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Lesson 27, Section 11.2: Ellipses - Properties, Equations, and Foci, Slides of Analytical Geometry and Calculus

An in-depth explanation of ellipses, including their definition, properties, and equations. It covers the concepts of foci, vertices, major and minor axes, and the importance of the numbers a, b, and c. The document also includes exercises to find the vertices and foci of specific ellipses and their standard equations. This resource is ideal for students studying geometry or mathematics.

What you will learn

  • What is the definition of an ellipse?
  • How do you find the vertices and foci of an ellipse given its equation?
  • What are the properties of an ellipse, including the major and minor axes and the foci?

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MA 15400 Lesson 27 Section 11.2
Ellipses
1
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points
(the foci) in the plane is a positive constant.
The midpoint of the two foci is the center of the ellipse. The center C of the above ellipse is
marked.
There are two axes: The major axis is the longer of the two and contains the foci. Its endpoints
are the vertices of the ellipses. The minor axis is the shorter of the two and its endpoints are
simply known as the endpoints of the minor axis.
There are three important numbers associated with an ellipse: a, b, and c. These numbers
correspond the distances involving the center, the foci, the vertices, and the endpoints of the
minor axis.
Points
and
F F
are the foci
(plural of focus). The sum of
the distances from any point of
an ellipse (such as
P or H) to
each focus is a constant value.
For example, if
10, then
10
FP F P
FH F H
+ = .
Notice the foci always lie
‘inside’ the ellipse.
C
The major axis of the ellipse at the left
is the segment from
to
V V
, which are
called the vertices. Vertices are
always the endpoints of the major axis.
The minor axis (below left) is the
segment form
to
M M
. In this ellipse
the major axis is horizontal and the
minor axis is vertical.
The a value is the distance between the
center a vertex. The b value is the distance
between the center and an endpoint of the
minor axis. The c value is the distance
between the center and a focus.
Length of major axis = 2a
Length of minor axis = 2b
Length between 2 foci = 2
c
Minor Axis
pf3
pf4
pf5

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Ellipses

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (the foci ) in the plane is a positive constant.

The midpoint of the two foci is the center of the ellipse. The center C of the above ellipse is marked.

There are two axes: The major axis is the longer of the two and contains the foci. Its endpoints are the vertices of the ellipses. The minor axis is the shorter of the two and its endpoints are simply known as the endpoints of the minor axis.

There are three important numbers associated with an ellipse: a, b, and c. These numbers correspond the distances involving the center, the foci, the vertices, and the endpoints of the minor axis.

Points F and F ′^ are the foci (plural of focus). The sum of the distances from any point of an ellipse (such as P or H ) to each focus is a constant value. For example, if 10, then 10

FP F P

FH F H

Notice the foci always lie ‘inside’ the ellipse.

C

The major axis of the ellipse at the left is the segment from V to V ′^ , which are called the vertices. Vertices are always the endpoints of the major axis. The minor axis (below left) is the segment form M to M ′^. In this ellipse the major axis is horizontal and the minor axis is vertical.

The a value is the distance between the center a vertex. The b value is the distance between the center and an endpoint of the minor axis. The c value is the distance between the center and a focus. Length of major axis = 2 a Length of minor axis = 2 b Length between 2 foci = 2 c

Minor Axis

Ellipses

Standard Equation of an Ellipse with Center at the Origin : 2 2 2 2 2 2 1 2 2 1 Where 0

x y x y or a b b a a b

Length of major axis = 2 a Length of minor axis = 2 b

Length between foci = 2 c and

c = a − b

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

A

x^2 36

y^2 9

= 1 B

x^2 16

y^2 49

C 15 x^2 + 9 y^2 = 45

Notice, the a and b can be ‘switched’. The a^2 always is the larger number and is associated with the major axis. If the ellipse has a horizontal major axis, the a^2 is under the x^2. If the ellipse has a vertical major axis, the a^2 is under the y^2.

When writing ordered pairs for vertices, foci, and endpoints of the minor axis; keep in mind whether to add/subtract the a , b , or c from the x or the y.

x

y

x

y

x

y

Ellipses

F x^2 + 36 y^2 + 6 x − 72 y + 9 = 0

G 9( x − 6)^2 + 32( y − 1)^2 = 144

x

y

x

y

Note: As the foci get closer to the center of the ellipse, the ellipse becomes more ‘circular’. Then the foci converge on the center and becomes the center, the ellipse is a circle.

Ellipses

Find an equation for the ellipse with the following information.

H V(±4, 0), M(0,±3) I V(0, ±5), M(± 2, 0)

J V(-1, 1) V' (-1, -3) K V(-1, 2) V'(9, 2)

M(-2, -1) M'(0, -1) M(4, 5) M'(4, -1)

(Sketches will help determine the equations.)

x

y

x

y