Ellipses

An ellipse is the set of all points P such that the sum of the distances between P and two distinct fixed points, called the foci is a constant

The line through the foci intersects the ellipse at two points, the vertices. The line through the vertices is the major axis.

The line perpendicular to the major axis at the center intersects the ellipse at two points called the co-vertices. The line segment joining these points is the minor axis.

• Horizontal Ellipse:

– Foci:

• Vertical Ellipse:

– Foci:

2 2

2 2

( ) ( ) 1x h y ka b

(- , ) ( , )c h k and c h k

2 2

2 2

( ) ( ) 1x h y kb a

( , - ) ( , )h c k and h c k

Identify the center, foci, vertices, co-vertices, length of the major axis, and length of the minor axis.

2 2( 2) ( -3) 125 16x y

Center: (-2, 3)

Vertices: (-7, 3) and (3, 3)*biggest number is under the “x” so we add & sub 5 to the x-coordinate

Co-vertices: (-2, 7) and (-2, -1)*smallest number is under the “y” so we add and sub 4 to the y-coordinate

Foci: c2 = a2 - b2

c2 = 25 – 16 c2 = 9 c = ±3

*Move 3 units left and right from the center to locate the foci.

(1,3) and (-5,3)

Length of major axis: 10

Length of minor axis: 8

Identify the center, foci, vertices, co-vertices, length of the major axis, and length of the minor axis.

2 2( 2) ( 3) 19 4

x y

Center: (-2, 3)Vertices: (1,3) and (-5,3)Co-vertices: (-2, 5) and (-2, 1)Foci: c2 = a2 - b2

c2 = 9 - 4 c2 = 5 c = ±√5 (-2 ±√5, 3)

Length of major axis: 6Length of minor axis:4

Graphing the ellipse

1. Put equation in standard form 2. Graph the center (h, k)3. Graph the foci (look at the equation to determine

your direction)4. Graph a units and –a units from the center to get

the end points (horizontally if under x, vertically if under y)

5. Connect the end points!

Graph 16x2 + 9y2 = 144

To graph:1. Put in standard form.

2. Plot the center (0,0)3. Plot the endpoints of the horizontal axis. Endpoints at (-3,0) and (3,0)

2 2

19 16x y

4. Plot the endpoints of the vertical axis. Endpoints at (0,4) and (0,-4)5. Connect endpoint of axes with smooth curve6. Which way is the major axis in this problem? Vertical because 16>9 and 16 is under the “y” Locate the foci: c2 = b 2 - a2

c2 = 16 - 9 c2 = 7 c = ±√7 Where are the foci? (0, √7) and (0,-√7)

Length of Major Axis is 6.Length of Minor Axis is 4.

2 2( 2) ( 3) 19 4

x y