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Ellipses and Circles
An ellipse is the set of points such that the sum of the distances from any
point on the ellipse to two other fixed points is constant. The two fixed
points are called the foci (plural of focus) of the ellipse.
Figure 3.1: The sum of the distances d1 + d2 is the same for any point on the
ellipse.
The line segment containing the foci of an ellipse with both endpoints on the
ellipse is called the major axis. The endpoints of the major axis are
called the vertices. The point halfway between the foci is the center
of the ellipse. The line segment perpendicular to the major axis and passing
through the center, with both endpoints on the ellipse, is the minor axis.
The standard equation of an ellipse with a horizontal major axis is the
following: 
+ 
= 1. The center is
at (h, k). The length of the major axis is 2a, and the length of the minor
axis is 2b. The distance between the center and either focus is c, where
c2 = a2 - b2. Here a > b > 0.
+ = 1. The center is
at (h, k). The length of the major axis is 2a, and the length of the minor
axis is 2b. The distance between the center and either focus is c, where
c2 = a2 - b2. Here a > b > 0.
The standard equation of an ellipse with a vertical major axis is the following:
+ 
= 1. The center is at (h, k).
The length of the major axis is 2a, and the length of the minor axis is 2b.
The distance between the center and either focus is c, where c2 = a2 - b2. Here a > b > 0.
+ = 1. The center is at (h, k).
The length of the major axis is 2a, and the length of the minor axis is 2b.
The distance between the center and either focus is c, where c2 = a2 - b2. Here a > b > 0.
The eccentricity of an ellipse is e = 
. For any ellipse, 0 < e < 1. The eccentricity of an ellipse is basically a measure of the "ovalness"
of an ellipse. It is the ratio of the distance between the foci and the length
of the major axis. If the foci are very near the center of an ellipse, the
ellipse is nearly circular, and e is close to zero. If the foci are
relatively far away from the center, the ellipse is shaped more like an oval,
and e is closer to one.
. For any ellipse, 0 < e < 1. The eccentricity of an ellipse is basically a measure of the "ovalness"
of an ellipse. It is the ratio of the distance between the foci and the length
of the major axis. If the foci are very near the center of an ellipse, the
ellipse is nearly circular, and e is close to zero. If the foci are
relatively far away from the center, the ellipse is shaped more like an oval,
and e is closer to one.
Circles
A circle is the collection of points equidistant from a fixed point. The
fixed point is called the center. The distance from the center to any point
on the circle is the radius of the circle, and a segment containing the
center whose endpoints are both on the circle is a diameter of the circle.
The radius, r, equals one-half the diameter, d.
Figure 3.2: A circle
The standard equation for a circle is (x - h)2 + (y - k)2 = r2. The
center is at (h, k). The radius is r.
In a way, a circle is a special case of an ellipse. Consider an ellipse whose
foci are both located at its center. Then the center of the ellipse is the
center of the circle, a = b = r, and e = = 0.
= 0.
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