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Homework, Page 653

Homework, Page 653. Find the vertices and foci of the ellipse. 1. . Homework, Page 653. Find the vertices and foci of the ellipse. 5. . Homework, Page 653. Match the graph to the equation given that all of the ticks represent one unit. 9. a. . Homework, Page 653.

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Homework, Page 653

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  1. Homework, Page 653 Find the vertices and foci of the ellipse. 1.

  2. Homework, Page 653 Find the vertices and foci of the ellipse. 5.

  3. Homework, Page 653 Match the graph to the equation given that all of the ticks represent one unit. 9. a.

  4. Homework, Page 653 Sketch the graph of the ellipse by hand. 13.

  5. Homework, Page 653 Graph the ellipse using a function grapher. 17.

  6. Homework, Page 653 Find an equation for the ellipse that satisfies the given conditions. 21.

  7. Homework, Page 653 Find an equation for the ellipse that satisfies the given conditions. 25.

  8. Homework, Page 653 Find an equation for the ellipse that satisfies the given conditions. 29.

  9. Homework, Page 653 Find an equation for the ellipse that satisfies the given conditions. 33.

  10. Homework, Page 653 Find center, vertices, and foci of the ellipse. 37.

  11. Homework, Page 653 Graph the ellipse using a parametric grapher. 41.

  12. Homework, Page 653 Prove the graph of the equation is an ellipse, and find its vertices, foci, and eccentricity. 45.

  13. Homework, Page 653 Write an equation for the ellipse. 49.

  14. Homework, Page 653 53. The Moon’s apogee is 252,710 mi, and perigee is 221,463 mi. Assuming the Moon’s orbit is elliptical, with the Earth at one focus, calculate and interpret a, b, c, and e.

  15. Homework, Page 653 57. The sun grazers pass within a Sun’s diameter of the solar surface. What can you conclude about a – c for orbits of sun grazers? (From problem #54, we know that the diameter of the Sun is about 1.392 Gm.)

  16. Homework, Page 653 Solve the system of equations algebraically and support your answer graphically. 61.

  17. Homework, Page 653 65. The distance from a focus of an ellipse to the closer vertex is a(1 + e) where a is a semimajor axis and e is the eccentricity. Justify your answer

  18. Homework, Page 653 69. A. (4, 2) B. (4, 3) C. (4, 4) D. (4, 5) E. (4, 6)

  19. 8.3 Hyperbolas

  20. 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 in astronomy, optics, and navigation.

  21. 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 line through the center and perpendicular to the focal axis is the conjugate axis. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola. The line collinear with the focal axis and connecting the vertices is the transverse axis.

  22. Hyperbola

  23. Hyperbola

  24. Hyperbola with Center (0,0)

  25. Hyperbola Centered at (0,0)

  26. Example Finding the Vertices and Foci of a Hyperbola

  27. Example Sketching a Hyperbola by Hand

  28. Example Finding an Equation of a Hyperbola

  29. Hyperbola with Center (h,k)

  30. Hyperbola with Center (h,k)

  31. Example Locating Key Points of a Hyperbola

  32. Example Locating a Point Using Hyperbolas Radio signals are sent simultaneously from three transmitters located at O, Q, and R. R is 80 miles due north of O and Q is 100 miles due east of O. A ship receives the transmission from O 323.27 μsec after the signal from R and 258.61 μsec after the signal from Q. What is the ship’s bearing and distance from O?

  33. Example Locating a Point Using Hyperbolas

  34. Eccentricity of a Hyperbola

  35. Homework • Homework Assignment #18 • Review Section 8.3 • Page 663, Exercises: 1 – 65(EOO), skip 53

  36. 8.4 Translations and Rotations of Axes

  37. Quick Review

  38. Quick Review Solutions

  39. What you’ll learn about • Second-Degree Equations in Two Variables • Translating Axes versus Translating Graphs … and why You will see ellipses, hyperbolas, and parabolas as members of the family of conic sections rather than as separate types of curves.

  40. Example Plotting Conics

  41. Example Writing Conic Equations From Graphs

  42. Translation-of-Axes Formulas

  43. Translation of Cartesian Coordinate Axes

  44. Example Finding Coordinates of a Point in a Translated Coordinate System Using the point P (x, y) and the translation information, find the coordinates of P in the translated x’y’ coordinate system. 20.

  45. Example Translation Formula

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