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:

Equations2 2

2 2

( ) ( ) 1x h y ka b

2 2

2 2

( ) ( ) 1x h y kb a

Horizontal

Vertical

2

2 2 2

is always the biggest terma

b c a

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

2 2( 2) ( 1) 125 16x y

Horizontal

Example 2

2 2( 1) ( 4) 19 49x y

Vertical

Example 3

2 2( 1) 11 4x y

Vertical

Example 4

2 2( 5) ( 3) 116 25x y

Vertical

Example 5

2 2( 1) ( 5) 136 25x y

Horizontal

Example 1

• Find the foci and length of major and minor axes for the ellipse with this equation:

• 16x2 + 4y2 = 144 (1st divide all terms by 144)

144144

1444

14416 22

yx

1369

22

yx

Since 36 > 9, then a2 = 36 and b2 = 9

a = 6; b = 3

b2 = a2 – c2

b2 = 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• 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)

3636

36)3(9

36)2( 22

yx

Example Continued

3636

36)3(9

36)2( 22

yx

14

)3(36

)2( 22

yx

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 10-6