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The Ellipse

The Ellipse. Definition. Ellipse – is the set of all points in a plane such that the sum of the distances from two fixed points is constant. The two fixed points are called the foci of the ellipse. Graph of an Ellipse. Deriving the Formula. Note Write with distance formula Simplify.

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The Ellipse

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  1. The Ellipse

  2. Definition Ellipse – is the set of all points in a plane such that the sum of the distances from two fixed points is constant. The two fixed points are called the foci of the ellipse.

  3. Graph of an Ellipse

  4. Deriving the Formula Note Write withdistance formula Simplify

  5. Major Axis on y-Axis • Standard form of equation becomes • In both cases • Length of major axis = 2a • Length of minor axis = 2b

  6. Table of information Ellipse with center at (0,0).

  7. What if the ellipse is not centered at the origin? • If it is centered at any point, (h,k), the ellipse is translated. It is moved right “h” units and up “k” units from the origin. • Consider: • The center is at (2,-3); the distance from the center to the right & left endpoint = 2; the distance to the top & bottom endpoint = 1 • Since a>b, the horizontal axis will be the major axis and the focal points will be on that axis of the ellipse.

  8. Graph

  9. What changes if the Ellipse is not centered on the origin?

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

  11. What is the equation of an ellipse, centered at the origin with a horizontal axis=10 and vertical axis=8?

  12. Find the Equation 1. Consider an ellipse with • Center at (0,3) • Minor axis of length 4 • Foci at (0,0) and (0,6) • What is the equation? 2. Find the coordinates of the center and foci and the lengths of the major and minor axes of the ellipse with equation

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

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

  15. Eccentricity • What limitations can we place on c in relationship to a? • c < a • What limitationsdoes this put on • When e is close to 0, graph is almost a circle • When e close to 1, the graph is long and thin

  16. 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

  17. Whispering Gallery 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! At Chicago Museumof Science andIndustry

  18. Elliptical Orbits Planets travel in elliptical orbits around the sun.

  19. Elliptical Orbits MeanDist • Perihelion • Distance from focus to closest approach • Aphelion • Distance from focus to farthest reach • Mean Distance • Half the majoraxis

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