The document discusses two geometrical theorems about parabolas: 1. It proves that the tangents drawn from the extremities of a focal chord intersect at right angles on the directrix. 2. It proves that the angle of incidence of a line parallel to the axis of a parabola is equal to the angle of reflection. This establishes that a line and its reflection are equally inclined to the normal and tangent of the parabola.

1. Geometrical Theorems about
Parabola

2. Geometrical Theorems about
(1) Focal Chords
Parabola

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. 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. 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. 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. 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. 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. 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. 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. 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. (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 axis
1 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 axis
1 Show tangent at P is y  px  ap 2
2 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 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. (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. 2ap  0  ap  a 
2
d PS 
2 2

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. 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. 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. 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. 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. 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. 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