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Conic Sections
The EllipsePart A
Ellipse
• Another conicsection formedby a plane intersecting acone
• Ellipse formed when
Definition of Ellipse
• Set of all points in the plane …– ___________ of distances from two fixed
points (foci) is a positive _____________
Definition of Ellipse
• Definition demonstrated by using two tacks and a length of string to draw an ellipse
Graph of an EllipseNote various parts
of an ellipse
Note various parts of an ellipse
Deriving the Formula
• Note– Why?
• Write withdist. formula
• Simplify
( , )P x y
1 2( , ) ( , ) 2d P F d P F a
Major Axis on y-Axis
• Standard form of equation becomes
• In both cases– Length of major axis = _______– Length of __________ axis = 2b–
2 2
2 21
x y
b aa b
2 2 2c a b
Using the Equation
• Given an ellipse with equation
• Determine foci
• Determine values for a, b, and c
• Sketch the graph
2 2
136 49
x y
Find the Equation
• Given that an ellipse …– Has its center at (0,0)– Has a minor axis of length 6– Has foci at (0,4) and (0,-4)
• What is the equation?
Ellipses with Center at (h,k)
• When major axis parallelto x-axis equation can be shown to be
Ellipses with Center at (h,k)
• When major axis parallelto y-axis equation can be shown to be
Find Vertices, Foci
• Given the following equations, find the vertices and foci of these ellipses centered at (h, k)
2 2( 6) ( 2)1
25 81
x y
2 29 6 36 36 0x y x y
Find the Equation
• Consider an ellipse with– Center at (0,3)– Minor axis of length 4– Focci at (0,0) and (0,6)
• What is the equation?
Assignment
• Ellipses A
• 1 – 43 Odd
Conic Sections
EllipseThe Sequel
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
• Eccentricity
Eccentricity
• What limitations can we place on c in relationship to a?– _________________
• What limitationsdoes this put on
• When e is close to 0, graph __________
• When e close to 1, graph ____________
?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
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 ________________
• Aphelion – Distance from _______ to farthest reach
• Mean Distance– Half the
___________
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
Conic Sections
EllipsePart 3
Additional Ellipse Elements
• Recall that the parabola had a directrix
• The ellipse has _________ directrices– They are related to the eccentricity– Distance from center to directrix =
Directrices of An Ellipse
• An ellipse is the locus of points such that – The ratio of the distance to the nearer focus to
…– The distance to the nearer directrix …– Equals a constant that
is less than one.
• This constant is the _______________.
Directrices of An Ellipse
• Find the directrices of the ellipse defined by
2 2
149 35
x y
Additional Ellipse Elements
• The latus rectum is the distance across the ellipse ______________________– There is one at each focus.
Latus Rectum
• Consider the length of the latus rectum
• Use the equation foran ellipse and solve for the y valuewhen x = c– Then double that
distance
Try It Out
• Given the ellipse
• What is the length of the latus rectum?
• What are the lines that are the directrices?
2 23 2
116 9
x y
Graphing An Ellipse On the TI
• Given equation of an ellipse– We note that it is not a
function– Must be graphed in two portions
• Solve for y
2 23 2
125 36
x y
Graphing An Ellipse On the TI
• Use both results
Area of an Ellipse
• What might be the area of an ellipse?
• If the area of a circle is
…how might that relate to the area of the ellipse?– An ellipse is just a unit circle that has been
stretched by a factor A in the x-direction, and a factor B in the y-direction
Area of an Ellipse
• Thus we could conclude that the area of an ellipse is
• Try it with
• Check with a definite integral (use your calculator … it’s messy)
2 2
136 25
x y
Assignment
• Ellipses C
• Exercises from handout 6.2
• Exercises 69 – 74, 77 – 79
• Also find areas of ellipse described in 73 and 79