11X1 T12 08 geometrical theorems

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1. Geometrical Theorems aboutParabola 2. Geometrical Theorems about (1) Focal ChordsParabola 3. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect at right angles on the directrix. 4. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect at right angles on the directrix.1 Prove pq 1 5. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q. 6. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q.pq 1 7. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q.pq 1 Tangents are perpendicular to each other 8. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q.pq 1 Tangents are perpendicular to each other3 Show that the point of intersection,T , of the tangents isa p q , apq 9. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q.pq 1 Tangents are perpendicular to each other3 Show that the point of intersection,T , of the tangents isa p q , apq y apq 10. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q.pq 1 Tangents are perpendicular to each other3 Show that the point of intersection,T , of the tangents isa p q , apq y apq y a pq 1 11. Geometrical Theorems about(1) Focal Chords Parabola e.g. Prove that the tangents drawn from the extremities of a focal chordintersect 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 thetangent at Q is q.pq 1 Tangents are perpendicular to each other3 Show that the point of intersection,T , of the tangents isa p q , apq y apq y a pq 1 Tangents meet on the directrix 12. (2) Reflection Property 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. (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. (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. (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. (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. (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 axis1 Show tangent at P is y px ap 2 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 axis1 Show tangent at P is y px ap 22 tangent meets y axis when x = 0 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 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 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 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 2 d PS 22 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 1p 1 2 a 2 a p 2 1 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 1p 1 2 a 2 a p 2 1 d SK 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 1p 1 2 a 2 a p 2 1 d SK SPK is isosceles two = sides 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 1p 1 2 a 2 a p 2 1 d SK SPK is isosceles two = sides SPK SKP (base 's isosceles ) 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 12 a2 a p 2 1 d SK SPK is isosceles two = sides SPK SKP (base 's isosceles )SKP CPB (corresponding 's , SK || CP) 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 1p 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. 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 1p 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