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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.

4x2 + y2 + z2 − 24x − 8y + 4z + 55 = 0

To determine

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

Explanation

Given data:

Surface equation is 4x2+y2+z224x8y+4z+55=0 .

Formula used:

Consider the standard equation of an ellipsoid.

x2a2+y2b2+z2c2=1 (1)

Consider the given surface equation.

4x2+y2+z224x8y+4z+55=0 (2)

Rearrange the equation.

4(x26x+9)+(y28y+16)+(z2+4z+4)=55+36+16+44(x3)2+(y4)2+(z+2)2=1

(x3)2(14)+(y4)2+(z+2)2=1 (3)

Find the center by using equation (3).

(x3)2=0x3=0x=3

(y4)2=0y4=0y=4

(z+2)2=0z+2=0z=2

As the computed expression (3) satisfies the equation of ellipsoid, which is centered at (3,4,2) .

Thus, the surface equation 4x2+y2+z224x8y+4z+55=0 is an ellipsoid.

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

a2=14a=14a=±0.5

The value of a is ±0.5 , So the points along the x-axis is (2.5,4,2) and (3.5,4,2) .

b2=1b=1b=±1

The value of b is ±1 , So the points along the y-axis is (3,1,2) and (3,1,2) .

c2=1c=1c=±1

The value of c is ±1 , So the points along the y-axis is (3,4,1) and (3,4,1) .

Case i:

Let x=k .

Substitute k for x in equation (3),

(k3)2(14)+(y4)2+(z+2)2=1

(y4)2+(z+2)2=1(k3)2(14) (4)

Modify equation (4) for k=3.5 ,

(y4)2+(z+2)2=1(3.53)2(14)(y4)2+(z+2)2=1(0

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Chapter 12 Solutions

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