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