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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
