5/28/2013
10.3
The ellipse
An ellipse is the collection of points in the plane the
sum of whose distances from two fixed points,
called the foci, is a constant.
Minor Axis
y
P = (x, y)
V1
F1
Major Axis
F2
x
V2
Center
Theorem Equation of an Ellipse;
Center at (0, 0); Foci at (+ c, 0); Major Axis
along the x-Axis
y
F1=(-c, 0)
The major axis is the x-axis; the vertices are
at (-a, 0) and (a, 0).
(0, b)
F2=(c, 0)
x
V2=(a, 0)
V1 = (-a, 0)
(0, -b)
y
Theorem Equation of an Ellipse;
V2= (0, a)
Center at (0, 0); Foci at (0, + c); Major Axis
along the y-Axis
F2 = (0, c)
(-b, 0)
(b, 0)
x
The major axis is the y-axis; the vertices are
at (0, -a) and (0, a).
F1= (0, -c)
V1= (0, -a)
1
5/28/2013
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:
Major axis is the _______________ so equation is
of the form
x2 y2

1
b2 a2
Distance from center to focus is _____ so c = _____
Distance from center to vertex is _____ so a = _____
Ellipse with Major Axis Parallel to the x-Axis where
a > b and b2 = a2 - c2.
Ellipse with Major Axis Parallel to the y-Axis
where a > b and b2 = a2 - c2.
y
y
(h, k + a)
(h + c, k)
(h - c, k)
(h, k + c)
Major axis
(h - a, k)
(h, k)
(h + a, k)
(h, k)
(h, k - c)
x
x
Major axis
(h, k - a)
Center:
Major axis parallel to the x-axis
Vertices:
Foci:
2
5/28/2013
Find the center, foci and vertices for the following ellipse;
4
x 2  9 y 2  6 x  18 y  9  0
2
8
6
4
2
0
2
4
Find the center, foci and vertices for the following ellipse;
4x 2  y 2  8 x  6 y  5
3