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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 Homework, Page 55 Find an equation for each circle. 1.Center (–2, 3); radius 3

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 2 Homework, Page 55 Find an equation for each circle. 3.Center (0, 3); radius 12

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 3 Homework, Page 55 Graph, if possible. Find center and radius. 5.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4 Homework, Page 55 Graph, if possible. Find center and radius. 7.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5 Homework, Page 55 Graph, if possible. Find center and radius. 9.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6 Homework, Page 55 Graph, if possible. Find center and radius. 11.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7 Homework, Page 55 Graph, if possible. Find center and radius. 13. Empty graph.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8 Homework, Page 55 Graph, if possible. Find center and radius. 15.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9 Homework, Page 55 Find an equation of the line tangent to the circle at P. 17.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 10 Homework, Page 55 Determine if A is inside, on, or outside the circle. 19.C = (2, –1); r = 3; A = (3, 2)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11 Homework, Page 55 Determine if A is inside, on, or outside the circle. 21.C = (0, 0); r = 4; A = (2, 2)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12 Homework, Page 55 Find an equation of each circle. 23.Center (3, 5); tangent to the x-axis

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 13 Homework, Page 55 Find an equation of each circle. 25.Tangent to the x-axis, the y-axis, and the line y = 5. (two answers)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 14 Homework, Page 55 Find an equation of each circle. 27.Center on the line y = 1 – 2x, tangent to the y-axis at (0, 3)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15 Homework, Page 55 29.Find an equation of the circle containing (–9, 2), (–1, 2), (–1, 6), and (–9, 6)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 8.1 Conic Sections and Parabolas

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 17 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.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18 A Right Circular Cone (of two nappes)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19 Conic Sections

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20 Degenerate Conic Sections

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 21 Second-Degree (Quadratic) Equations in Two Variables

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22 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.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 23 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 = –p x = –p Axisy-axisx-axis Focal lengthpp Focal width|4p||4p|

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24 Graphs of x 2 =4py

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25 Graphs of y 2 = 4px

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26 Example Finding an Equation of a Parabola

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 27 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|

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 28 Example Finding an Equation of a Parabola

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 29 Example Graphing a Parabola with a Calculator

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 30 Example Solving a word Problem About Parabolas 62.Stein Glass, Inc. makes parabolic headlights for a variety of automobiles. If one of its headlights has a parabolic surface generated by the parabola x 2 = 12y, where should the light bulb be placed?

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 31 Homework Homework Assignment #21 Review Section: 7.1 Page 641, Exercises: 1 – 69 (EOO)

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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 8- 33 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 34 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 35 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.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 36 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.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 37 Key Points on the Focal Axis of an Ellipse

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 38 Ellipse - Additional Terms The major axis is the chord connecting the vertices of the ellipse. The semimajor axis is the distance from the center of the ellipse and to one of the vertices. The minor axis is the chord perpendicular to the major axis and passing through the center of the ellipse. The semiminor axis is the distance from the center of the ellipse to one end of the minor axis, sometimes called a minor vertex.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 39 Ellipse with Center (0,0)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 40 Pythagorean Relation

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 41 Example Finding the Vertices and Foci of an Ellipse

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 42 Example Finding an Equation of an Ellipse

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 43 Ellipse with Center (h,k)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 44 Ellipse with Center (h,k)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 45 Example Locating Key Points of an Ellipse

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 46 Example Finding the Equation of an Ellipse

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 47 Example Graphing an Ellipse Using Parametric Equations

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 48 Example Proving an Ellipse

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 49 Elliptical Orbits Around the Sun

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 50 Eccentricity of an Ellipse

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