3. Chord of Contact
y x 2 4ay
We know the coordinates
of an external point (T)
x
4. Chord of Contact
y x 2 4ay
We know the coordinates
of an external point (T)
x
T x0 , y0
5. Chord of Contact
y x 2 4ay
We know the coordinates
of an external point (T)
From this external point,
x two tangents can be drawn
meeting the parabola at P
T x0 , y0 and Q.
6. Chord of Contact
y x 2 4ay
P(2ap, ap 2 )
We know the coordinates
of an external point (T)
Q(2aq, aq 2 ) From this external point,
x two tangents can be drawn
meeting the parabola at P
T x0 , y0 and Q.
7. Chord of Contact
y x 2 4ay
P(2ap, ap 2 )
We know the coordinates
of an external point (T)
Q(2aq, aq 2 ) From this external point,
x two tangents can be drawn
meeting the parabola at P
T x0 , y0 and Q.
8. Chord of Contact
y x 2 4ay
P(2ap, ap 2 )
We know the coordinates
of an external point (T)
Q(2aq, aq 2 ) From this external point,
x two tangents can be drawn
meeting the parabola at P
T x0 , y0 and Q.
The line joining these two
points is called the chord
of contact.
9. Chord of Contact
y x 2 4ay
P(2ap, ap 2 )
We know the coordinates
of an external point (T)
Q(2aq, aq 2 ) From this external point,
x two tangents can be drawn
meeting the parabola at P
T x0 , y0 and Q.
The line joining these two
points is called the chord
of contact.
12. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
13. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
14. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
4 But T is x0 , y0
15. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
4 But T is x0 , y0
x0 a p q
x0
pq
a
16. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
4 But T is x0 , y0
x0 a p q y0 apq
x0
pq
a
17. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
4 But T is x0 , y0
x0 a p q y0 apq
x0
pq
a
x0 x
PQ is 2 y 2 y0
a
18. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
4 But T is x0 , y0
x0 a p q y0 apq
x0
pq
a
x0 x
PQ is 2 y 2 y0
a
Hence the chord of contact is x0 x 2a y0 y
19. (1) Parametric approach
1 Show that PQ has equation p q x 2 y 2apq
2 Show the two tangents have equations
px y ap 2 0 and qx y aq 2 0
3 Show that T is the point a p q , apq
4 But T is x0 , y0
x0 a p q y0 apq
x0
pq
a
x0 x notice
PQ is 2 y 2 y0
a similarity
to tangent
Hence the chord of contact is x0 x 2a y0 y
22. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
23. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
P x1 , y1 lies on the line with equation x0 x 2a y0 y
24. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
P x1 , y1 lies on the line with equation x0 x 2a y0 y
2 Show that QT has equation xx2 2a y y2
25. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
P x1 , y1 lies on the line with equation x0 x 2a y0 y
2 Show that QT has equation xx2 2a y y2
T lies on QT x0 x2 2a y0 y2
26. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
P x1 , y1 lies on the line with equation x0 x 2a y0 y
2 Show that QT has equation xx2 2a y y2
T lies on QT x0 x2 2a y0 y2
Q x2 , y2 lies on the line with equation x0 x 2a y0 y
27. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
P x1 , y1 lies on the line with equation x0 x 2a y0 y
2 Show that QT has equation xx2 2a y y2
T lies on QT x0 x2 2a y0 y2
Q x2 , y2 lies on the line with equation x0 x 2a y0 y
Hence the chord of contact is x0 x 2a y0 y
28. (2) Cartesian approach
1 Show that PT has equation xx1 2a y y1
T lies on PT x0 x1 2a y0 y1
P x1 , y1 lies on the line with equation x0 x 2a y0 y
2 Show that QT has equation xx2 2a y y2
T lies on QT x0 x2 2a y0 y2
Q x2 , y2 lies on the line with equation x0 x 2a y0 y
Hence the chord of contact is x0 x 2a y0 y
Exercise 9H; 1c, 2d, 3, 6, 8, 10, 14