Transcript No Slide Title

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
F2
Major Axis
V2
x
Theorem Equation of an Ellipse; Center at
(0, 0); Foci at (+ c, 0); Major Axis
along the x-Axis
An equation of the ellipse with center at
(0, 0) and foci at (- c, 0) and (c, 0) is
x2 y2
2
2
2
 2  1 where a  b  0 and b  a  c
2
a b
The major axis is the x-axis; the vertices are
at (-a, 0) and (a, 0).
2
2
x
y
 2 1
2
a
b
b a c
2
2
2
y
F1=(-c, 0)
(0, b)
F2=(c, 0)
x
V2=(a, 0)
V1 = (-a, 0)
(0, -b)
Theorem Equation of an Ellipse; Center at
(0, 0); Foci at (0, + c); Major Axis
along the y-Axis
An equation of the ellipse with center at
(0, 0) and foci at (0, - c) and (0, c) is
2
2
x y
2
2
2
 2  1 where a  b  0 and b  a  c
2
b a
The major axis is the y-axis; the vertices are
at (0, -a) and (0, a).
2
2
y
x
y
 2 1
2
b
a
b a c
2
2
V2= (0, a)
2
(-b, 0)
F2 = (0, c)
(b, 0)
x
F1= (0, -c)
V1= (0, -a)
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 and using a graphing
utility.
Center: (0, 0)
Major axis is the y-axis, so equation is of the form
2
2
x
y
 2 1
2
b a
Distance from center to focus is 5, so c = 5
Distance from center to vertex is 7, so a = 7
b  a  c  7  5  49  25  24
2
2
2
2
2
2
2
x
y
2
and
a

7
,
b

24


1
2
2
b a
2
2
2
2
x
y
 2 1
24 7
x
y

1
24 49
(0, 7)
FOCI
5
(  24,0)
( 24,0)
5
0
5
(0, -7)
5
Ellipse with Major Axis Parallel to the x-Axis
where a > b and b2 = a2 - c2.
Equation
2
2
 x  h   y  k   1
2
2
a
b
y
(h - c, k)
Center
(h, k)
Foci
Vertices
(h + c, k) (h + a, k)
(h + c, k)
Major axis
(h - a, k)
(h, k)
(h + a, k)
x
Ellipse with Major Axis Parallel to the y-Axis
where a > b and b2 = a2 - c2.
Equation
2
2
 x  h   y  k   1
2
2
b
a
y
Center
(h, k)
Foci
Vertices
(h, k + c) (h, k + a)
(h, k + a)
(h, k + c)
(h, k)
(h, k - c)
x
Major axis
(h, k - a)
Find the center, major axis, foci, and vertices of
4 x 2  9 y 2  32 x  36 y  64  0
4 x  32 x  9 y  36 y  64
2
2
4 x 2  8 x  _   9 y 2  4 y  _   64







2
2
 8   16
  4  4
 
 
 2
 2 
4 x 2  8 x  16  9 y 2  4 y  4  64  64  36
4 x  4  9 y  2  36
2
 x  4
9
2
2
y  2


4
2
1
 x  4   y  2  1
9 2
4 2
 x  h   y  k   1
2
a
2
2
b
2
Center: (h, k) = (-4, 2)
Major axis parallel to the x-axis
c  a b  94  5
2
2
2
Vertices: (h + a, k) = (-4 + 3, 2) or (-7, 2) and (-1, 2)
Foci: (h + c, k) = ( 4  5, 2) or
( 4  5, 2) and ( 4  5, 2)
(-4, 4)
4
V(-1, 2)
V(-7, 2) F(-6.2, 2) F(-1.8, 2)
2
C (-4, 2)
8
6
4
2
0
(-4, 0)
2
4