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
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 )
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
Link to Animated
Web Page
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
( 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
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