Hyperbola
Steps for this constuction
1) Construct the line y=3.
2) Construct two points on the line you just created. Label the left point A and the right point B.
3) Construct a third point not on the line. Label this point Z
4) Now use the conic section button to construct a hyperbola.
5) Construct the points of intersection of the hyperbola and the line y=3. Label these two points C and D. Left point should be C and the right point should be D
6) Hide the line line y=3.
7) Construct the midpoint of segment AB. Label the midpoint E.
8) Construct segment EA and make sure it is labeled as "c". Find the length of segment c.
9) Construct segment ED and make sure it is labeled as "a". Find the length of segment a.
10) Calculate "sqrt(c^2-a^2)". Make sure this value is labeled "b".
11) Construct a line perpendicular to segment c through the vertex C, and another line perpendicular to segment ED through D.
12) Construct a circle with center at point C and radius length of "b" units.
13) Construct the intersections of the circle and the perpendicular line through point C. Label those points F and G respectively. F should be the top point and G the bottom point. Hide the circle.
14) Construct perpendicular lines to line FG through points F and G.
15) Construct a perpendicular line to segment c through point E. Label the points of intersections with the lines in step 14. Label them H and I. H should be the top point and I the bottom point.
16) Find the length of EH. Notice its length should be the same as the value in step 10 for b. Label the length b.
17) Construct segment EF and find its length. Notice that its length is the same as the length of segment c.
18) Move point A or B to make sure the relationships are consistent.
19) Now construct a point on the hyperbola and label it X.
20) Construct segments AX and BX and find their lengths.
21) Find the value of "abs(AX-BX).
22) Move point X around to make sure that difference is constant. Move point Z and then move point X again to see if the relationship holds.