Transcript 10.4 Ellipses

What you should learn:
10.4 Ellipses
Goal 1 Graph and write equations of Ellipses.
Goal 2
Identify the Vertices,
co-vertices, and Foci
of the ellipse.
10.4 Ellipses
• An Ellipse is a set of points such that the
distance between that point and two fixed
points called Foci remains constant
P
d1
f1
d2
f2
d1 + d2 = constant
10.4 Ellipses
• The line that goes through the Foci is the Major
Axis.
• The midpoint of that segment between the foci is the
Center of the ellipse (c)
• The intersection of the major axis and the ellipse
itself results in two points, the Vertices (v)
• The line that passes through the center and is
perpendicular to the major axis is called the Minor
Axis
• The intersection of the minor axis and the ellipse
results in two points known as co-vertices
10.4 Ellipses
Graphing and Writing Equations of Ellipses
An ellipse is the set of all points P such that the sum of the distances
between P and two distinct fixed points, called the foci, is a constant.
The line through the foci intersects the ellipse at
two points, the vertices.
The line segment joining the vertices is the
major axis, and its midpoint is the center of the
ellipse.
•
d1
•
focus
•
P
d2
focus
d 1 + d 2 = constant
The line perpendicular to the major axis at the center intersects the ellipse at
two points called co-vertices.
The line segment that joins these points is the minor axis of the ellipse.
The two types of ellipses we will discuss are those with a horizontal major axis
and those with a vertical major axis.
10.4 Ellipses
y
y
vertex:
(–a, 0)
•
•
focus:
(c, 0)
• co-vertex:
•
co-vertex:
(–b, 0)
•
focus:
(0, –c)
minor
axis
(0, –b)
Ellipse with horizontal major axis
x 2 y 2= 1
+
a2 b2
•
co-vertex:
(b, 0)
•
major
axis
•
focus:
(–c, 0)
vertex:
(a, 0)
x
focus:
(0, c)
•
•
•
co-vertex:
(0, b)
vertex: (0, a)
major
axis
x
minor
axis
• vertex: (0, –a)
Ellipse with vertical major axis
x 2 y 2= 1
+
b2 a2
10.4 Ellipses
cv1
v1
F1
c
F2
v2
cv2
10.4 Ellipses
Example of ellipse with vertical
major axis
10.4 Ellipses
Example of ellipse with horizontal
major axis
10.4 Ellipses
Standard Form for Elliptical
Equations
Equation
x2 y 2
 2 1
2
a
b
Major Axis Minor Axis
(length is 2a) (length is 2b)
Vertices CoVertices
Horizontal Vertical
(a,0) (-a,0)
(0,b) (0,-b)
(0,a) (0,-a)
(b,0) (-b,0)
Note that a is the biggest number!!!
x2 y2
 2 1
2
Vertical
b
a
Horizontal
10.4 Ellipses
• The foci lie on the major axis at the
points:
• (c,0) (-c,0) for horizontal major axis
• (0,c) (0,-c) for vertical major axis
2
• Where c
=
2
a
–
2
b
10.4 Ellipses
WRITING EQUATIONS
Write the equation of an ellipse with center (0,0)
that has a vertex at (0,7) & co-vertex at (-3,0)
• Since the vertex is on the y-axis (0,7) a = 7
• The co-vertex is on the x-axis (-3,0) b=3
• The ellipse has a vertical major axis & is of the form
2
2
x
y
 2 1
2
b
a
2
2
x
y

1
9 49
10.4 Ellipses
IDENTIFYING PARTS
Given the equation 9x2 + 16y2 = 144
Identify: foci, vertices, & co-vertices
• First put the equation in standard form:
2
2
9 x 16y
144


144 144 144
2
2
x
y

1
16 9
10.4 Ellipses
2
2
x
y

1
16 9
• From this we know the major axis is
horizontal & a = 4, b = 3
• So the vertices are (4,0) & (-4,0)
•
the co-vertices are (0,3) & (0,-3)
• To find the foci we use c2 = a2 – b2
•
c2 = 16 – 9
•
c = √7
• So the foci are at (√7,0) (-√7,0)
10.4 Ellipses
Reflection on the Section
How can you tell from the equation of an ellipse
whether the major axis is horizontal or vertical?
Write the equation in standard form:
if the larger denominator is under the x, it is horizontal.
Under the y, it is vertical.
assignment
Page 612
# 19 – 67 odd
10.4 Ellipses