Henry County High School Mrs. Pennebaker

 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.

x y From each point in the plane, the sum of the distances to the foci is a constant. Example: f1f1 f2f2 d2d2 d1d1 foci Point A: d 1 +d 2 = c Point B: d 1 +d 2 = c B A d1d1 d2d2

Center y f1f1 f2f2 foci x Major axis Minor axis

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

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 f 1 and f 2 to the edges of the ellipse. This is known as the major axis. Locate the midpoint between f 1 and f 2. 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.

 Both variables are squared.  Equation:  What makes the ellipse different from the circle?  What makes the ellipse different from the parabola?

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.

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.

To graph: 6. Use the following formula to help locate the foci: c 2 = a 2 - b 2 if a>b or c 2 = b 2 – a 2 if b>a **Challenge question: Why are we using this formula to locate the foci? Draw a diagram and justify your answer.**

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 f 1 and f 2 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.

To graph: 7. Identify the length of the major and minor axes.

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)

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

Major axis Center Minor axis

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: c 2 = a 2 - b 2 if a>b or c 2 = b 2 – a 2 if b>a c 2 = a 2 - b 2 c 2 = 25 – 16 c 2 = 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)

Foci f1f1 f2f2 Length of Major Axis is 10. Length of Minor Axis is 8.

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)

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: c 2 = b 2 - a 2 c 2 = c 2 = 7 c = ±√7 Where are the foci? (0, √7) and (0,-√7)

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

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

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: c 2 = a 2 - b 2 c 2 = c 2 = 5 c = ±√5 Where are the foci? (-2 ±√5, 3)

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

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.

1) How can you tell if the graph of an equation will be a line, parabola, circle, or an ellipse? 2) What’s the standard form of an ellipse? 3) What are the steps for graphing an ellipse? 4) What’s the standard form of a parabola? 5) What’s the standard form of a circle? 6) How are the various equations similar and different?