11X1 T12 07 chord of contact

Chord of Contact

Chord of Contacty

x

2 4x ay

Chord of Contacty

x

2 4x ay

We know the coordinates of an external point (T)

Chord of Contacty

x

2 4x ay

0 0,T x y

We know the coordinates of an external point (T)

Chord of Contacty

x

2 4x ay

0 0,T x y

We know the coordinates of an external point (T)

From this external point, two tangents can be drawn meeting the parabola at P and Q.

Chord of Contacty

x

2 4x ay

2(2 , )Q aq aq

2(2 , )P ap ap

0 0,T x y

We know the coordinates of an external point (T)

From this external point, two tangents can be drawn meeting the parabola at P and Q.

Chord of Contacty

x

2 4x ay

2(2 , )Q aq aq

2(2 , )P ap ap

0 0,T x y

We know the coordinates of an external point (T)

From this external point, two tangents can be drawn meeting the parabola at P and Q.

Chord of Contacty

x

2 4x ay

2(2 , )Q aq aq

2(2 , )P ap ap

0 0,T x y

We know the coordinates of an external point (T)

From this external point, two tangents can be drawn meeting the parabola at P and Q.The line joining these two points is called the chord of contact.

Chord of Contacty

x

2 4x ay

2(2 , )Q aq aq

2(2 , )P ap ap

0 0,T x y

We know the coordinates of an external point (T)

From this external point, two tangents can be drawn meeting the parabola at P and Q.The line joining these two points is called the chord of contact.

(1) Parametric approach

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

4 0 0But is ,T x y

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

4 0 0But is ,T x y

axqp

qpax

0

0

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

4 0 0But is ,T x y

axqp

qpax

0

0

apqy 0

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

4 0 0But is ,T x y

axqp

qpax

0

0

00 22 is yyaxxPQ

apqy 0

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

4 0 0But is ,T x y

axqp

qpax

0

0

00 22 is yyaxxPQ

Hence the chord of contact is yyaxx 00 2

apqy 0

(1) Parametric approach

1 Show that has equation 2 2PQ p q x y apq

22 2

Show the two tangents have equations0 and 0px y ap qx y aq

3 Show that is the point ,T a p q apq

4 0 0But is ,T x y

axqp

qpax

0

0

00 22 is yyaxxPQ

Hence the chord of contact is yyaxx 00 2

apqy 0

notice similarity to tangent

(2) Cartesian approach

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

yyaxxyxP 0011 2equation with lineon the lies ,

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

2 2 2Show that has equation 2QT xx a y y

yyaxxyxP 0011 2equation with lineon the lies ,

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

2 2 2Show that has equation 2QT xx a y y

T lies on QT 2020 2 yyaxx

yyaxxyxP 0011 2equation with lineon the lies ,

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

2 2 2Show that has equation 2QT xx a y y

T lies on QT 2020 2 yyaxx

yyaxxyxP 0011 2equation with lineon the lies ,

yyaxxyxQ 0022 2equation with lineon thelies ,

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

2 2 2Show that has equation 2QT xx a y y

T lies on QT 2020 2 yyaxx

yyaxxyxP 0011 2equation with lineon the lies ,

yyaxxyxQ 0022 2equation with lineon thelies ,

Hence the chord of contact is yyaxx 00 2

(2) Cartesian approach

1 1 1Show that has equation 2PT xx a y y

T lies on PT 1010 2 yyaxx

2 2 2Show that has equation 2QT xx a y y

T lies on QT 2020 2 yyaxx

yyaxxyxP 0011 2equation with lineon the lies ,

yyaxxyxQ 0022 2equation with lineon thelies ,

Hence the chord of contact is yyaxx 00 2

Exercise 9H; 1c, 2d, 3, 6, 8, 10, 14