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Homework, Page 55

Homework, Page 55. Find an equation for each circle. 1. Center (–2, 3); radius 3. Homework, Page 55. Find an equation for each circle. 3. Center (0, 3); radius 12. Homework, Page 55. Graph, if possible. Find center and radius. 5. . Homework, Page 55.

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Homework, Page 55

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  1. Homework, Page 55 Find an equation for each circle. 1. Center (–2, 3); radius 3

  2. Homework, Page 55 Find an equation for each circle. 3. Center (0, 3); radius 12

  3. Homework, Page 55 Graph, if possible. Find center and radius. 5.

  4. Homework, Page 55 Graph, if possible. Find center and radius. 7.

  5. Homework, Page 55 Graph, if possible. Find center and radius. 9.

  6. Homework, Page 55 Graph, if possible. Find center and radius. 11.

  7. Homework, Page 55 Graph, if possible. Find center and radius.13. Empty graph.

  8. Homework, Page 55 Graph, if possible. Find center and radius.15.

  9. Homework, Page 55 Find an equation of the line tangent to the circle at P. 17.

  10. Homework, Page 55 Determine if A is inside, on, or outside the circle. 19. C = (2, –1); r = 3; A = (3, 2)

  11. Homework, Page 55 Determine if A is inside, on, or outside the circle. 21. C = (0, 0); r = 4; A = (2, 2)

  12. Homework, Page 55 Find an equation of each circle. 23. Center (3, 5); tangent to the x-axis

  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)

  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)

  15. Homework, Page 55 29. Find an equation of the circle containing (–9, 2), (–1, 2), (–1, 6), and (–9, 6)

  16. 8.1 Conic Sections and Parabolas

  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.

  18. A Right Circular Cone (of two nappes)

  19. Conic Sections

  20. Degenerate Conic Sections

  21. Second-Degree (Quadratic) Equations in Two Variables

  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.

  23. Parabolas with Vertex (0,0) • Standard equation x2 = 4pyy2 = 4px • Opens Upward or To the right or to the downward left • Focus (0,p) (p,0) • Directrix y = –p x = –p • Axis y-axis x-axis • Focal length pp • Focal width |4p| |4p|

  24. Graphs of x2=4py

  25. Graphs of y2 = 4px

  26. Example Finding an Equation of a Parabola

  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) • Directrix y = k–px = h–p • Axis x = h y = k • Focal length pp • Focal width |4p| |4p|

  28. Example Finding an Equation of a Parabola

  29. Example Graphing a Parabola with a Calculator

  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 x2 = 12y, where should the light bulb be placed?

  31. Homework • Homework Assignment #21 • Review Section: 7.1 • Page 641, Exercises: 1 – 69 (EOO)

  32. 8.2 Ellipses

  33. Quick Review

  34. Quick Review Solutions

  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.

  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.

  37. Key Points on the Focal Axis of an Ellipse

  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 semiminoraxis is the distance from the center of the ellipse to one end of the minor axis, sometimes called a minor vertex.

  39. Ellipse with Center (0,0)

  40. Pythagorean Relation

  41. Example Finding the Vertices and Foci of an Ellipse

  42. Example Finding an Equation of an Ellipse

  43. Ellipse with Center (h,k)

  44. Ellipse with Center (h,k)

  45. Example Locating Key Points of an Ellipse

  46. Example Finding the Equation of an Ellipse

  47. Example Graphing an Ellipse Using Parametric Equations

  48. Example Proving an Ellipse

  49. Elliptical Orbits Around the Sun

  50. Eccentricity of an Ellipse

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