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