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Conic Sections

Ellipses and Circles

Problems

Problems

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 %: The sum of the distances d 1 + d 2 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 c 2 = a 2 - b 2 . 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 c 2 = a 2 - b 2 . 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.

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 %: A circle

The standard equation for a circle is (x - h)2 + (y - k)2 = r 2 . 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 .

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