11x1 t12 08 geometrical theorems (2011)

Geometrical Theorems about

Parabola

Geometrical Theorems about

Parabola (1) Focal Chords

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.

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

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.

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

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

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

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

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

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

(2) Reflection Property

(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.

(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.

(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

(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)

(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

(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

(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

(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

(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

22202 aapapdPS

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

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

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

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

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

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

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