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Conic Sections

Conic Sections

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Conic Sections

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  1. Conic Sections The EllipsePart A

  2. Ellipse • Another conicsection formedby a plane intersecting acone • Ellipse formed when

  3. Definition of Ellipse • Set of all points in the plane … • ___________ of distances from two fixed points (foci) is a positive _____________

  4. Definition of Ellipse • Definition demonstrated by using two tacks and a length of string to draw an ellipse

  5. Graph of an Ellipse Note various parts of an ellipse

  6. Deriving the Formula • Note • Why? • Write withdist. formula • Simplify

  7. Major Axis on y-Axis • Standard form of equation becomes • In both cases • Length of major axis = _______ • Length of __________ axis = 2b

  8. Using the Equation • Given an ellipse with equation • Determine foci • Determine values for a,b, and c • Sketch the graph

  9. Find the Equation • Given that an ellipse … • Has its center at (0,0) • Has a minor axis of length 6 • Has foci at (0,4) and (0,-4) • What is the equation?

  10. Ellipses with Center at (h,k) • When major axis parallelto x-axis equation can be shown to be

  11. Ellipses with Center at (h,k) • When major axis parallelto y-axis equation can be shown to be

  12. Find Vertices, Foci • Given the following equations, find the vertices and foci of these ellipses centered at (h, k)

  13. Find the Equation • Consider an ellipse with • Center at (0,3) • Minor axis of length 4 • Focci at (0,0) and (0,6) • What is the equation?

  14. Assignment • Ellipses A • 1 – 43 Odd

  15. Conic Sections EllipseThe Sequel

  16. Eccentricity • A measure of the "roundness" of an ellipse not so round very round

  17. Eccentricity • Given measurements of an ellipse • c = distance from center to focus • a = ½ thelength of the major axis • Eccentricity

  18. Eccentricity • What limitations can we place on c in relationship to a? • _________________ • What limitationsdoes this put on • When e is close to 0, graph __________ • When e close to 1, graph ____________

  19. Finding the Eccentricity • Given an ellipse with • Center at (2,-2) • Vertex at (7,-2) • Focus at (4,-2) • What is the eccentricity? • Remember that

  20. Using the Eccentricity • Consider an ellipse with e = ¾ • Foci at (9,0) and (-9,0) • What is the equationof the ellipse in standardform?

  21. Acoustic Property of Ellipse • Sound waves emanating from one focus will be reflected • Off the wall of the ellipse • Through the opposite focus

  22. Whispering Gallery • At Chicago Museumof Science andIndustry The Whispering Gallery is constructed in the form of an ellipsoid, with a parabolic dish at each focus. When a visitor stands at one dish and whispers, the line of sound emanating from this focus reflects directly to the dish/focus at the other end of the room, and to the other person!

  23. Elliptical Orbits • Planets travel in elliptical orbits around the sun • Or satellites around the earth

  24. Elliptical Orbits • Perihelion • Distance from focus to ________________ • Aphelion • Distance from _______ to farthest reach • Mean Distance • Half the___________ MeanDist

  25. Elliptical Orbits • The mean distance of Mars from the Sun is 142 million miles. • Perihelion = 128.5 million miles • Aphelion = ?? • Equation for Mars orbit? Mars

  26. Assignment • Ellipses B • 45 – 63 odd

  27. Conic Sections EllipsePart 3

  28. Additional Ellipse Elements • Recall that the parabola had a directrix • The ellipse has _________ directrices • They are related to the eccentricity • Distance from center to directrix =

  29. Directrices of An Ellipse • An ellipse is the locus of points such that • The ratio of the distance to the nearer focus to … • The distance to the nearer directrix … • Equals a constant that is less than one. • This constant is the _______________.

  30. Directrices of An Ellipse • Find the directrices of the ellipse defined by

  31. Additional Ellipse Elements • The latus rectum is the distance across the ellipse ______________________ • There is one at each focus.

  32. Latus Rectum • Consider the length of the latus rectum • Use the equation foran ellipse and solve for the y valuewhen x = c • Then double that distance

  33. Try It Out • Given the ellipse • What is the length of the latus rectum? • What are the lines that are the directrices?

  34. Graphing An Ellipse On the TI • Given equation of an ellipse • We note that it is not a function • Must be graphed in two portions • Solve for y

  35. Graphing An Ellipse On the TI • Use both results

  36. Area of an Ellipse • What might be the area of an ellipse? • If the area of a circle is…how might that relate to the area of the ellipse? • An ellipse is just a unit circle that has been stretched by a factor A in the x-direction, and a factor B in the y-direction

  37. Area of an Ellipse • Thus we could conclude that the area of an ellipse is • Try it with • Check with a definite integral (use your calculator … it’s messy)

  38. Assignment • Ellipses C • Exercises from handout 6.2 • Exercises 69 – 74, 77 – 79 • Also find areas of ellipse described in 73 and 79