Download - 11X1 T12 07 chord of contact (2011)
Chord of Contact
Chord of Contact
y
x
2 4x ay
Chord of Contact
y
x
2 4x ay
We know the coordinates
of an external point (T)
Chord of Contact
y
x
2 4x ay
0 0,T x y
We know the coordinates
of an external point (T)
Chord of Contact
y
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 Contact
y
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 Contact
y
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 Contact
y
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 Contact
y
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
2
2 2
Show the two tangents have equations
0 and 0px y ap qx y aq
(1) Parametric approach
1 Show that has equation 2 2PQ p q x y apq
2
2 2
Show the two tangents have equations
0 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
2
2 2
Show the two tangents have equations
0 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
2
2 2
Show the two tangents have equations
0 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
a
xqp
qpax
0
0
(1) Parametric approach
1 Show that has equation 2 2PQ p q x y apq
2
2 2
Show the two tangents have equations
0 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
a
xqp
qpax
0
0
apqy 0
(1) Parametric approach
1 Show that has equation 2 2PQ p q x y apq
2
2 2
Show the two tangents have equations
0 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
a
xqp
qpax
0
0
00 22 is yya
xxPQ
apqy 0
(1) Parametric approach
1 Show that has equation 2 2PQ p q x y apq
2
2 2
Show the two tangents have equations
0 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
a
xqp
qpax
0
0
00 22 is yya
xxPQ
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
2
2 2
Show the two tangents have equations
0 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
a
xqp
qpax
0
0
00 22 is yya
xxPQ
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 line on 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 line on 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 line on 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 line on the lies ,
yyaxxyxQ 0022 2equation with line on 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 line on the lies ,
yyaxxyxQ 0022 2equation with line on the lies ,
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 line on the lies ,
yyaxxyxQ 0022 2equation with line on the lies ,
Hence the chord of contact is yyaxx 00 2
Exercise 9H; 1c, 2d, 3, 6, 8, 10, 14
