Practice 1B Find the Standard Equation Vertices: Minor Axis of Length 2

10.2Ellipses

Definition of an Ellipse An ellipseis the set of all points the sum of whose distances from two distinct fixed points (foci) is constant.

Example 1Find the standard form of the equation of the ellipse having foci at and and a major axis of length 6

Practice 1b Find the standard equation Vertices: Minor axis of length 2

Example 2 Sketch the ellipse given by and identify the vertices.

2. Sketch the ellipse given by and identify the vertices

EXAMPLE 3 Write the equation of the ellipse in standard form and sketch the graph of the ellipse.

Example 4 Find the center,vertices,and foci of the ellipse

7. Find the center,vertices,and foci of the ellipse given by

5. Find standard form of equation

Center: Foci:

6. Find standard form of equation

Vertices: Endpoints of the minor axis:

Definition of Eccentricity

The eccentricityeof an ellipse is given by the ratio

7. Find standard form of equation

Find an equation of the ellipse with vertices and eccentricity

8. Find the standard form of the equation of the specified ellipse.

10. The moon travels about the earth in an elliptical orbit with the earth at one focus,The major and minor axes of the orbit have lengths of 768,806 kilometers and 767,746 kilometers,respectively. Find the greatest and least distances (the apogee and perigee) from the earth’s center to the moon’s center.

11.Satellite Orbit The first artificial satellite to orbit earth was Sputnik I(launched by Russia in 1957). Its highest point above earth’s surface was 938 kilometers,and its lowest point was 212 kilometers. The radius of earth is 6378 kilometers. Find the eccentricity of the orbit.