Constructing a Hyperbola Adaptation
Goal
Find the locus (set of all points) that will create a hyperbola.
Definition: A hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points is constant.
Directions
1) Set the constant difference, d, of our hyperbola to be 2.
2) Draw a circle centered at A with radius 3.
We want to locate the points whose distances to A and B always have a difference of d, or in this case 2. We have all the points 3 away from point A. 5 is 2 away from 3, and 1 is also 2 away from 3.
2) Create another circle centered at B and with radius 5.
3) Mark the intersection of the circles as point C and point D.
4) Calculate CB-CA. Calculate DB-DA. What can you conclude about points C and D?
6) Create another circle centered at B with radius 1.
7) Mark the intersection of this new circle with the circle centered at A.
8) Explain to the person next to you why this new point, E, must be on the hyperbola.
9) Unselect the circles so that they do not show up, but keep points A-E visible.
Repeat as follows:
10) Draw a circle centered at A with radius 4.
11) Draw a circle centered at B with radius 6 (4+2).
12) Draw a circle centered at B with radius 2 (4-2).
13) Mark all intersection points.
14) Unselect all circles.
Finding all points:
15) Draw a circle centered at A with variable radius a.
16) Draw a circle centered at B with variable radius a+d.
17) Draw a circle centered at B with variable radius a-d.
18) Adjust the a-value so that you see four clear intersection points.
19) Mark all intersection points and turn on "trace" for each of these four points. You may need to zoom in to get the precise intersection point.
20) Play the a-slider to see the full hyperbola.
What name might we give the two fixed points?
What questions do you have about this activity?