Slide 8- 1

Chapter 8 Analytic Geometry in Two and Three Dimensions

8.1 Conic Sections and Parabolas

Slide 8- 4 Quick Review

Slide 8- 5 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.

Slide 8- 6 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.

Slide 8- 7 A Right Circular Cone (of two nappes)

Slide 8- 8 Conic Sections and Degenerate Conic Sections

Slide 8- 9 Conic Sections and Degenerate Conic Sections (cont’d) Animation

Slide Second-Degree (Quadratic) Equations in Two Variables

Slide Structure of a Parabola

Slide Graphs of x 2 =4py

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|

Slide Graphs of y 2 = 4px

Slide Example Finding an Equation of a Parabola

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|

Slide Example Finding an Equation of a Parabola

8.2 Ellipses

Slide Quick Review

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.

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.

Slide Key Points on the Focal Axis of an Ellipse

Slide Ellipse with Center (0,0)

Slide Pythagorean Relation

Slide Example Finding the Vertices and Foci of an Ellipse

Slide Example Finding an Equation of an Ellipse

Slide Ellipse with Center (h,k)

Slide Ellipse with Center (h,k)

Slide Example Locating Key Points of an Ellipse

Slide Elliptical Orbits Around the Sun

Slide Eccentricity of an Ellipse

8.3 Hyperbolas

Slide Quick Review

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.

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.

Slide Hyperbola

Slide Hyperbola

Slide Hyperbola with Center (0,0)

Slide Hyperbola Centered at (0,0)

Slide Example Finding the Vertices and Foci of a Hyperbola 1

Slide Example Finding an Equation of a Hyperbola

Slide Hyperbola with Center (h,k)

Slide Hyperbola with Center (h,k)

Slide Example Locating Key Points of a Hyperbola

Slide Eccentricity of a Hyperbola

8.4 Translations and Rotations of Axes

Slide Quick Review

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.

Slide Translation-of-Axes Formulas

Slide Example Translation Formula

Slide Rotation-of-Axes Formulas

Slide Rotation of Cartesian Coordinate Axes

Slide Example Rotation of Axes

Slide Example Rotation of Axes

Slide Coefficients for a Conic in a Rotated System

Slide Angle of Rotation to Eliminate the Cross- Product Term

Slide Discriminant Test

Slide Conics and the Equation Ax 2 +Bxy+Cy 2 +Dx+Ey+F=0

8.5 Polar Equations of Conics

Slide Quick Review

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.

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.)

Slide Focus-Directrix Eccentricity Relationship

Slide The Geometric Structure of a Conic Section

Slide A Conic Section in the Polar Plane

Slide Three Types of Conics for r = ke/(1+ecosθ)

Slide Polar Equations for Conics

Slide Example Writing Polar Equations of Conics

Slide Example Identifying Conics from Their Polar Equations

Slide Semimajor Axes and Eccentricities of the Planets

Slide Ellipse with Eccentricity e and Semimajor Axis a

8.6 Three-Dimensional Cartesian Coordinate System

Slide Quick Review

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.

Slide The Point P(x,y,z) in Cartesian Space

Slide The Coordinate Planes Divide Space into Eight Octants

Slide Distance Formula (Cartesian Space)

Slide Midpoint Formula (Cartesian Space)

Slide Example Calculating a Distance and Finding a Midpoint

Slide Standard Equation of a Sphere

Slide Drawing Lesson

Slide Drawing Lesson (cont’d)

Slide Example Finding the Standard Equation of a Sphere

Slide Equation for a Plane in Cartesian Space

Slide The Vector v =

Slide Vector Relationships in Space

Slide Equations for a Line in Space

Slide Example Finding Equations for a Line

Slide Chapter Test

Slide Chapter Test

Slide Chapter Test Solutions

Slide Chapter Test Solutions