Transcript Conic Sections The Ellipse Part A

Conic Sections
The Ellipse
Part A
Ellipse
• Another conic
section formed
by a plane
intersecting a
cone
• 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 Ellipse
Note various parts
of an ellipse
Deriving the Formula
• Note d ( P, F1 )  d ( P, F2 )  2a
– Why?
• Write with
dist. formula
• Simplify
P ( x, y )
Major Axis on y-Axis
• Standard form of
equation becomes
2
2
x
y
 2 1
2
b
a
ab
• In both cases
– Length of major axis = _______
– Length of __________ axis = 2b
2
2
2
– c  a b
Using the Equation
• Given an ellipse with equation
• Determine foci
• Determine values for
a, b, and c
• Sketch the graph
x2 y 2

1
36 49
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 parallel
to x-axis equation can be
shown to be
Ellipses with Center at (h,k)
• When major axis parallel
to 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
( x  6) ( y  2)

1
25
81
x 2  9 y 2  6 x  36 y  36  0
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
Ellipse
The Sequel
Eccentricity
• A measure of the "roundness" of an ellipse
not so round
very 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 limitations
does this put on
c
e
a
?
• When e is close to 0, graph __________
• When e close to 1, graph ____________
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 c  a  b
2
2
2
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?
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 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!
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
___________
Mean
Dist
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
Ellipse
Part 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
x2 y 2

1
49 35
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 for
an ellipse and
solve for the y value
when x = c
– Then double that
distance
Try It Out
• Given the ellipse
 x  3
16
2
y  2


9
2
1
• What is the length of the latus rectum?
• What are the lines that are the directrices?
Graphing An Ellipse On the TI
• Given equation of an ellipse
 x  3   y  2   1
25
36
– We note that it is not a
function
– Must be graphed in two portions
• Solve for y
2
2
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
2
2
x
y

1
36 25
• Check with a definite integral (use your
calculator … it’s messy)
Assignment
•
•
•
•
Ellipses C
Exercises from handout 6.2
Exercises 69 – 74, 77 – 79
Also find areas of ellipse
described in 73 and 79