Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 8 Analytic Geometry in Two and Three Dimensions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.1 Conic Sections and Parabolas
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4 Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6 What you’ll learn about Conic Sections Geometry of a Parabola Translations of Parabolas Reflective Property of a Parabola … and why Conic sections are the paths of nature: Any free-moving object in a gravitational field follows the path of a conic section.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7 Parabola A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus) in the plane.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8 A Right Circular Cone (of two nappes)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9 Conic Sections and Degenerate Conic Sections
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Conic Sections and Degenerate Conic Sections (cont’d)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Second-Degree (Quadratic) Equations in Two Variables
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Structure of a Parabola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Graphs of x 2 =4py
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Parabolas with Vertex (0,0) Standard equationx 2 = 4pyy 2 = 4px Opens Upward or To the right or to the downward left Focus(0,p)(p,0) Directrixy = -px = -p Axisy-axisx-axis Focal lengthpp Focal width|4p||4p|
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Graphs of y 2 = 4px
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of a Parabola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of a Parabola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Parabolas with Vertex (h,k) Standard equation (x-h) 2 = 4p(y-k)(y-k) 2 = 4p(x-h) Opens Upward or To the right or to the downward left Focus(h,k+p)(h+p,k) Directrixy = k-px = h-p Axisx = hy = k Focal lengthpp Focal width|4p||4p|
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of a Parabola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of a Parabola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.2 Ellipses
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Geometry of an Ellipse Translations of Ellipses Orbits and Eccentricity Reflective Property of an Ellipse … and why Ellipses are the paths of planets and comets around the Sun, or of moons around planets.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Ellipse An ellipse is the set of all points in a plane whose distance from two fixed points in the plane have a constant sum. The fixed points are the foci (plural of focus) of the ellipse. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the ellipse intersects its axis are the vertices of the ellipse.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Key Points on the Focal Axis of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Ellipse with Center (0,0)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Pythagorean Relation
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Vertices and Foci of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Vertices and Foci of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Ellipse with Center (h,k)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Ellipse with Center (h,k)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Locating Key Points of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Locating Key Points of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Elliptical Orbits Around the Sun
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Eccentricity of an Ellipse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.3 Hyperbolas
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Geometry of a Hyperbola Translations of Hyperbolas Eccentricity and Orbits Reflective Property of a Hyperbola Long-Range Navigation … and why The hyperbola is the least known conic section, yet it is used astronomy, optics, and navigation.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference. The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola with Center (0,0)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola Centered at (0,0)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Vertices and Foci of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Vertices and Foci of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Equation of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola with Center (h,k)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Hyperbola with Center (h,k)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Locating Key Points of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Locating Key Points of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Eccentricity of a Hyperbola
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.4 Translations and Rotations of Axes
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Second-Degree Equations in Two Variables Translating Axes versus Translating Graphs Rotation of Axes Discriminant Test … and why You will see ellipses, hyperbolas, and parabolas as members of the family of conic sections rather than as separate types of curves.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Translation-of-Axes Formulas
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Translation Formula
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Translation Formula
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Rotation-of-Axes Formulas
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Rotation of Cartesian Coordinate Axes
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Rotation of Axes
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Rotation of Axes
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Coefficients for a Conic in a Rotated System
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Angle of Rotation to Eliminate the Cross- Product Term
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Discriminant Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Conics and the Equation Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.5 Polar Equations of Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Eccentricity Revisited Writing Polar Equations for Conics Analyzing Polar Equations of Conics Orbits Revisited … and why You will learn the approach to conics used by astronomers.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Focus-Directrix Definition Conic Section A conic section is the set of all points in a plane whose distances from a particular point (the focus) and a particular line (the directrix) in the plane have a constant ratio. (We assume that the focus does not lie on the directrix.)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Focus-Directrix Eccentricity Relationship
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Geometric Structure of a Conic Section
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A Conic Section in the Polar Plane
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Three Types of Conics for r = ke/(1+ecosθ)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Polar Equations for Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Writing Polar Equations of Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Writing Polar Equations of Conics
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Identifying Conics from Their Polar Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Identifying Conics from Their Polar Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Semimajor Axes and Eccentricities of the Planets
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Ellipse with Eccentricity e and Semimajor Axis a
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.6 Three-Dimensional Cartesian Coordinate System
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Three-Dimensional Cartesian Coordinates Distances and Midpoint Formula Equation of a Sphere Planes and Other Surfaces Vectors in Space Lines in Space … and why This is the analytic geometry of our physical world.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Point P(x,y,z) in Cartesian Space
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Coordinate Planes Divide Space into Eight Octants
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Distance Formula (Cartesian Space)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Midpoint Formula (Cartesian Space)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Calculating a Distance and Finding a Midpoint
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Calculating a Distance and Finding a Midpoint
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Standard Equation of a Sphere
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Drawing Lesson
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Drawing Lesson (cont’d)
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Standard Equation of a Sphere
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding the Standard Equation of a Sphere
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Equation for a Plane in Cartesian Space
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Vector v =
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Vector Relationships in Space
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Equations for a Line in Space
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Equations for a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding Equations for a Line
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Chapter Test Solutions