10.3 Ellipse

Objective

• To determine the relationship between the equation of an ellipse and its foci and endpoints.

• To graph an ellipse.

Defintion

• The set of all point P in a plane such that the sum of the distances from P to two fixed points is a constant.

• Each of the fixed point is a focus.

Standard form of the equation of an ellipse with

center at the origin and foci on the x-axis

2 2

2 21

x y

a b

The major axis is 2a units long and contains the foci (-c,0) and (c,0). The endpoints of the major axis (-a,0) and (a,0) are often called the vertices.

The shorter segment from –b to b is the minor axis. It is 2b units long. The center is the intersection of the major and minor axis.

Standard form of the equation of the ellipse with center at the origin and foci

on the y-axis2 2

2 21

x y

b a

The major axis is 2a units long and contains the foci (0,-c) and (0,c). The endpoints of the major axis (0,-a) and (0,a) are often called the vertices. The shorter segment from –b to b is the minor axis. It is 2b units long. The center is the intersection of the major and minor axis.

The standard of an ellipse

• Horizontal • Vertical

2 2

2 21

x h y k

a b

2 2

2 21

x h y k

b a

Where h is the horizontal shift and k is the vertical shift.

a > b and c2 = a2 – b2

Ellipse

Determine an equation of an ellipse in standard form with foci (1, 7) and (1, -3) if the length of the minor

axis is 8.

Example 1

Determine the center, endpoints of the major and minor axes, and the foci of

the ellipse with the equation:

2 24 10 8 7 0x y x y

Example 2

Application

The first artificial satellite to orbit earth was Sputnik I. Its orbit was elliptical with the center of Earth at one focus. The major and minor axes of the orbit had lengths of 13906 kilometers and 13, 887 kilometers, respectively. Find the greatest and smallest distances form Earth’s center to the satellilte. Use these to find the least distance and greatest distance of the satellite form Earth’s surface in this orbit.

Eccentricity –e-

•Equals the ratio of the distance between the center and a focus to the distance between the center and the corresponding vertex.

cea

A conic is an ellipse if and only if 0 < e < 1.

The closer e is to zero, the more circular the ellipse. The close e is to 1, the more elongated the ellipse.