8.3 Technology: Loci and Conics
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1 8.3 Technology: Loci and Conics The diagram shows a double cone. The two cones have one point in common. The intersection of a double cone and a plane is called a conic section or a conic. The circle, ellipse, parabola, and hyperbola are the cross sections that can be formed when a double cone is sliced by a plane. The different cross sections are formed by changing the angle and location of the plane. Circle Ellipse Parabola Hyperbola In this section, the locus definitions of an ellipse, a hyperbola, and a parabola will be used to construct geometric models of these conics using The Geometer s Sketchpad. The Ellipse An ellipse is the set or locus of points P in the plane such that the sum of the distances from P to two fixed points, F 1 and F 2, is a constant. F 1 P + F 2 P = k One method of constructing this locus using The Geometer s Sketchpad is described in the following steps. 1. Construct a line segment in the lower portion of the screen. Label the endpoints F1 and F2. 2. Construct line segment CD in the upper portion of the screen. CD must be longer than F1F Technology: Loci and Conics MHR 601
2 3. With line segment CD selected, choose Point on Object from the Construct menu. Label this point E. 4. Select points C and E. From the Construct menu, choose Segment. 5. Select points D and E. From the Construct menu, choose Segment. 6. Select point F1 and segment CE. From the Construct menu, choose Circle By Center And Radius. 7. Select point F2 and segment DE. From the Construct menu, choose Circle By Center And Radius. If necessary, drag point E so that the two circles intersect. 8. Select the two circles. From the Construct menu, choose Point At Intersection. Label the points P1 and P2. 9. Select F1 and P1. From the Construct menu, choose Segment. 10. Select F2 and P1. From the Construct menu, choose Segment. 11. Select line segment CD. From the Display menu, choose Hide Segment. 12. Select line segment CE, line segment F1P1, and the circle with centre F1. Right click and choose a colour to be applied to these three objects. 13. Select line segment ED, line segment F2P1, and the circle with centre F2. Right click and choose a different colour to be applied to these three objects. 14. Select points P1 and P2. From the Display menu, choose Trace Points. Drag point E along line segment CD to generate the locus of points. Describe the resulting shape. 602 MHR Chapter 8
3 15. Explain why the sum of F1P1 + F2P1 = CD. Hint: Check the colour used for F1P1 and CE, and then the colour used for F2P1 and DE. 16. Test the construction by changing the length of the line segment joining F1 and F2. Describe how you would change the length of this line segment in order to get ellipses that are wider and narrower. 17. Extension a) Select point E and line segment CD. You may need to choose Show All Hidden from the Display menu. From the Edit menu, choose Action Button. Choose the Animation sub-option. Change the first drop box to once and the third to slowly. Click on Animate to close the window. Double click on to show an ellipse. Experiment with the placement of points F1 and F2 and the length of line segment CD to obtain wider and narrower ellipses. b) From the Graph menu, choose Show Grid. Place F1 and F2 on the x-axis equal distances from the origin. When the construction is completed, what is the distance between the points of intersection of the locus and the x-axis? How is this distance related to the length of line segment CD? c) Select point P1 and point P2. From the Edit menu, choose Trace Points. This will remove the trace feature. Delete the Animate button. Select point P1 and point E in that order. From the Construct menu, choose Locus. Repeat for P2 and point E. The result is a locus of points for each of P1 and P2. What happens to this locus of points if you drag point F1? The Hyperbola The absolute value of a real number is its distance from zero on a real number line. Since distance is always positive, the absolute value of a number is always positive. For a real number represented by x, the absolute value is written x, which means the positive value of x. 8.3 Technology: Loci and Conics MHR 603
4 The absolute value of 2 is 2, or 2 = 2. Similarly 4 9 = 5 = 5 And 9 4 = 5 = 5 A hyperbola is the set or locus of points P in the plane such that the absolute value of the difference of the distances from P to two fixed points, F 1 and F 2, is a constant. F 1 P F 2 P = k One method of constructing this locus using The Geometer s Sketchpad is described in the following steps. 18. Change the Line Tool to line and construct a line through points A and B near the top of the screen. Place points A and B as close to the centre of the top of the screen as possible. 19. From the Construct menu, choose Point On Object. Label this point C. If necessary, move C so that it is not between A and B. 20. Change the Line Tool to line segment. Near the bottom of your screen, construct a line segment. Name the endpoints F1 and F Measure the lengths of AB and F1F2. If necessary drag A or B so that AB is the shorter segment. 22. Construct line segments AC and BC. The difference between the lengths of these two line segments will always equal the length of line segment AB and will serve as the constant for the locus. 23. Select point F1 and line segment AC. From the Construct menu, choose Circle By Center And Radius. 24. Select point F2 and line segment BC. From the Construct menu, choose Circle By Center And Radius. 25. Select the two circles. From the Construct menu, choose Point At Intersection. Label these two points P1 and P Select points P1 and F1. From the Construct menu, choose Segment. 27. Select points P1 and F2. From the Construct menu, choose Segment. 604 MHR Chapter 8
5 28. Select and measure the line segments AC, BC, P1F1, and P1F2. From the Measure menu, choose Calculate. Use the calculator to find m(ac) m(bc) and m(p1f1) m(p1f2). 29. Drag point C along the line through A and B. Notice which segments and calculated values are equal. 30. Select P1 and P2. From the Display menu, choose Trace Points. 31. Select the two circles. From the Display menu, choose Hide Circles. 32. Drag point C along the line through A and B. For the best effect, start at one side of the screen and drag point C to the other side slowly. 33. Describe the locus of points generated by this construction. 34. Extension a) Change the length of line segment F1F2. How can you make the curves flatten? Can you get a similar effect by changing the length of line segment AB? b) Select point C and the line through A and B. From the Edit menu, choose Action Button. Choose the Animation sub-option. Change the first drop box to once and the third drop box to slowly. Click on Animate. Double click on. c) Select Point P1 and point P2. From the Edit menu, choose Trace Points. This will remove the trace feature. Delete the Animate button. Select point P1 and point C in that order. From the Construct menu, choose Locus. Repeat for P2 and point C. The result is a locus of points for each of P1 and P2. What happens to this locus of points if you drag point F1? What happens when the length of F1F2 is shorter than the length of AB? 8.3 Technology: Loci and Conics MHR 605
6 The Parabola A parabola is the set or locus of points P in the plane such that the distance from P to a fixed point F equals the distance from P to a fixed line l. PF = PD The fixed point F is called the focus. The fixed line l is called the directrix. One method of constructing this locus using The Geometer s Sketchpad is described in the following steps. 35. Construct a line AB near the bottom of the screen. Label this line d for directrix. 36. Construct a point not on line d. Label it F, for focus. 37. Select point A and line d. From the Construct menu, choose Perpendicular Line. 38. With this perpendicular line selected, choose Point On Object from the Construct menu. Label this point D. 39. Select points A and D. Change the line tool to a segment tool. From the Construct menu, choose Segment. 40. Select point F and segment AD. From the Construct menu, choose Circle By Center And Radius. This creates a set of points a fixed distance from the focus. 41. We now create a set of points that are the same distance from line d. Select points A and D in that order. From the Transform menu, choose Mark Vector A D. 42. Select line d. From the Transform menu, choose Translate. Choose the option By Marked Vector. Press OK. 43. Select this new line through D and the circle. From the Construct menu, choose Point At Intersection. Label these points P and Q. 44. With points P and Q selected, choose Trace Points from the Display menu. 45. Select the circle and the line through D and P. From the Display menu, choose Hide Objects. 606 MHR Chapter 8
7 46. Drag point D back and forth along line AD. The path of the points P and Q represents the locus of points that are equidistant from point F and line d. 47. Describe the locus of points you constructed. 48. To test this construction, place point F at different positions. Describe the locus when the focus, point F, is closer to the directrix, line AB. Describe the locus when point F is farther away. 49. Extension a) To start with a grid, choose Show Grid from the Graph menu. Construct the directrix as a line parallel to and below the x-axis. Place point F on the y-axis the same number of units above the origin as the directrix is below the x-axis. When the construction is complete, where is the vertex of the parabola? b) Select point D and line AD. From the Edit menu, choose Action Button. Choose the Animation sub-option. Change the first drop box to once and the third drop box to slowly. Click on Animate. Double click to get the locus of points. c) Select points P and Q. From the Edit menu, choose Trace Points. This will remove the tracing of points P and Q. Click on points P and D. From the Construct menu, choose Locus. A locus of points will be created. Then select points Q and D. Again choose Locus from the Construct menu to complete the shape on the screen. Drag point F to new locations and watch the locus of points change. Describe the effect that the position of F has on the locus. 8.3 Technology: Loci and Conics MHR 607
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