1. Ellipses
Objectives:
Write the standard equation for an ellipse given sufficient information
Given an equation of an ellipse, graph it and label the center, vertices, co-vertices, and foci

2. Definition of Ellipse
An ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed points, F1 and F2, called the foci, is a constant. P F1 F2
F1P + F2P = 2a

3. Standard Equation of an Ellipse
Horizontal Major Axis:
F1(–c, 0)
F2 (c, 0) y x
V1(–a, 0)
V2 (a, 0)
(0, b)
(0, –b) O x2 a2 y2 b2 +
= 1
a2 > b2
a2 – b2 = c2
length of major axis: 2a
length of minor axis: 2b

4. Standard Equation of an Ellipse
Vertical Major Axis: x2 b2 y2 a2 +
= 1
F1(0, –c)
F2 (0, c) y x
V1(0, –a)
V2 (0, a)
(b, 0)
(–b, 0) O
a2 > b2
a2 – b2 = c2
length of major axis: 2a
length of minor axis: 2b

5. Example 1
Write the standard equation for an ellipse with foci at (-8,0) and (8,0) and with a major axis of 20. Sketch the graph. -8 -6 -4 -2 2 4 6 8
length of major axis: 2a
2a = 20,
so a = 10
a2 – b2 = c2
102 – b2 = 82
b2 =
b2 = 36,
so b = 6 x2
100 y2 36 +
= 1

6. Example 2 Find the vertices and co-vertices of the ellipse. x2 16 y249 +
= 1
vertices:
(0,7) and (0,-7)
co-vertices:
(4,0) and (-4,0)

7. Example 3 Write the standard equation of the ellipse. x2 16 y2 64 +
length of major axis: 2a
2a = 16,
so a = 8 -8 -6 -4 -2 2 4 6 8
length of minor axis: 2b
2b = 8,
so b = 4 x2 16 y2 64 +
= 1

8. Practice
Write the standard equation for an ellipse with foci at (5,0) and (-5,0) and with vertices at (9,0) and (-9,0). Sketch the graph.

9. Standard Equation of a Translated Ellipse
Horizontal Major Axis:
(x – h)2 a2
(y – k)2 b2 +
= 1
a2 > b2
a2 – b2 = c2
length of major axis: 2a length of minor axis: 2b

10. Standard Equation of a Translated Ellipse
Vertical Major Axis:
(x – h)2 b2
(y – k)2 a2 +
= 1
a2 > b2
a2 – b2 = c2
length of major axis: 2a length of minor axis: 2b

11. Example 1
An ellipse is defined by the equation 4x2 + 9y2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse.
4x2 – 16x + 9y2 + 18y = 11
4(x2 – 4x) + 9(y2 + 2y) = 11
4(x2 – 4x + 4) + 9(y2 + 2y + 1) = (4) + 9(1)
4(x – 2)2 + 9(y + 1)2 = 36

12. Example 1
An ellipse is defined by the equation 4x2 + 9y2 – 16x + 18y = 11. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci. Sketch the graph of the ellipse. -6 -4 -2 2 4 6
center: (2,-1)
a2 = 9, so a = 3
vertices: (-1,-1) and (5,-1)
b2 = 4, so b = 2
co-vertices: (2,1) and (2,-3)
a2 – b2 = c2
9 - 4 = c2

13. Practice Write the standard equation for the ellipse
9x2 + 16y2 – 36x – 64y – 44 = 0. Identify the center, vertices, co-vertices, and foci.