Transcript of "11X1 T11 08 geometrical theorems (2010)"
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
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
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
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