Ellipse
Construction
Section 10-3
Pages 744-752
Objectives
• You will be able to write equations for ellipses
• You will be able to graph ellipses with certain
properties
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 is the long axis drawn down the
middle. Its length is 2a
• Minor axis is the short axis drawn down the
middle. Its length is 2b
Ellipse Construction
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:
Eccentricity
• Eccentricity is a measure of ovalness of the
ellipse.
• It is given the symbol “e”
• 0<e<1
c
e
a
Eccentricity Visually
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) ( y  1)

1
25
16
2
2
Example 2
( x  4) ( y  1)

1
9
49
2
2
Example 3
y ( x  1)

1
4
1
2
2
Example 4
( y  5) ( x  3)

1
16
9
2
2
Example 5
( y  5) ( x  3)

1
25
36
2
2
Example 1
• Find the foci and length of major and minor axes for the ellipse
with this equation:
• 16x2 + 4y2 = 144
2
2
16 x
4y
144


144 144 144
2
2
x
y

1
9 36
(1st divide all terms by 144)
Since 36 > 9, then a2 = 36 and b2 = 9
a = 6; b = 3
c2 = a2 – b2
c2 = 36 – 9 = 27, so c = 5.2
Major Axis = 2a = 12 units
Minor Axis = 2b = 6 units
Foci are at (0, 5.2) and (0, -5.2)
Example 2
• Given the following equation, find the length of the
major and minor axes, plus location of foci
• x2 + 9y2 – 4x + 54y + 49 = 0
• (x2 – 4x) + (9y2 + 54y) = -49
• (x2 – 4x) + 9(y2 + 6y) = -49
• (x2 – 4x + 4) + 9(y2 + 6y +9) = -49 + 4 + 81
• (x – 2)2 + 9(y + 3)2 = 36 (Now divide by 36)
( x  2)
9( y  3)
36


36
36
36
2
2
Example Continued
( x  2)
9( y  3)
36


36
36
36
2
2
( x  2)
( y  3)

1
36
4
2
2
a2 = 36 , so a = 6
b2 = 4, so b = 2
c2 = a2 – b2 = 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 12-3