11x1 t12 08 geometrical theorems (2011)

29
Geometrical Theorems about Parabola

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Page 1: 11X1 T12 08 geometrical theorems (2011)

Geometrical Theorems about

Parabola

Page 2: 11X1 T12 08 geometrical theorems (2011)

Geometrical Theorems about

Parabola (1) Focal Chords

Page 3: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

(2) Reflection Property

Page 13: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

(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: 11X1 T12 08 geometrical theorems (2011)

22202 aapapdPS

Page 23: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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: 11X1 T12 08 geometrical theorems (2011)

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