11X1 T12 08 geometrical theorems (2011)

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  1. 1. Geometrical Theorems about Parabola
  2. 2. Geometrical Theorems about(1) Focal Chords Parabola
  3. 3. Geometrical Theorems about (1) Focal ChordsParabolae.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 ChordsParabolae.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 ChordsParabolae.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 ChordsParabolae.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 ChordsParabolae.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 1Tangents are perpendicular to each other
  8. 8. Geometrical Theorems about (1) Focal ChordsParabolae.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 1Tangents 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 ChordsParabolae.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 1Tangents 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 ChordsParabolae.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 1Tangents 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 ChordsParabolae.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 1Tangents 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 1Tangents meet on the directrix
  12. 12. (2) Reflection Property
  13. 13. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent.
  14. 14. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent.
  15. 15. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB
  16. 16. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB(angle of incidence = angle of reflection)
  17. 17. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB(angle of incidence = angle of reflection) Data: CP || y axis
  18. 18. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB(angle of incidence = angle of reflection) Data: CP || y axis1 Show tangent at P is y px ap 2
  19. 19. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB(angle of incidence = angle of reflection) Data: CP || y axis1 Show tangent at P is y px ap 22 tangent meets y axis when x = 0
  20. 20. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB(angle of incidence = angle of reflection) Data: CP || y axis1 Show tangent at P is y px ap 22 tangent meets y axis when x = 0 K is 0,ap 2
  21. 21. (2) Reflection PropertyAny line parallel to the axis of the parabola is reflected towards thefocus.Any line from the focus parallel to the axis of the parabola is reflectedparallel to the axis.Thus a line and its reflection are equally inclined to the normal, as wellas to the tangent. Prove: SPK CPB(angle of incidence = angle of reflection) Data: CP || y axis1 Show tangent at P is y px ap 22 tangent meets y axis when x = 0 K is 0,ap 2 d SK a ap 2
  22. 22. 2ap 0 ap a 2d PS 22
  23. 23. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1
  24. 24. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1 d SK
  25. 25. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1 d SKSPK is isosceles two = sides
  26. 26. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1 d SKSPK is isosceles two = sides SPK SKP (base s isosceles )
  27. 27. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1 d SKSPK is isosceles two = sides SPK SKP (base s isosceles ) SKP CPB(corresponding s , SK || CP)
  28. 28. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1 d SKSPK is isosceles two = sides SPK SKP (base s isosceles ) SKP CPB(corresponding s , SK || CP) SPK CPB
  29. 29. 2ap 0 ap a 2d PS 2 2 4a 2 p 2 a 2 p 4 2a 2 p 2 a 2 a p4 2 p2 1p 12a 2 a p 2 1 d SKSPK is isosceles two = sides SPK SKP (base s isosceles ) SKP CPB(corresponding s , SK || CP) SPK CPBExercise 9I; 1, 2, 4, 7, 11, 12, 17, 18, 21