Algebra II / Mr. Hansen
3/1/2000 (due 3/17/2000)

Name: ________________________

Part 1.

Using a piece of graph paper, carefully construct an ellipse. You may use the thumbtack method demonstrated in class (be sure to use a cardboard backing!) or any other suitable method. For example, you may wish to trace an ellipse printed in a math textbook, or you could trace the outline of a candy dish if you happen to have one that is a true ellipse. Make an ellipse that is big enough to give good accuracy and room for you to work comfortably (at least 3 or 4 inches for the semimajor axis).

Part 2.

Carefully measure the semimajor axis (a), semiminor axis (b), and focal radius (c). Give your answer in graph paper units (most likely decimal inches), and show that a² = b² + c². (In class, we proved that this equation holds for all ellipses; you merely need to show that it holds for your values of a, b, and c.) Note: If you constructed an ellipse, you can measure c based on your thumbtack points. However, if you traced an ellipse, you must calculate c from the equation.
a = _______________
b = _______________
c = _______________
Proof that a² = b² + c² for these values (show work):

Part 3.

Using a straightedge, draw an x-axis and a y-axis on your ellipse, except do not place the origin at the center of the ellipse. Make one axis parallel to the major axis of the ellipse, and the other one parallel to the minor axis of the ellipse. Do this any way you wish, so long as your origin is not located at the center of the ellipse. Now write the equation of your ellipse (in inches), using standard form. No work is required for this step.
Write equation here:

Part 4.

Using the coordinate system you just created, label the 4 extreme points of the ellipse (i.e., the points at the ends of the major and minor axes) as A, B, C, and D. Then mark the foci (F₁ and F₂). Finally, pick any point P on the ellipse (i.e., any point other than A, B, C, and D) and label P on your diagram. Show all the coordinates below:
A = ( ______ , ______ )
B = ( ______ , ______ )
C = ( ______ , ______ )
D = ( ______ , ______ )
F₁ = ( ______ , ______ )
F₂ = ( ______ , ______ )
P = ( ______ , ______ )

Part 5.

Prove that A, B, C, D, and P satisfy the equation you found in Part 3. Because of roundoff error, they may not match precisely. Show your work:
A:

B:

C:

D:

P:

Part 6.

Using a straightedge, draw the line L that is tangent to the ellipse at P. (Remember, from geometry, that tangent means that the line touches at P but nowhere else.) Using the "rise over run" method, compute the tangent line’s slope as accurately as you can. Show your work:

Part 7.

Find a calculus student, hand him your equation from Part 3, and ask him to compute the slope of the tangent line at point P. Although any calculus student will do, HappyCal students may be especially willing to help you since they will earn a few points in my class. Here is a roster of HappyCal students for your convenience: Mat Brown, C.B. Buente, Bill Clausen, Braxton Collier, Will Felder, Dan Hammond, Eerik Hantsoo, Caesar Maasry, Fareed Melham, Tarit Mitra, Max Murphey, James Rohrbach, Riley Soles, Ryu Yoshida, Colin Zima. Write the name of the calculus student you contacted: ___________ Slope computed by that person: ___________ Show calculus work here:

Part 8.

Using a straightedge, carefully draw segments PF₁ and PF₂ on your diagram. Using a protractor, measure the acute angle between PF₁ and line L. Record your answer here, in degrees: _________ Now measure the acute angle between PF₂ and line L. Record your answer here, in degrees: _________ What do you notice?

Part 9.

Pick two other random points on the ellipse, with nothing special about them, and label them Q and R (no need to show coordinates). Please choose them at places where the curvature is noticeably different (for example, you could put Q near one end of the major axis, where the ellipse is curving rather sharply, and R near a "flatter spot" elsewhere on the ellipse. Carefully draw a line M that is tangent to the ellipse at Q and a line N that is tangent to the ellipse at R. Use arcs to mark the following four angles on your diagram: (1) the acute angle between line M and QF₁, (2) the acute angle between line M and QF₂, (3) the acute angle between line N and RF₁, and (4) the acute angle between line N and RF₂. Use a protractor to measure each of these angles. What can you say about Ð 1, Ð 2, Ð 3, and Ð 4?

Part 10.

Imagine now that your diagram represents an elliptical room (or a room that has an elliptical cross section, such as one of the old meeting rooms—now used as a statue gallery—in the U.S. Capitol building). If somebody is standing at point F₁ and speaking softly, what can you conjecture about the sound waves generated by that person? Write your answer using a complete sentence or two.