1. Warm-Up
Write the standard equation of the circle with the given radius and center.
1) 9; (0,0)
2) 1; (0,5)
3) 4; (-8,-1)
4) 5; (4,2)

2. 10.4 Ellipses What are the major parts of an ellipse?
What are the standard forms of an ellipse equation?
How do you find the value of the foci c when given a & b?
How do you graph an ellipse?
How do you find the equation of an ellipse from a graph?

3. 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

4. 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

5. 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

6. 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

7. 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)

8. 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

9. 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.

10. What are the major parts of an ellipse?
Vertices, co-vertices, foci, center, major & minor axis
What are the standard forms of an ellipse equation?
How do you find the value of the foci c when given a & b?
c2= a2 – b2
How do you graph an ellipse?
Plot the vertices, the co-vertices and draw the ellipse.
How do you find the equation of an ellipse from a graph?
a = vertex of major axis goes into the denominator under major axis letter, and b = co-vertex of minor axis goes into the denominator under the minor axis letter in the formula.

11. Assignment 10.4
p.612, 19-67
Skip 41, 47

12. Warm-Up
Write the standard equation for an ellipse with foci at (-5,0) and (5,0) and with a major axis of 18. Sketch the graph.

13. 10.4 Day 2 Ellipses
What is the standard equation for an ellipse if the vertex has been translated?

14. 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

15. 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

16. 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

17. Example 1
Write an equation of the ellipse with foci at (3,5) and (3,−1) and vertices at (3,6) and (3,−2).
Plot the given points and make a rough sketch. The ellipse has a vertical axis, so its equation is in the form of: -6 -4 -2 2 4 6
Find the center which is halfway between the vertices.
Find a and c
a = 6−2=4 c = 5−2=3
Find b using b2= a2−c2

18. What is the standard equation for an ellipse if the vertex has been translated?

19. Assignment 10.4 day 2
p. 612, even
p. 628, 17, 18, 24, 25