sketch it.

y2 = x2 + 4z2 + 4

To determine

To classify: The given surface equation and sketch the surface equation.

Explanation

Given data:

Surface equation is y2=x2+4z2+4 .

Formula used:

Consider the standard equation of hyperboloid of two sheets.

x2a2+y2b2z2c2=1 (1)

Consider the given surface equation.

y2=x2+4z2+4 (2)

Rearrange the equation.

x2+y24z2=4x24+y244z24=1

x24+y24z21=1 (3)

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

Thus, the surface equation y2=x2+4z2+4 is a hyperboloid of two sheets.

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

a2=4a=4a=±2

The value of a is ±2 .

b2=4b=4b=±2

The value of b is ±2 .

c2=1c=1c=±1

The value of c is ±1 .

Case i:

Let x=k .

Substitute k for x in equation (2),

y2=k2+4z2+4y24z2=4+k2

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

Case ii:

Let z=k

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Chapter 12 Solutions

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P-38RECh-12 P-1PCh-12 P-2PCh-12 P-5P

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