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ellipse

ellipse. Jeff Bivin -- LZHS. Ellipse. The set of all points whose sum of the distances from two fixed points (foci) are constant. Jeff Bivin -- LZHS. Equation of Ellipse. Ellipse. Definition of major and minor axis. minor axis. major axis. Jeff Bivin -- LZHS. Ellipse. major axis.

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ellipse

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  1. ellipse Jeff Bivin -- LZHS

  2. Ellipse • The set of all points whose sum of the distances from two fixed points (foci) are constant. Jeff Bivin -- LZHS

  3. Equation of Ellipse

  4. Ellipse • Definition of major and minor axis minor axis major axis Jeff Bivin -- LZHS

  5. Ellipse major axis Note that foci are ALWAYS on major axis minor axis Jeff Bivin -- LZHS

  6. Ellipse • Definition of vertices • and co-vertices co-verticies verticies verticies endpoint ofmajor axis co-verticies endpoint ofminor axis Jeff Bivin -- LZHS

  7. Ellipse Distance from center to vertex = a Distance from center to co-vertex = b Distance from center to foci = c b c a Length of major axis = 2a Length of minor axis = 2b Eccentricity = c/a directrices a2/c or a/e Jeff Bivin -- LZHS

  8. Eccentricity • Reminder: eccentricity describes the roundness of a conic • For an ellipse: 0<e<1 An eccentricity close to zero means the ellipse is more rounded. See Definition of Ellipse to look at eccentricities

  9. Graph the following Ellipse Center: (0,0) a = 3 in x direction b = 2 in y direction (0, 2) 2 (0, 0) (-3, 0) (3, 0) 3 3 2 (0, -2) Jeff Bivin -- LZHS

  10. Graph the following Ellipse Foci: Eccentricity Directrices (0, 2) 2 (0, 0) (-3, 0) (3, 0) 3 3 2 (0, -2) Jeff Bivin -- LZHS

  11. Center (0,0) major axis = 6 • a = 3 minor axis = 4 • b = 2 vertices (±3,0) • c = co-vertices (0,±2) • foci = • e = • directrices y= x=

  12. Graph the following Ellipse 4x2 + 8x + 9y2 + 54y + 52 = 3 (4x2 + 8x ) + (9y2 + 54y ) = 3 - 52 4(x2 + 2x + 12) + 9(y2 + 6y + 32) = -49 + 4 + 81 4(x + 1)2 + 9(y + 3)2 = 36 9 4 Jeff Bivin -- LZHS

  13. Graph the following Ellipse Center: (-1, 5) a = 7 in y direction b = 4 in x direction (-1, 12) 7 (-1, 5) (-5, 5) (3, 5) 4 4 7 (-1, -2) Jeff Bivin -- LZHS

  14. Graph the Ellipse Center (-1,5) a = 7 b = 4 (-1, 12) foci Major Axis: length = 14 Minor Axis: length = 8 7 (-1, 5) Vertices(-1, 12) & (-1, -2) (-5, 5) (3, 5) 4 4 Co-vertices(-5, 5) & (3, 5) 7 (-1, -2) Jeff Bivin -- LZHS

  15. Graph the Ellipse: Center: (2, -3) a = 5 in x direction b = 3 in y direction (2, 0) 3 (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6) Jeff Bivin -- LZHS

  16. Graph the Ellipse: Major Axis: length = 10 Center: (2,-3) Minor Axis: length = 6 vertices (-3, -3) & (7, -3) co-vertices (2, 0) & (2, -6) a = 5 b = 3 foci (2, 0) c = 4 3 (-2, -3) (6, -3) (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6) Jeff Bivin -- LZHS

  17. Graph the Ellipse: directrices foci (2, 0) 3 (-2, -3) (6, -3) (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6) Jeff Bivin -- LZHS

  18. Center ( ) major axis • a minor axis • b vertices ( ) • c co-vertices ( ) • foci ( ) • e • directrices y= x=

  19. Directrices • The ellipse has twodirectrices • They are related to the eccentricity • Distance from center to directrix =

  20. Eccentricityof 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 eccentricity.

  21. eccentricity = • directrices:

  22. Directrices (directrix) • The directrices of an ellipse are given by

  23. Directrices of An Ellipse • Find the directrices of the ellipse defined by D: x=a2 /c x=49/ c= c=

  24. b2 = 16. This is the smaller of the two numbers in the denominator. a2 = 25. This is the larger of the two numbers in the denominator. Graph : 25x2 + 16y2 = 400. Solution We begin by expressing the equation in standard form. Because we want 1 on the right side, we divide both sides by 400. The equation is the standard form of an ellipse’s equation with a2 = 25 and b2 = 16. Because the denominator of the y2 term is greater than the denominator of the x2 term, the major axis is vertical.

  25. Let's look at how the values of a,b,h,and k affect the ellipse • Note the directrices

  26. Graph the following Ellipse Center: (-1, -3) a = 3 in x direction b = 2 in y direction (-1, -1) 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5) Jeff Bivin -- LZHS

  27. Graph the Ellipse: a = 3 b = 2 Major Axis: length = 6 Minor Axis: length = 4 vertices(-4, -3) & (2, -3) Co-vertices(-1, -5) & (-1, -1) foci (-1, -1) 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5) Jeff Bivin -- LZHS

  28. Graph the following Ellipse 9x2 + 36x + 4y2 - 40y - 100 = 88 (9x2 + 36x ) + (4y2 - 40y ) = 88 + 100 9(x2 + 4x + 22) + 4(y2 - 10y + (-5)2) = 188 + 36 + 100 9(x + 2)2 + 4(y - 5)2 = 324 36 81 Jeff Bivin -- LZHS

  29. Graph the following Ellipse Center: (-2, 5) a = 9 in y direction b = 6 in x direction (-2, 14) 9 (-2, 5) (-8, 5) (4, 5) 6 6 9 (-2, -4) Jeff Bivin -- LZHS

  30. Graph the Ellipse: C(-2,5) a = 9 b = 6 Major Axis: length = 18 (-2, 14) Minor Axis: length = 12 foci 9 (-2, 5) (-8, 5) (6, 5) 6 6 9 vertices(-2, -4) & (-2, 14) co-vertices(-8, 5) & (4, 5) (-2, -4) Jeff Bivin -- LZHS

  31. Now let’s go backwards! • Given the following information, write the equation of the ellipse in standard form. • Focus point ( • Directrices at x =

  32. One more time • Focus point ( • Directrices at y =

  33. Graph the ellipse on a graphing calculator: Rearrange to “y= “ form Graph both parts.

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