1. 10.3 The Ellipse

2. An ellipse is the locus of all points in a plane such that the sum of the distances from two fixed points, called the foci, is a constant.
Minor Axis y
P = (x, y)
Major Axis x V1 F1 F2 V2

3. Minor Axis P = (x, y) Major Axis V1 F1 F2 V2 y x
In the figure below, F1 and F2 are the foci. The foci are located on the major axis.
Minor Axis y
P = (x, y)
Major Axis x V1 F1 F2 V2

4. The major axis and the minor axis intersect in a point
The major axis and the minor axis intersect in a point. This is the center (h, k) of the ellipse. The vertices are on the major axis and the co-vertices are on the minor axis.
The segment from the center to the vertex on the major axis is called the semi-major axis and the length of this segment is symbolized as a.
The segment from the center to the co-vertex on the minor axis is called the semi-minor axis and the length of this segment is symbolized as b. y a b x F1 F2 V1 V2

5. The distance from the center to the foci along the major axis as c.
The distances a, b, and c have a relationship that can be derived from the definition of ellipse. y x F1 F2 V1 V2 c c

6. The major axis is the x-axis; the vertices are at (-a, 0) and (a, 0).
An equation of the ellipse with center at (0, 0) and foci at (- c, 0) and (c, 0) is
The largest denominator will determine whether the major axis is horizontal or vertical. If the largest denominator is under the x term, it is horizontal.
The major axis is the x-axis; the vertices are at (-a, 0) and (a, 0).

7. y
F2=(c, 0)
F1=(-c, 0)
(0, b) x V1
= (-a, 0)
V2=(a, 0)
(0, -b)

8. An equation of the ellipse with center at (0, 0) and foci at (0, - c) and (0, c) is
Watch where a square appears this time!
The major axis is the y-axis; the vertices are at (0, -a) and (0, a).

9. y
V2= (0, a)
F2 = (0, c)
(-b, 0)
(b, 0) x
F1= (0, -c)
V1= (0, -a)

10. Distance from center to focus is 5, so c = 5
Find an equation of the ellipse with center at the origin, one focus at (0, 5), and a vertex at (0, -7). Graph the equation by hand.
Center: (0, 0)
Major axis is the y-axis, so equation is of the form
Distance from center to focus is 5, so c = 5
Distance from center to vertex is 7, so a = 7

12. (0, 7)
FOCI
(0, -7)

13. Ellipse with Major Axis Parallel to the x-Axis where a > b>0 and Center at (h, k)y
(h + c, k)
(h - c, k)
Major axis
(h - a, k)
(h, k)
(h + a, k) x

14. Ellipse with Major Axis Parallel to the y-Axis where a > b>0 and Center at (h, k).
(h, k + a)
(h, k + c)
(h, k)
(h, k - c) x
Major axis
(h, k - a)

15. Here comes the General Form!
Ax2 + Cy2 + Dx + Ey + F = 0

17. Center: (h, k) = (-4, 2)
Major axis parallel to the x-axis
Vertices: (h + a, k) = (-4 + 3, 2) or (-7, 2) and (-1, 2)
Foci:

18. (-4, 4)
V(-1, 2)
V(-7, 2)
F(-6.2, 2)
F(-1.8, 2)
C (-4, 2)
(-4, 0)