# 11X1 T12 08 geometrical theorems (2011)

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- 1. Geometrical Theorems about Parabola
- 2. Geometrical Theorems about(1) Focal Chords Parabola
- 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. 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. 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. 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. 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. 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. 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. 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. 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. (2) Reflection Property
- 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. (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. (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. (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. (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. (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. (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. (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. (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. 2ap 0 ap a 2d PS 22
- 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. 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. 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. 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. 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. 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. 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