Transcript Conic Sections The Ellipse Part A Ellipse • Another conic section formed by a plane intersecting a cone • Ellipse formed when     90  

Conic Sections
The Ellipse
Part A
Ellipse
• Another conic
section formed
by a plane
intersecting a
cone
• Ellipse formed
when     90


Definition of Ellipse
• Set of all points in the plane …
 Sum of distances from two fixed points
(foci) is a positive constant
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Geogebra
Example
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
2
2
x
y
 2 1
2
a
b
ab
P( x, y)
Deriving the Formula
d ( P, F1 )  d ( P, F2 )  2a
•
• Consider P at (0, b)
 Isosceles
P( x, y)
triangle
 Legs = a
a
• And
a b c
2
2
2
a
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 = 2a
 Length of minor axis = 2b

c  a b
2
2
2
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Using the Equation
• Given an ellipse with equation
• Determine foci
• Determine values for
a, b, and c
• Sketch the graph
2
2
x
y

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
( x  h) 2 ( y  k ) 2

1 a  b
2
2
a
b
Ellipses with Center at (h,k)
• When major axis parallel
to y-axis equation can be
shown to be
( x  h) 2 ( y  k ) 2

1 a  b
2
2
b
a
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
x2  9 y 2  6x  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