11X1 T12 07 chord of contact

1. Chord of Contact

2. Chord of Contact
y
x
2
4
x
a
y


3. Chord of Contact
y
x
2
4
x
a
y

We know the coordinates
of an external point ( T)

4. Chord of Contact
y
x
2
4
x
a
y


00
, Tx
y
We know the coordinates
of an external point ( T)

5. Chord of Contact
y
x
2
4
x
a
y


00
, Tx
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.

6. Chord of Contact
y
x
2
4
x
a
y

2
(2 , ) Qaqaq
2
(2 , )
P
ap ap

00
, Tx
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.

7. Chord of Contact
y
x
2
4
x
a
y

2
(2 , ) Qaqaq
2
(2 , )
P
ap ap

00
, Tx
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.

8. Chord of Contact
y
x
2
4
x
a
y

2
(2 , ) Qaqaq
2
(2 , )
P
ap ap

00
, Tx
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.

9. Chord of Contact
y
x
2
4
x
a
y

2
(2 , ) Qaqaq
2
(2 , )
P
ap ap

00
, Tx
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.

10. (1) Parametric approach

11. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq



12. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 

13. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point , T a p q apq


14. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point , T a p q apq

4

00
But is , Tx
y

15. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point , T a p q apq

4

00
But is , Tx
y

a
x
qp
q
p
a
x
0
0





16. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point ,
Ta p q apq

4

00
But is , Tx
y

a
x
qp
q
p
a
x
0
0




apqy


0

17. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point ,
Ta p q apq

4

00
But is , Tx
y

a
x
qp
q
p
a
x
0
0




0
0
22 is
yy
a
x
x
PQ
apqy


0

18. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point ,
Ta p q apq

4

00
But is , Tx
y

a
x
qp
q
p
a
x
0
0




0
0
22 is
yy
a
x
x
PQ
Hence the chord of contact is


yya
x
x


0 0
2
apqy


0

19. (1) Parametric approach
1


Show that has equation 2 2
P
Qpqx
y
apq


2
22
Show the two tangents have equations
0 and 0 px y ap qx y aq

 
3




Show that is the point ,
Ta p q apq

4

00
But is , Tx
y

a
x
qp
q
p
a
x
0
0




0
0
22 is
yy
a
x
x
PQ
Hence the chord of contact is


yya
x
x


0 0
2
apqy


0
notice
similarity
to tangent

20. (2) Cartesian approach

21. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy



22. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x




23. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x







yya
x
x
y
x
P



0 0 11
2 equation with lineon the lies ,

24. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x



2


22
Show that has equation 2 QT xx a
yy






yya
x
x
y
x
P



0 0 11
2 equation with lineon the lies ,

25. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x



2


22
Show that has equation 2 QT xx a
yy


T
lies on
QT


20 20
2
yya
x
x







yya
x
x
y
x
P



0 0 11
2 equation with lineon the lies ,

26. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x



2


22
Show that has equation 2 QT xx a
yy


T
lies on
QT


20 20
2
yya
x
x







yya
x
x
y
x
P



0 0 11
2 equation with lineon the lies ,




yya
x
x
y
x
Q



0 0 22
2 equation with lineon thelies ,

27. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x



2


22
Show that has equation 2 QT xx a
yy


T
lies on
QT


20 20
2
yya
x
x







yya
x
x
y
x
P



0 0 11
2 equation with lineon the lies ,




yya
x
x
y
x
Q



0 0 22
2 equation with lineon thelies ,
Hence the chord of contact is


yya
x
x


0 0
2

28. (2) Cartesian approach
1

11
Show that has equation 2
P
Txxa
yy


T
lies on
PT


10 10
2
yya
x
x



2


22
Show that has equation 2 QT xx a
yy


T
lies on
QT


20 20
2
yya
x
x







yya
x
x
y
x
P



0 0 11
2 equation with lineon the lies ,




yya
x
x
y
x
Q



0 0 22
2 equation with lineon thelies ,
Hence the chord of contact is


yya
x
x


0 0
2
Exercise 9H; 1c, 2d, 3, 6, 8, 10, 14