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# 11X1 T12 08 geometrical theorems (2011)

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### 11X1 T12 08 geometrical theorems (2011)

1. 1. Geometrical Theorems about Parabola
2. 2. Geometrical Theorems about(1) Focal Chords Parabola
3. 3. Geometrical Theorems about (1) Focal Chords Parabolae.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 Parabolae.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 Parabolae.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 Parabolae.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 Parabolae.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 Parabolae.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 Parabolae.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 Parabolae.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 Parabolae.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 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 axis 1 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 axis 1 Show tangent at P is y  px  ap 2 2 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 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 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 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  2d PS  2 2
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  1 p  1 2 a 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  1 p  1 2 a 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  1 p  1 2 a 2  a  p 2  1  d SK SPK 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  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  2d 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  2d 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  2d 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