1. Conic Sections Ellipse The Sequal

2. Deriving the Formula Consider P at (0, b)  Isosceles triangle  Legs = a And a a

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

4. Eccentricity Given measurements of an ellipse  c = distance from center to focus  a = ½ the length of the major axis Eccentricity

5. What limitations can we place on c in relationship to a?  c < a What limitations does this put on When e is close to 0, graph almost a circle When e close to 1, graph long and thin

6. 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

7. 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?

8. 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

9. 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!

10. Elliptical Orbits Planets travel in elliptical orbits around the sun  Or satellites around the earth

11. Elliptical Orbits Perihelion  Distance from focus to closest approach Aphelion  Distance from focus to farthest reach Mean Distance  Half the major axis Mean Dist

12. 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

13. Assignment Ellipses B 45 – 63 odd