11 x1 t11 08 geometrical theorems (2012)
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Transcript of 11 x1 t11 08 geometrical theorems (2012)
![Page 1: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/1.jpg)
Geometrical Theorems about
Parabola
![Page 2: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/2.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
![Page 3: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/3.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
![Page 4: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/4.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
![Page 5: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/5.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q.
![Page 6: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/6.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q. 1pq
![Page 7: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/7.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q. 1pq
Tangents are perpendicular to each other
![Page 8: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/8.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q. 1pq
Tangents are perpendicular to each other
3
Show that the point of intersection, , of the tangents is
,
T
a p q apq
![Page 9: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/9.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q. 1pq
Tangents are perpendicular to each other
3
Show that the point of intersection, , of the tangents is
,
T
a p q apq apqy
![Page 10: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/10.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q. 1pq
Tangents are perpendicular to each other
3
Show that the point of intersection, , of the tangents is
,
T
a p q apq apqy
1 pqay
![Page 11: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/11.jpg)
Geometrical Theorems about
Parabola (1) Focal Chords
e.g. Prove that the tangents drawn from the extremities of a focal chord
intersect at right angles on the directrix.
1 Prove 1pq
2 Show that the slope of the tangent at P is p, and the slope of the
tangent at Q is q. 1pq
Tangents are perpendicular to each other
3
Show that the point of intersection, , of the tangents is
,
T
a p q apq apqy
1 pqay
Tangents meet on the directrix
![Page 12: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/12.jpg)
(2) Reflection Property
![Page 13: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/13.jpg)
(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.
![Page 14: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/14.jpg)
(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.
![Page 15: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/15.jpg)
(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
![Page 16: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/16.jpg)
(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)
![Page 17: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/17.jpg)
(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: || axisCP y
![Page 18: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/18.jpg)
(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
1 2Show tangent at is P y px ap
(angle of incidence = angle of reflection)
Data: || axisCP y
![Page 19: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/19.jpg)
(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
1 2Show tangent at is P y px ap
2
(angle of incidence = angle of reflection)
tangent meets y axis when x = 0
Data: || axisCP y
![Page 20: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/20.jpg)
(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
1 2Show tangent at is P y px ap
2
(angle of incidence = angle of reflection)
tangent meets y axis when x = 0
2,0 is apK
Data: || axisCP y
![Page 21: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/21.jpg)
(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
1 2Show tangent at is P y px ap
2
(angle of incidence = angle of reflection)
tangent meets y axis when x = 0
2,0 is apK
2
SKd a ap
Data: || axisCP y
![Page 22: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/22.jpg)
22202 aapapdPS
![Page 23: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/23.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
![Page 24: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/24.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
SKd
![Page 25: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/25.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
SKd
is isosceles two = sidesSPK
![Page 26: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/26.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
SKd
is isosceles two = sidesSPK
(base 's isosceles )SPK SKP
![Page 27: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/27.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
SKd
is isosceles two = sidesSPK
(base 's isosceles )SPK SKP
(corresponding 's , )SKP CPB SK ||CP
![Page 28: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/28.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
SKd
is isosceles two = sidesSPK
(base 's isosceles )SPK SKP
(corresponding 's , )SKP CPB SK ||CP
CPBSPK
![Page 29: 11 x1 t11 08 geometrical theorems (2012)](https://reader034.fdocuments.net/reader034/viewer/2022042715/5597df2f1a28ab69388b4744/html5/thumbnails/29.jpg)
2 2 2 4 2 2 2
4 2
22
2
4 2
2 1
1
1
a p a p a p a
a p p
a p
a p
22202 aapapdPS
SKd
is isosceles two = sidesSPK
(base 's isosceles )SPK SKP
(corresponding 's , )SKP CPB SK ||CP
CPBSPK
Exercise 9I; 1, 2, 4, 7, 11, 12, 17, 18, 21