Conic Sections: The Ellipse

1. Conic Sections:The Ellipse Colleen Beaudoin January, 2009

2. Ellipse • Review: The geometric definition relies on a cone and a plane intersecting it • Algebraic definition: a set of points in the plane such that the sum of the distances from two fixed points, called foci, remains constant.

3. Ellipse From each point in the plane, the sum of the distances to the foci is a constant. Example: B A d1 d2 d1 d2 x f1 f2 Point A: d1+d2 = c foci Point B: d1+d2 = c y

4. Items referenced on the graph of an ellipse: Center Major axis x Minor axis f1 f2 foci y

5. Hands-on Activity At your table is paper, corkboard, string, and tacks. Follow the directions on your handout to complete the activity.

6. Ellipse Definitions Algebraic Definition of an Ellipse: a set of points in the plane such that the sum of the distances from two fixed points, called foci, remains constant. • What remains constant in your sketch? • The points where you placed the tacks are known as the foci. Draw a line through f1 and f2 to the edges of the ellipse. This is known as the major axis. Locate the midpoint between f1 and f2. Is this the center of the ellipse? Will that always be the case? • What inference can you draw from the data? • Does the data support the definition? Explain.

7. Facts: Ellipse Equation • Both variables are squared. • Equation: • What makes the ellipse different from the circle? • What makes the ellipse different from the parabola?

8. where the center is at (h,k) and |2a | is the length of the horizontal axis and |2b| is the of the length of the vertical axis. Procedure to graph: 1. Put in standard form (above): x squared term + y squared term = 1 2. Plot the center (h,k) 3. Plot the endpoints of the horizontal axis by moving “a” units left and right from the center.

9. where the center is at (h,k) and |2a | is the length of the horizontal axis and |2b| is the of the length of the vertical axis. To graph: 4. Plot the endpoints of the vertical axis by moving “b” units up and down from the center. Note: Steps 3 and 4 locate the endpoints of the major and minor axes. 5. Connect endpoint of axes with smooth curve.

10. where the center is at (h,k) and |2a | is the length of the horizontal axis and |2b| is the of the length of the vertical axis. To graph: 6. Use the following formula to help locate the foci: c2 = a2 - b2 if a>b or c2 = b2 – a2 if b>a **Challenge question: Why are we using this formula to locate the foci? Draw a diagram and justify your answer.**

11. where the center is at (h,k) and |2a | is the length of the horizontal axis and |2b| is the of the length of the vertical axis. To graph: 6. (continued) Move “c” units left and right form the center if the major axis is horizontal OR Move “c” units up and down form the center if the major axis is vertical Label the points f1 and f2 for the two foci. Note: It is not necessary to plot the foci to graph the ellipse, but it is common practice to locate them.

12. where the center is at (h,k) and |2a | is the length of the horizontal axis and |2b| is the of the length of the vertical axis. To graph: 7. Identify the length of the major and minor axes.

13. Exp. 1: Graph To graph: 1. Put in standard form (set = 1) Done 2. Plot the center (h,k) (-2,3) 3. Plot the endpoints of the horizontal axis by moving “a” units left and right from the center. Endpoints at (-7,3) and (3,3)

14. Exp. 1: Graph 4. Plot the endpoints of the vertical axis by moving “b” units up and down from the center. Endpoints at (-2,7) and (-2,-1) 5. Connect endpoint of axes with smooth curve

15. Major axis Center Minor axis

16. Exp. 1: Graph 6. Which way is the major axis in this problem (horizontal or vertical)? Horizontal because 25>16 and 25 is under the “x” Use the following formula to help locate the foci: c2 = a2 - b2 if a>b or c2 = b2 – a2 if b>a c2 = a2 - b2 c2 = 25– 16 c2 = 9 c= ±3 Move 3 units left and right from the center to locate the foci. Where are the foci? (-5,3) and (1,3)

17. Foci f1 f2 Length of Major Axis is 10. Length of Minor Axis is 8.

18. Exp. 2: Graph 16x2 + 9y2 = 144 To graph: 1. Put in standard form. 2. Plot the center (0,0) 3. Plot the endpoints of the horizontal axis. Endpoints at (-3,0) and (3,0)

19. Exp. 2: Graph 16x2 + 9y2 = 144 4. Plot the endpoints of the vertical axis. Endpoints at (0,4) and (0,-4) 5. Connect endpoint of axes with smooth curve 6. Which way is the major axis in this problem? Vertical because 16>9 and 16 is under the “y” Locate the foci: c2 = b 2 - a2 c2 = 16 - 9 c2 = 7 c= ±√7 Where are the foci? (0, √7) and (0,-√7)

20. Length of Major Axis is 8. Length of Minor Axis is 6.

21. Exp. 3: Graph 4x2 + 9y2 + 16x – 54y +61 = 0. 1. Put in standard form. (Hint: Complete the square.) 4x2 + 16x + 9y2 – 54y = -61 4(x2 + 4x ) + 9(y2 – 6y ) = -61 +4 +9 +16 + 81 4(x + 2)2 + 9(y – 3)2 = 36 2. Plot the center (-2,3) 3. Plot the endpoints of the horizontal axis. Endpoints at (-5,3) and (1,3)

22. Exp. 3: Graph 4x2 + 9y2 + 16x – 54y +61 = 0. 4. Plot the endpoints of the vertical axis. Endpoints at (-2,5) and (-2,1) 5. Connect endpoint of axes with smooth curve 6. Which way is the major axis in this problem? Horizontal Locate the foci: c2 = a2 - b2 c2 = 9 - 4 c2 = 5 c= ±√5 Where are the foci? (-2 ±√5, 3)

23. Length of Major Axis is 6. Length of Minor Axis is 4.

24. Challenge Question Given the following information, write the equation of the ellipse. Sketch and find the foci. Center is (4,-3), the major axis is vertical and has a length of 12, and the minor axis has a length of 8.

25. Review • How can you tell if the graph of an equation will be a line, parabola, circle, or an ellipse? • What’s the standard form of an ellipse? • What are the steps for graphing an ellipse? • What’s the standard form of a parabola? • What’s the standard form of a circle? • How are the various equations similar and different?