Circles and Ellipses

1. Circles and Ellipses Lesson 8.2

2. Slicing these cones with a plane at different angles produces different conic sections.

3. For example, you can describe a circle as a locus of points that are a fixed distance from a fixed point. • Definition of a Circle • A circle is a locus of points P in a plane, that are a constant distance, r, from a fixed point, C. Symbolically, PC r. The fixed point is called the center and the constant distance is called the radius.

4. The locus describes a circle with radius 4 and center (0, 0).

6. Example B • A circle has center (3, 2) and is tangent to the line y 2x+1. Write the equation of the circle.

7. Now find the point of intersection of this line with the tangent line by solving the system of equations. You should find that the point of intersection, which is the point of tangency, is (0.6, 0.2). You can now find the radius.

9. An ellipse is like a circle, except that it involves two fixed points called foci instead of just one point at the center. You can construct an ellipse by tying a string around two pins and tracing a set of points, as shown. The sum of the distances, d1+d2, is the same for any point on the ellipse.

13. The length of half of the horizontal axis of an ellipse is the horizontal scale factor, a, and the length of half of the vertical axis is the vertical scale factor, b.

14. The segment that forms the longer dimension of an ellipse and contains the foci is the major axis. The segment along the shorter dimension is the minor axis.

15. If you connect an endpoint of the minor axis to the foci, you form two congruent right triangles. Explain why the distance from a focus to an end of the minor axis is the same as half the length of the major axis. • Because the two triangles are congruent, d1 =d2. Because d1 and d2 sum to the length of the major axis, each is half the length of the major axis.

16. From these facts you can conclude that the distance, c, between the center and a focus is related to a and b through the Pythagorean Theorem. • To find the coordinates of the foci, add or subtract c from the appropriate coordinate of the center.

17. Example C • Graph an ellipse that is centered at the origin, with a vertical major axis of 6 units and a minor axis of 4 units. Where are the foci?

20. A Slice of Life • The beam of a flashlight is close to the shape of a cone. A sheet of paper held in front of the flashlight shows different slices, or sections, of the cone of light. Work with a partner, then share results with your group.

21. Step 1 Draw a pair of coordinate axes at the center of your graph paper. Follow the Procedure Note and trace an ellipse. • Step 2 Write an equation that fits the data as closely as possible. Find the lengths of both the major and minor axes. Use the values in your equation to locate the foci. Finally, verify your equation by selecting any two pairs of points on the ellipse and checking that the sum of the distances to the foci is constant.

22. Eccentricity is a measure of how elongated an ellipse is. Eccentricity is defined as the ratio c/a , for an ellipse with a horizontal major axis, or c/b, for an ellipse with a vertical major axis. If the eccentricity is close to 0, then the ellipse looks almost like a circle. The higher the ratio, the more elongated the ellipse.

23. Step 3 Use your flashlight to make ellipses with different eccentricities. Trace three different ellipses. Calculate the eccentricity of each one and label it on your paper. What is the range of possible values for the eccentricity of an ellipse? • Step 4 Continue to tilt your flashlight until the eccentricity becomes too large and you no longer have an ellipse. What shape can you trace now?