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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.
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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
Notice the foci always lie ‘inside’ the ellipse.
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
Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.
x^2 36
y^2 9
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)
(Sketches will help determine the equations.)
x
y
x
y