Conic Sections Ellipse The Sequal. Deriving the Formula Consider P at (0, b) Isosceles triangle ...
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Transcript of Conic Sections Ellipse The Sequal. Deriving the Formula Consider P at (0, b) Isosceles triangle ...
Conic Sections
EllipseThe Sequal
Deriving the Formula
•
• Consider P at (0, b) Isosceles
triangle Legs = a
• And
( , )P x y
1 2( , ) ( , ) 2d P F d P F a
2 2 2a b c
a a
Eccentricity
• A measure of the "roundness" of an ellipse
very roundnot so round
Eccentricity
• Given measurements of an ellipse c = distance from center to focus a = ½ the
length of the major axis
• Eccentricityc
ea
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
?c
ea
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 2 2 2c a b
Using the Eccentricity
• Consider an ellipse with e = ¾ Foci at (9,0) and (-9,0)
• What is the equationof the ellipse in standardform?
Acoustic Property of Ellipse
• Sound waves emanating from one focus will be reflected Off the wall of the ellipse Through the opposite focus
View AnimationView Animation
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!
Elliptical Orbits
• Planets travel in elliptical orbits around the sun Or satellites around the earth
Elliptical Orbits
• Perihelion Distance from focus to closest approach
• Aphelion Distance from focus to farthest reach
• Mean Distance Half the major
axis
MeanDist
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
Assignment
• Ellipses B
• 45 – 63 odd