11X1 T11 08 geometrical theorems (2010)

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11X1 T11 08 geometrical theorems (2010)

  1. 1. Geometrical Theorems about Parabola
  2. 2. Geometrical Theorems about (1) Focal Chords Parabola
  3. 3. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix.
  4. 4. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1
  5. 5. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q.
  6. 6. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q. pq  1
  7. 7. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q. pq  1 Tangents are perpendicular to each other
  8. 8. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q. pq  1 Tangents are perpendicular to each other 3 Show that the point of intersection,T , of the tangents is a  p  q  , apq
  9. 9. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q. pq  1 Tangents are perpendicular to each other 3 Show that the point of intersection,T , of the tangents is a  p  q  , apq y  apq
  10. 10. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q. pq  1 Tangents are perpendicular to each other 3 Show that the point of intersection,T , of the tangents is a  p  q  , apq y  apq  y  a  pq  1
  11. 11. Geometrical Theorems about (1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 1 Prove pq  1 2 Show that the slope of the tangent at P is p, and the slope of the tangent at Q is q. pq  1 Tangents are perpendicular to each other 3 Show that the point of intersection,T , of the tangents is a  p  q  , apq y  apq  y  a  pq  1 Tangents meet on the directrix
  12. 12. (2) Reflection Property
  13. 13. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent.
  14. 14. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent.
  15. 15. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB
  16. 16. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB (angle of incidence = angle of reflection)
  17. 17. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB (angle of incidence = angle of reflection) Data: CP || y axis
  18. 18. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB (angle of incidence = angle of reflection) Data: CP || y axis 1 Show tangent at P is y  px  ap 2
  19. 19. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB (angle of incidence = angle of reflection) Data: CP || y axis 1 Show tangent at P is y  px  ap 2 2 tangent meets y axis when x = 0
  20. 20. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB (angle of incidence = angle of reflection) Data: CP || y axis 1 Show tangent at P is y  px  ap 2 2 tangent meets y axis when x = 0  K is 0,ap 2 
  21. 21. (2) Reflection Property Any line parallel to the axis of the parabola is reflected towards the focus. Any line from the focus parallel to the axis of the parabola is reflected parallel to the axis. Thus a line and its reflection are equally inclined to the normal, as well as to the tangent. Prove: SPK  CPB (angle of incidence = angle of reflection) Data: CP || y axis 1 Show tangent at P is y  px  ap 2 2 tangent meets y axis when x = 0  K is 0,ap 2  d SK  a  ap 2
  22. 22. 2ap  0  ap  a  2 d PS  2 2
  23. 23. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1
  24. 24. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1  d SK
  25. 25. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1  d SK SPK is isosceles  two = sides 
  26. 26. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1  d SK SPK is isosceles  two = sides  SPK  SKP (base 's isosceles  )
  27. 27. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1  d SK SPK is isosceles  two = sides  SPK  SKP (base 's isosceles  ) SKP  CPB (corresponding 's  , SK || CP)
  28. 28. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1  d SK SPK is isosceles  two = sides  SPK  SKP (base 's isosceles  ) SKP  CPB (corresponding 's  , SK || CP)  SPK  CPB
  29. 29. 2ap  0  ap  a  2 d PS  2 2  4a 2 p 2  a 2 p 4  2a 2 p 2  a 2  a p4  2 p2  1 p  1 2 a 2  a  p 2  1  d SK SPK is isosceles  two = sides  SPK  SKP (base 's isosceles  ) SKP  CPB (corresponding 's  , SK || CP)  SPK  CPB Exercise 9I; 1, 2, 4, 7, 11, 12, 17, 18, 21

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