Coordinate-Free Definition of Ellipse
An ellipse is the set of all points P in a plane
such that the sum of all distances of P from two fixed points
in the plane is constant
Each of the two fixed points, F’ and F, is called a focus,
and together they are
called foci
The line segment V’V through the foci is called the major axis
The perpendicular bisector B’B of the major axis is called the minor axis
Each end of the major axis, V’ and V, is called a vertex
The midpoint of the line segment F’F is called the center of the ellipse
Using the coordinate-free definition of an ellipse,
we can derive the following equations with respect to the x- and y-axis
of a rectangular coordinate system
Standard Equations of an Ellipse with Center (0,0)
1]
x intercepts:
(vertices) y intercepts:
Foci:
and
Major axis length =
Minor axis length =
2]
x intercepts:
(vertices) y intercepts:
Foci:
and
Major axis length =
Minor axis length =
Both graphs are symmetric with respect to the x axis, y axis, and origin
The major axis is always longer than the minor axis.
See Examples 1 – 2, pages 865 – 867, of the textbook
Elliptic forms occur in nature
orbits in space of satellites and planets
shapes of galaxies
gears and cams
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