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Conic Sections Ellipse The Sequal 1 Eccentricity • A measure of the "roundness" of an ellipse not so round very round 2 Eccentricity • Given measurements of an ellipse c = distance from center to focus a = ½ the length of the major axis • Eccentricity c e a 3 Eccentricity • What limitations can we place on c in relationship to a? c< a • What limitations does this put on c e a ? • When e is close to 0, graph almost a circle • When e close to 1, graph long and thin 4 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 c a b 2 2 2 5 Using the Eccentricity • Consider an ellipse with e = ¾ Foci at (9,0) and (-9,0) • What is the equation of the ellipse in standard form? 6 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 7 Whispering Gallery • At Chicago Museum of Science and Industry 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! 8 Elliptical Orbits • Planets travel in elliptical orbits around the sun Or satellites around the earth 9 Elliptical Orbits • Perihelion Distance from focus to closest approach • Aphelion Distance from focus to farthest reach • Mean Distance Mean Dist Half the major axis 10 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 11 Assignment • Ellipses B • 45 – 63 odd 12