Section 9-4
Ellipses

Objectives
• I can write equations for ellipses
•
I can graph ellipses with certain properties
• I can Complete the Square on equations to find Standard Format

Ellipse
• An ellipse is a set of points in a plane such that the sum of the distances from the two foci is a constant .
• Major axis length is 2a
• Minor axis is 2b

Ellipse Axis

Basic Ellipse Construction
Major Axis = 2a; Minor Axis = 2b, Foci are a distance of “c” from the center point.

Standard Ellipse Equations
• Major Axis is horizontal
• Center at (h, k)
• Standard equation is:
• Major Axis is vertical
• Center at (h, k)
• Standard equation is:

Equations
( a
2
)
2

( y
 b
2 k )
2
( b
2
)
2

( y
 a
2 k )
2

1

1
Horizontal
Vertical a
2
is always the biggest term b
2  c
2  a
2

Graphing Ellipses
• Get equation into standard format
• Determine whether horizontal or vertical??
• Determine key numbers “a”, “b”, and “c” from the equation and using pyth thm.
• Graph center (h, k)
• Graph major axis (2a)
• Graph minor axis (2b)
• Sketch in ellipse shape, then plot foci points on the major axis (“c”)

Example 1
( x

2)
2

( y

1)
25 16
2

1
Horizontal

Example 2
( x

1)
2

( y

4)
9 49
2
 
1
Vertical

Example 3
( x

1)
2
1 y
2
 
4
1
Vertical

Example 4
( x

5)
2

( y

3)
16 25
2
 
1
Vertical

Example 5
( x

1)
2

( y

5)
36 25
2

1
Horizontal

Example 1
• Find the foci and length of major and minor axes for the ellipse with this equation:
• 16x 2 + 4y 2 = 144 (1 st divide all terms by 144)
16 x
2
144

4 y
144
2

144
144 x
2
9 y
2

36

1
Since 36 > 9, then a 2 = 36 and b 2 = 9 a = 6; b = 3 b 2 = a 2 – c 2 b 2 = 36 – 9 = 27, so b = 5.2
Major Axis = 2a = 12 units
Minor Axis = 2b = 6 units
Foci are at (0, 5.2) and (0, -5.2)

Completing the Square
• Given the following equation, find the length of the major and minor axes, plus location of foci
• x 2 + 9y 2 – 4x + 54y + 49 = 0
• (x 2 – 4x) + (9y 2 + 54y) = -49
• (x 2 – 4x) + 9(y 2 + 6y) = -49
• (x 2 – 4x + 4) + 9(y 2 + 6y +9) = -49 + 4 + 81
• (x – 2) 2 + 9(y + 3) 2 = 36 (Now divide by 36)
( x

2 )
2
36

9 ( y

3 )
2
36

36
36

Example Continued
( x

2 )
2
( x
36

2 )
2
36

9 ( y

3 )
2

( y
36

3 )
2
4


1
36
36 a 2 = 36 , so a = 6 b 2 = 4, so b = 2 c 2 = a 2 – b 2 = 36 – 4 = 32, so c = 5.7
Major Axis = 2a = 12 units
Minor Axis = 2b = 4 units
Foci at (-5.7, 0) and (5.7, 0)

Homework
• Worksheet 10-6