11x1 t12 07 chord of contact (2011)
TRANSCRIPT
![Page 1: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/1.jpg)
Chord of Contact
![Page 2: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/2.jpg)
Chord of Contact
y
x
2 4x ay
![Page 3: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/3.jpg)
Chord of Contact
y
x
2 4x ay
We know the coordinates
of an external point (T)
![Page 4: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/4.jpg)
Chord of Contact
y
x
2 4x ay
0 0,T x y
We know the coordinates
of an external point (T)
![Page 5: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/5.jpg)
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.
![Page 6: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/6.jpg)
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.
![Page 7: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/7.jpg)
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.
![Page 8: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/8.jpg)
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.
![Page 9: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/9.jpg)
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.
![Page 10: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/10.jpg)
(1) Parametric approach
![Page 11: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/11.jpg)
(1) Parametric approach
1 Show that has equation 2 2PQ p q x y apq
![Page 12: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/12.jpg)
(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
![Page 13: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/13.jpg)
(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
![Page 14: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/14.jpg)
(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
![Page 15: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/15.jpg)
(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
![Page 16: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/16.jpg)
(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
![Page 17: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/17.jpg)
(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
![Page 18: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/18.jpg)
(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
![Page 19: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/19.jpg)
(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
![Page 20: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/20.jpg)
(2) Cartesian approach
![Page 21: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/21.jpg)
(2) Cartesian approach
1 1 1Show that has equation 2PT xx a y y
![Page 22: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/22.jpg)
(2) Cartesian approach
1 1 1Show that has equation 2PT xx a y y
T lies on PT 1010 2 yyaxx
![Page 23: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/23.jpg)
(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 ,
![Page 24: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/24.jpg)
(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 ,
![Page 25: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/25.jpg)
(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 ,
![Page 26: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/26.jpg)
(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 ,
![Page 27: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/27.jpg)
(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
![Page 28: 11X1 T12 07 chord of contact (2011)](https://reader031.vdocuments.us/reader031/viewer/2022030401/58ad9ec21a28abde5e8b58a7/html5/thumbnails/28.jpg)
(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