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Calculus Volume 3

2.6 Quadric Surfaces

Calculus Volume 32.6 Quadric Surfaces
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  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Chapter Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Chapter Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Chapter Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Chapter Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

Learning Objectives

  • 2.6.1. Identify a cylinder as a type of three-dimensional surface.
  • 2.6.2. Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
  • 2.6.3. Use traces to draw the intersections of quadric surfaces with the coordinate planes.

We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.

Identifying Cylinders

The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.

In the two-dimensional coordinate plane, the equation x2+y2=9x2+y2=9 describes a circle centered at the origin with radius 3.3. In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the z-axis (Figure 2.75), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the z-axis passing through circle x2+y2=9x2+y2=9 in the xy-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.

This figure a 3-dimensional coordinate system. It has a right circular center with the z-axis through the center. The cylinder also has points labeled on the x and y axis at (3, 0, 0) and (0, 3, 0).
Figure 2.75 In three-dimensional space, the graph of equation x2+y2=9x2+y2=9 is a cylinder with radius 33 centered on the z-axis. It continues indefinitely in the positive and negative directions.

Definition

A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder. The parallel lines are called rulings.

From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure 2.76).

This figure has a 3-dimensional surface that begins on the y-axis and curves upward. There is also the x and z axes labeled.
Figure 2.76 In three-dimensional space, the graph of equation z=x3z=x3 is a cylinder, or a cylindrical surface with rulings parallel to the y-axis.

Example 2.55

Graphing Cylindrical Surfaces

Sketch the graphs of the following cylindrical surfaces.

  1. x2+z2=25x2+z2=25
  2. z=2x2yz=2x2y
  3. y=sinxy=sinx

Solution

  1. The variable yy can take on any value without limit. Therefore, the lines ruling this surface are parallel to the y-axis. The intersection of this surface with the xz-plane forms a circle centered at the origin with radius 55 (see the following figure).
    This figure is the 3-dimensional coordinate system. It has a right circular cylinder on its side with the y-axis in the center. The cylinder intersects the x-axis at (5, 0, 0). It also has two points of intersection labeled on the z-axis at (0, 0, 5) and (0, 0, -5).
    Figure 2.77 The graph of equation x2+z2=25x2+z2=25 is a cylinder with radius 55 centered on the y-axis.
  2. In this case, the equation contains all three variables x,y,x,y, and zz so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).
    This figure has a surface in the first octant. The cross section of the solid is a parabola.
    Figure 2.78
  3. In this equation, the variable z can take on any value without limit. Therefore, the lines composing this surface are parallel to the z-axis. The intersection of this surface with the xy-plane outlines curve y=sinxy=sinx (see the following figure).
    This figure is a three dimensional surface. A cross section of the surface parallel to the x y plane would be the sine curve.
    Figure 2.79 The graph of equation y=sinxy=sinx is formed by a set of lines parallel to the z-axis passing through curve y=sinxy=sinx in the xy-plane.
Checkpoint 2.52

Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation z=y2.z=y2.

When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in Figure 2.80.

Definition

The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.

This figure has two images. The first image is a surface. A cross section of the surface parallel to the x z plane would be a sine curve. The second image is the sine curve in the x y plane.
Figure 2.80 (a) This is one view of the graph of equation z=sinx.z=sinx. (b) To find the trace of the graph in the xz-plane, set y=0.y=0. The trace is simply a two-dimensional sine wave.

Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in Figure 2.80, that the trace of the graph of z=sinxz=sinx in the xz-plane is useful in constructing the graph. The trace in the xy-plane, though, is just a series of parallel lines, and the trace in the yz-plane is simply one line.

Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.

Quadric Surfaces

We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.

Definition

Quadric surfaces are the graphs of equations that can be expressed in the form

Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.

When a quadric surface intersects a coordinate plane, the trace is a conic section.

An ellipsoid is a surface described by an equation of the form x2a2+y2b2+z2c2=1.x2a2+y2b2+z2c2=1. Set x=0x=0 to see the trace of the ellipsoid in the yz-plane. To see the traces in the xy- and xz-planes, set z=0z=0 and y=0,y=0, respectively. Notice that, if a=b,a=b, the trace in the xy-plane is a circle. Similarly, if a=c,a=c, the trace in the xz-plane is a circle and, if b=c,b=c, then the trace in the yz-plane is a circle. A sphere, then, is an ellipsoid with a=b=c.a=b=c.

Example 2.56

Sketching an Ellipsoid

Sketch the ellipsoid x222+y232+z252=1.x222+y232+z252=1.

Solution

Start by sketching the traces. To find the trace in the xy-plane, set z=0:z=0: x222+y232=1x222+y232=1 (see Figure 2.81). To find the other traces, first set y=0y=0 and then set x=0.x=0.

This figure has three images. The first image is an oval centered around the origin of the rectangular coordinate system. It intersects the x axis at -2 and 2. It intersects the y-axis at -3 and 3. The second image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -2 and 2 and the y-axis at -5 and 5. The third image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -3 and 3 and the y-axis at -5 and 5.
Figure 2.81 (a) This graph represents the trace of equation x222+y232+z252=1x222+y232+z252=1 in the xy-plane, when we set z=0.z=0. (b) When we set y=0,y=0, we get the trace of the ellipsoid in the xz-plane, which is an ellipse. (c) When we set x=0,x=0, we get the trace of the ellipsoid in the yz-plane, which is also an ellipse.

Now that we know what traces of this solid look like, we can sketch the surface in three dimensions (Figure 2.82).

This figure has two images. The first image is a vertical ellipse. There two curves drawn with dashed lines around the center horizontally and vertically to give the image a 3-dimensional shape. The second image is a solid elliptical shape with the center at the origin of the 3-dimensional coordinate system.
Figure 2.82 (a) The traces provide a framework for the surface. (b) The center of this ellipsoid is the origin.

The trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form x2a2+y2b2=zc,x2a2+y2b2=zc, then we call that surface an elliptic paraboloid. The trace in the xy-plane is an ellipse, but the traces in the xz-plane and yz-plane are parabolas (Figure 2.83). Other elliptic paraboloids can have other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation x2a2+z2c2=ybx2a2+z2c2=yb or y2b2+z2c2=xa.y2b2+z2c2=xa.

This figure is the image of a surface. It is in the 3-dimensional coordinate system on top of the origin. A cross section of this surface parallel to the x y plane would be an ellipse.
Figure 2.83 This quadric surface is called an elliptic paraboloid.

Example 2.57

Identifying Traces of Quadric Surfaces

Describe the traces of the elliptic paraboloid x2+y222=z5.x2+y222=z5.

Solution

To find the trace in the xy-plane, set z=0:z=0: x2+y222=0.x2+y222=0. The trace in the plane z=0z=0 is simply one point, the origin. Since a single point does not tell us what the shape is, we can move up the z-axis to an arbitrary plane to find the shape of other traces of the figure.

The trace in plane z=5z=5 is the graph of equation x2+y222=1,x2+y222=1, which is an ellipse. In the xz-plane, the equation becomes z=5x2.z=5x2. The trace is a parabola in this plane and in any plane with the equation y=b.y=b.

In planes parallel to the yz-plane, the traces are also parabolas, as we can see in the following figure.

This figure has four images. The first image is the image of a surface. It is in the 3-dimensional coordinate system on top of the origin. A cross section of this surface parallel to the x y plane would be an ellipse. A cross section parallel to the x z plane would be a parabola. A cross section of the surface parallel to the y z plane would be a parabola. The second image is the cross section parallel to the x y plane and is an ellipse. The third image is the cross section parallel to the x z plane and is a parabola. The fourth image is the cross section parallel to the y z plane and is a parabola.
Figure 2.84 (a) The paraboloid x2+y222=z5.x2+y222=z5. (b) The trace in plane z=5.z=5. (c) The trace in the xz-plane. (d) The trace in the yz-plane.
Checkpoint 2.53

A hyperboloid of one sheet is any surface that can be described with an equation of the form x2a2+y2b2z2c2=1.x2a2+y2b2z2c2=1. Describe the traces of the hyperboloid of one sheet given by equation x232+y222z252=1.x232+y222z252=1.

Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in Figure 2.85(a). In fact, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid. The builders are able to use straight steel beams in the construction, which makes the towers very strong while using relatively little material (Figure 2.85(b)).

This figure has two images. The first image is a sculpture made of parallel sticks, curved together in a circle with a hyperbolic cross section. The second image is a nuclear power plant. The towers are hyperbolic shaped.
Figure 2.85 (a) A sculpture in the shape of a hyperboloid can be constructed of straight lines. (b) Cooling towers for nuclear power plants are often built in the shape of a hyperboloid.

Example 2.58

Chapter Opener: Finding the Focus of a Parabolic Reflector

Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Figure 2.86). If the surface of a parabolic reflector is described by equation x2100+y2100=z4,x2100+y2100=z4, where is the focal point of the reflector?

This figure has two images. The first image is a picture of satellite dishes with parabolic reflectors. The second image is a parabolic curve on a line segment. The bottom of the curve is at point V. There is a line segment perpendicular to the other line segment through V. There is a point on this line segment labeled F. There are 3 lines from F to the parabola, intersecting at P sub 1, P sub 2, and P sub 3. There are also three vertical lines from P sub 1 to Q sub 1, from P sub 2 to Q sub 2, and from P sub 3 to Q sub 3.
Figure 2.86 Energy reflects off of the parabolic reflector and is collected at the focal point. (credit: modification of CGP Grey, Wikimedia Commons)

Solution

Since z is the first-power variable, the axis of the reflector corresponds to the z-axis. The coefficients of x2x2 and y2y2 are equal, so the cross-section of the paraboloid perpendicular to the z-axis is a circle. We can consider a trace in the xz-plane or the yz-plane; the result is the same. Setting y=0,y=0, the trace is a parabola opening up along the z-axis, with standard equation x2=4pz,x2=4pz, where pp is the focal length of the parabola. In this case, this equation becomes x2=100·z4=4pzx2=100·z4=4pz or 25=4p.25=4p. So p is 6.256.25 m, which tells us that the focus of the paraboloid is 6.256.25 m up the axis from the vertex. Because the vertex of this surface is the origin, the focal point is (0,0,6.25).(0,0,6.25).

Seventeen standard quadric surfaces can be derived from the general equation

Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0.

The following figures summarizes the most important ones.

This figure is of a table with two columns and three rows. The three rows represent the first 6 quadric surfaces: ellipsoid, hyperboloid of one sheet, and hyperboloid of two sheets. The equations and traces are in the first column. The second column has the graphs of the surfaces. The ellipsoid graph is a vertical oblong round shape. The hyperboloid of one sheet is circular on the top and the bottom and narrow in the middle. The hyperboloid in two sheets has two parabolic domes opposite of each other.
Figure 2.87 Characteristics of Common Quadratic Surfaces: Ellipsoid, Hyperboloid of One Sheet, Hyperboloid of Two Sheets.
This figure is of a table with two columns and three rows. The three rows represent the second 6 quadric surfaces: elliptic cone, elliptic paraboloid, and hyperbolic paraboloid. The equations and traces are in the first column. The second column has the graphs of the surfaces. The elliptic cone has two cones touching at the points. The elliptic paraboloid is similar to a cone but oblong. The hyperbolic paraboloid has a bend in the middle similar to a saddle.
Figure 2.88 Characteristics of Common Quadratic Surfaces: Elliptic Cone, Elliptic Paraboloid, Hyperbolic Paraboloid.

Example 2.59

Identifying Equations of Quadric Surfaces

Identify the surfaces represented by the given equations.

  1. 16x2+9y2+16z2=14416x2+9y2+16z2=144
  2. 9x218x+4y2+16y36z+25=09x218x+4y2+16y36z+25=0

Solution

  1. The x,y,x,y, and zz terms are all squared, and are all positive, so this is probably an ellipsoid. However, let’s put the equation into the standard form for an ellipsoid just to be sure. We have
    16x2+9y2+16z2=144.16x2+9y2+16z2=144.

    Dividing through by 144 gives
    x29+y216+z29=1.x29+y216+z29=1.

    So, this is, in fact, an ellipsoid, centered at the origin.
  2. We first notice that the zz term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We also note there are xx terms and yy terms that are not squared, so this quadric surface is not centered at the origin. We need to complete the square to put this equation in one of the standard forms. We have
    9x218x+4y2+16y36z+25=09x218x+4y2+16y+25=36z9(x22x)+4(y2+4y)+25=36z9(x22x+11)+4(y2+4y+44)+25=36z9(x1)29+4(y+2)216+25=36z9(x1)2+4(y+2)2=36z(x1)24+(y2)29=z.9x218x+4y2+16y36z+25=09x218x+4y2+16y+25=36z9(x22x)+4(y2+4y)+25=36z9(x22x+11)+4(y2+4y+44)+25=36z9(x1)29+4(y+2)216+25=36z9(x1)2+4(y+2)2=36z(x1)24+(y2)29=z.

    This is an elliptic paraboloid centered at (1,2,0).(1,2,0).
Checkpoint 2.54

Identify the surface represented by equation 9x2+y2z2+2z10=0.9x2+y2z2+2z10=0.

Section 2.6 Exercises

For the following exercises, sketch and describe the cylindrical surface of the given equation.

303.

[T] x2+z2=1x2+z2=1

304.

[T] x2+y2=9x2+y2=9

305.

[T] z=cos(π2+x)z=cos(π2+x)

306.

[T] z=exz=ex

307.

[T] z=9y2z=9y2

308.

[T] z=ln(x)z=ln(x)

For the following exercises, the graph of a quadric surface is given.

  1. Specify the name of the quadric surface.
  2. Determine the axis of symmetry of the quadric surface.
309.
This figure is a surface inside of a box. Its cross section parallel to the y z plane would be an upside down parabola. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
310.
This figure is a surface inside of a box. It is an elliptical cone. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
311.
This figure is a surface in the 3-dimensional coordinate system. There are two conical shapes facing away from each other. They have the x axis through the center.
312.
This figure is a surface in the 3-dimensional coordinate system. It is a parabolic surface with the x axis through the center.

For the following exercises, match the given quadric surface with its corresponding equation in standard form.

  1. x24+y29z212=1x24+y29z212=1
  2. x24y29z212=1x24y29z212=1
  3. x24+y29+z212=1x24+y29+z212=1
  4. z2=4x2+3y2z2=4x2+3y2
  5. z=4x2y2z=4x2y2
  6. 4x2+y2z2=04x2+y2z2=0
313.

Hyperboloid of two sheets

314.

Ellipsoid

315.

Elliptic paraboloid

316.

Hyperbolic paraboloid

317.

Hyperboloid of one sheet

318.

Elliptic cone

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

319.

x2+36y2+36z2=9x2+36y2+36z2=9

320.

−4x2+25y2+z2=100−4x2+25y2+z2=100

321.

−3x2+5y2z2=10−3x2+5y2z2=10

322.

3x2y26z2=183x2y26z2=18

323.

5y=x2z25y=x2z2

324.

8x25y210z=08x25y210z=0

325.

x2+5y2+3z215=0x2+5y2+3z215=0

326.

63x2+7y2+9z263=063x2+7y2+9z263=0

327.

x2+5y28z2=0x2+5y28z2=0

328.

5x24y2+20z2=05x24y2+20z2=0

329.

6x=3y2+2z26x=3y2+2z2

330.

49y=x2+7z249y=x2+7z2

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

331.

[T] x2+z2+4y=0,z=0x2+z2+4y=0,z=0

332.

[T] x2+z2+4y=0,x=0x2+z2+4y=0,x=0

333.

[T] −4x2+25y2+z2=100,x=0−4x2+25y2+z2=100,x=0

334.

[T] −4x2+25y2+z2=100,y=0−4x2+25y2+z2=100,y=0

335.

[T] x2+y24+z2100=1,x=0x2+y24+z2100=1,x=0

336.

[T] x2yz2=1,y=0x2yz2=1,y=0

337.

Use the graph of the given quadric surface to answer the questions.

This figure is a surface inside of a box. It is an oval solid on its side. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
  1. Specify the name of the quadric surface.
  2. Which of the equations—16x2+9y2+36z2=3600,9x2+36y2+16z2=3600,16x2+9y2+36z2=3600,9x2+36y2+16z2=3600, or 36x2+9y2+16z2=360036x2+9y2+16z2=3600—corresponds to the graph?
  3. Use b. to write the equation of the quadric surface in standard form.
338.

Use the graph of the given quadric surface to answer the questions.

This figure is a surface inside of a box. It is a parabolic solid opening up vertically. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
  1. Specify the name of the quadric surface.
  2. Which of the equations—36z=9x2+y2,9x2+4y2=36z,or36z=−81x2+4y236z=9x2+y2,9x2+4y2=36z,or36z=−81x2+4y2—corresponds to the graph above?
  3. Use b. to write the equation of the quadric surface in standard form.

For the following exercises, the equation of a quadric surface is given.

  1. Use the method of completing the square to write the equation in standard form.
  2. Identify the surface.
339.

x2+2z2+6x8z+1=0x2+2z2+6x8z+1=0

340.

4x2y2+z28x+2y+2z+3=04x2y2+z28x+2y+2z+3=0

341.

x2+4y24z26x16y16z+5=0x2+4y24z26x16y16z+5=0

342.

x2+z24y+4=0x2+z24y+4=0

343.

x2+y24z23+6x+9=0x2+y24z23+6x+9=0

344.

x2y2+z212z+2x+37=0x2y2+z212z+2x+37=0

345.

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points A(2,0,0),B(0,0,1),A(2,0,0),B(0,0,1), and C(12,11,12).C(12,11,12).

346.

Write the standard form of the equation of the ellipsoid centered at point P(1,1,0)P(1,1,0) that passes through points A(6,1,0),B(4,2,0)A(6,1,0),B(4,2,0) and C(1,2,1).C(1,2,1).

347.

Determine the intersection points of elliptic cone x2y2z2=0x2y2z2=0 with the line of symmetric equations x12=y+13=z.x12=y+13=z.

348.

Determine the intersection points of parabolic hyperboloid z=3x22y2z=3x22y2 with the line of parametric equations x=3t,y=2t,z=19t,x=3t,y=2t,z=19t, where t.t.

349.

Find the equation of the quadric surface with points P(x,y,z)P(x,y,z) that are equidistant from point Q(0,−1,0)Q(0,−1,0) and plane of equation y=1.y=1. Identify the surface.

350.

Find the equation of the quadric surface with points P(x,y,z)P(x,y,z) that are equidistant from point Q(0,2,0)Q(0,2,0) and plane of equation y=−2.y=−2. Identify the surface.

351.

If the surface of a parabolic reflector is described by equation 400z=x2+y2,400z=x2+y2, find the focal point of the reflector.

352.

Consider the parabolic reflector described by equation z=20x2+20y2.z=20x2+20y2. Find its focal point.

353.

Show that quadric surface x2+y2+z2+2xy+2xz+2yz+x+y+z=0x2+y2+z2+2xy+2xz+2yz+x+y+z=0 reduces to two parallel planes.

354.

Show that quadric surface x2+y2+z22xy2xz+2yz1=0x2+y2+z22xy2xz+2yz1=0 reduces to two parallel planes passing.

355.

[T] The intersection between cylinder (x1)2+y2=1(x1)2+y2=1 and sphere x2+y2+z2=4x2+y2+z2=4 is called a Viviani curve.

This figure is a surface inside of a box. It is a sphere with a right circular cylinder through the sphere vertically. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
  1. Solve the system consisting of the equations of the surfaces to find the equation of the intersection curve. (Hint: Find xx and yy in terms of z.)z.)
  2. Use a computer algebra system (CAS) to visualize the intersection curve on sphere x2+y2+z2=4.x2+y2+z2=4.
356.

Hyperboloid of one sheet 25x2+25y2z2=2525x2+25y2z2=25 and elliptic cone −25x2+75y2+z2=0−25x2+75y2+z2=0 are represented in the following figure along with their intersection curves. Identify the intersection curves and find their equations (Hint: Find y from the system consisting of the equations of the surfaces.)

This figure is a surface inside of a box. It is a hyperbolic paraboloid with a hyperbola of two sheets intersecting. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
357.

[T] Use a CAS to create the intersection between cylinder 9x2+4y2=189x2+4y2=18 and ellipsoid 36x2+16y2+9z2=144,36x2+16y2+9z2=144, and find the equations of the intersection curves.

358.

[T] A spheroid is an ellipsoid with two equal semiaxes. For instance, the equation of a spheroid with the z-axis as its axis of symmetry is given by x2a2+y2a2+z2c2=1,x2a2+y2a2+z2c2=1, where aa and cc are positive real numbers. The spheroid is called oblate if c<a,c<a, and prolate for c>a.c>a.

  1. The eye cornea is approximated as a prolate spheroid with an axis that is the eye, where a=8.7mm andc=9.6mm.a=8.7mm andc=9.6mm. Write the equation of the spheroid that models the cornea and sketch the surface.
  2. Give two examples of objects with prolate spheroid shapes.
359.

[T] In cartography, Earth is approximated by an oblate spheroid rather than a sphere. The radii at the equator and poles are approximately 39633963 mi and 39503950 mi, respectively.

  1. Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane z=0z=0 corresponds to the equator.
  2. Sketch the graph.
  3. Find the equation of the intersection curve of the surface with plane z=1000z=1000 that is parallel to the xy-plane. The intersection curve is called a parallel.
  4. Find the equation of the intersection curve of the surface with plane x+y=0x+y=0 that passes through the z-axis. The intersection curve is called a meridian.
360.

[T] A set of buzzing stunt magnets (or “rattlesnake eggs”) includes two sparkling, polished, superstrong spheroid-shaped magnets well-known for children’s entertainment. Each magnet is 1.6251.625 in. long and 0.50.5 in. wide at the middle. While tossing them into the air, they create a buzzing sound as they attract each other.

  1. Write the equation of the prolate spheroid centered at the origin that describes the shape of one of the magnets.
  2. Write the equations of the prolate spheroids that model the shape of the buzzing stunt magnets. Use a CAS to create the graphs.
361.

[T] A heart-shaped surface is given by equation (x2+94y2+z21)3x2z3980y2z3=0.(x2+94y2+z21)3x2z3980y2z3=0.

  1. Use a CAS to graph the surface that models this shape.
  2. Determine and sketch the trace of the heart-shaped surface on the xz-plane.
362.

[T] The ring torus symmetric about the z-axis is a special type of surface in topology and its equation is given by (x2+y2+z2+R2r2)2=4R2(x2+y2),(x2+y2+z2+R2r2)2=4R2(x2+y2), where R>r>0.R>r>0. The numbers RR and rr are called are the major and minor radii, respectively, of the surface. The following figure shows a ring torus for which R=2andr=1.R=2andr=1.

This figure is a surface inside of a box. It is a torus, a doughnut shape. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.
  1. Write the equation of the ring torus with R=2andr=1,R=2andr=1, and use a CAS to graph the surface. Compare the graph with the figure given.
  2. Determine the equation and sketch the trace of the ring torus from a. on the xy-plane.
  3. Give two examples of objects with ring torus shapes.
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