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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)