11 x1 t11 08 geometrical theorems
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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
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
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
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