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Conic Sections

Conic Sections. Ellipse The Sequal. Eccentricity. A measure of the "roundness" of an ellipse. not so round. very round. Eccentricity. Given measurements of an ellipse c = distance from center to focus a = ½ the length of the major axis Eccentricity. Eccentricity.

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Conic Sections

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  1. Conic Sections EllipseThe Sequal

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

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

  4. 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 almost a circle • When e close to 1, graph long and thin

  5. 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

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

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

  8. 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!

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

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

  11. 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

  12. Assignment • Ellipses B • 45 – 63 odd

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