Math 487 Lab #1. Wed 01/03/01

Many ways of defining curves which turn out to be conics.

A. Distance sum and differences.

Probably the most familiar definition of an ellipse is in terms of the sum of distances to two points. We start with this.

Set up the lengths and measures

• First we construct some segments that will provide the distances. In a new sketch, draw a LINE AB (NOT a segment) and construct a point C on the line.
• Hide the line and then construct segments AC and BC. Color them differently (say red for AC and blue for BC) and make them thick. Label segment AC as d1 and segment BC as d2.
• Measure the lengths of d1 and d2. Calculate the sum and difference of these lengths with the calculator. Also measure distance AB. Drag C to see what happens to the sum and difference.

.

Construct two circles and trace the intersections

In the sketch, draw two points and label them F1 and F2.

Select segment d1 and point F1 and construct the circle c1 with center F1 and radius d1. Do the same for F2 and d2 to construct circle c2.

Intersect the two circles and trace the two points of intersection as you drag point C between A and B.

For some positions of the points F1 and F2, the trace of the points of intersection will be an ellipse. What locations of C on line AB trace points on the ellipse?

For some positions of F1 and F2, a hyperbola is traced (note that both parts are traced). What locations of C on line AB trace points on the hyperbola?

Questions

1. Explain precisely what points P are traced on the "ellipse trace." The points P should satisfy a distance sum relation. Show what this is and why it is true.
2. Explain precisely what points P are traced on the "hyperbola trace." The points P should satisfy a distance difference relation. Show what this is and why it is true.
3. What are the relationships among F1, F2 and A and B that determine whether the trace is an ellipse or a hyperbola (or something else). In other words, what data about these points are needed in order to predict whether an ellipse or a hyperbola is drawn? (And how do we make the prediction?) Explain your reasoning.

B. Distance Ratio — two point case

New experiment with old figure

• In the very same figure you have already constructed, make one more measurement. Use the calculator to compute the ratio d1/d2 (or measure this directly by selecting the two segments).
• Now instead of dragging C, drag point B. Notice that the sum and difference measures change, but this ratio measure remains constant.
• Observe the trace as B is dragged and the ratio is held constant. What does this figure appear to be? Move C to a new location and drag B again to check the new trace. Again what does the figure appear to be?
• Move C as near the midpoint as possible, so that the ratio is as close to 1 as you can make it. Now drag B again. What does the locus appear to be?

Questions

1. When the ratio is 1, explain what figure is traced and why.
2. What is ratio is a real number e > 0 with e not equal to 1, explain why the set of points that is traced is the set of points P, so that the ratio PF1/PF2 = e, a constant. Introduce coordinates and use the distance formula to show that this equation is the equation of a circle. (We will find a more geometric reason later.)

C. Distance Ratio — point and line case

We will continue by creating another trace, this time with the ratio of the distance to a point and the distance to a line being a constant. We can do it by making a few additions to our current sketch.

• First, hide the points being previously traced. Select the two intersection points and then create Hide/Show buttons from the Edit ->Action Button menu.
• Next, construct the line (NOT SEGMENT) F1F2. Then construct the line through F2 perpendicular to line F1F2, and label this new line d.
• Next, construct the two points of intersection of line F1F2 with the circle c2. Then construct the lines through these two points parallel to d. The points on these two lines are the points that are at distance d2 from line d. Call these lines p1 and p2.
• Now hide the line F1F1, the circle c2, and their points of intersection (make Hide/Show buttons if you want).
• Again, taking e = ratio d1/d2, any point of intersection Q of c1 and either line p1 or p2 satisfies the relation (distance Q to F1)/(distance Q to d) = k.
• Construct the intersection of c1 with p1 and also with p2. Trace all four points of intersection.
• Drag B to keep k fixed and to trace out the set of all such Q.

Questions

1. For what values of e you get a trace that appears to be an ellipse, a parabola, and a hyperbola? Explain why for certain values of e you will get two parts to the trace with a gap in between.

D. Lines and conics

Carry out the Investigations 1 and 2 of Exploration 6.4, pages 89-93 of GTC.

Return to Math 445 Home Page.