1 2 2PF PF a

a

b

Center at(h,k)

An ellipse with major axis parallel to x-axis

c

Definition

2 2 2 a b c

Important Ideaa>b

a

b

(h,k)

c

DefinitionThe standard form of the equation of an ellipse when the major axis is parallel to the x-axis

2 2

2 2

( ) ( ) 1x h y ka b

An ellipse with major axis parallel to y-axis

a

b Center:at (h,k)

c

Definition

2 2 2 a b c

DefinitionThe standard form of the equation of an ellipse when the major axis is parallel to the y-axis

2 2

2 2

( ) ( ) 1

x h y kb a

Important IdeaThe direction of the major axis is determined by the larger denominator. The larger denominator is always a2 in the standard equation.

If the larger denominator is under the x term, the ellipse is “fat”; if the larger denominator is under the y term, the ellipse is “skinny”

Try ThisFor the following ellipse, find the coordinates of the center, foci, vertices, & endpoints of the minor axis. Then graph.

2 2( 4) 136 25x y

Solution 2 2( 4) 136 25x y

Center:(0,-4)Foci:

( 11, 4)

Vertices: (±6,-4) Minor Axes Ends(0,1),(0,-9)

Try ThisWrite an equation of the ellipse with Foci (3,2) and (3,-4) and whose major axes is 14 units long.

Solution

2 2( 3) ( 1) 140 49x y

How is the “roundness” of anellipse measured?

Try This

2 225 6 100 84 0x y x y

For the following ellipse, find the coordinates of the center, foci, vertices, & endpoints of the minor axis. Then graph.

Solution2 2( 3) ( 2) 1

25 1x y

Center:(3,2)Foci:(3 24,2)

Vertices: (-2,2) (8,2) Minor Axes Ends(3,3),(3,1)

Another

2 29 16 36 96 36 0 x y x y

For the following ellipse, find the coordinates of the center, foci, vertices, & endpoints of the minor axis. Then graph.

Solution2 2( 2) ( 3) 1

16 9

x y

Center:(2,-3)Foci:

(2 7, 3)

Vertices: (6,-3) (-2,-3) Minor Axes Ends(2,-6),(2,0)