11X1 T12 06 tangents & normals II
-
Upload
nigel-simmons -
Category
Education
-
view
417 -
download
3
Transcript of 11X1 T12 06 tangents & normals II
![Page 1: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/1.jpg)
Tangents & Normals(ii) Using Cartesian
![Page 2: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/2.jpg)
Tangents & Normals(ii) Using Cartesian
(1) Tangent
![Page 3: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/3.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay(1) Tangent
![Page 4: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/4.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
![Page 5: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/5.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
![Page 6: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/6.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
![Page 7: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/7.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
![Page 8: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/8.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
11when ,
2xdyx x
dx a
![Page 9: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/9.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
11when ,
2xdyx x
dx a
1slope of tangent is 2xa
![Page 10: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/10.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
11when ,
2xdyx x
dx a
1slope of tangent is 2xa
11
1 2xx
axyy
![Page 11: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/11.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
11when ,
2xdyx x
dx a
1slope of tangent is 2xa
11
1 2xx
axyy
211122 xxxayay
![Page 12: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/12.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
11when ,
2xdyx x
dx a
1slope of tangent is 2xa
11
1 2xx
axyy
211122 xxxayay
111 422 ayxxayay
![Page 13: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/13.jpg)
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
11when ,
2xdyx x
dx a
1slope of tangent is 2xa
11
1 2xx
axyy
211122 xxxayay
111 422 ayxxayay
11 2 yyaxx
![Page 14: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/14.jpg)
(2) Normal
![Page 15: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/15.jpg)
(2) Normaly
x
2 4x ay
![Page 16: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/16.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y
![Page 17: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/17.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y
![Page 18: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/18.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2xPa
![Page 19: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/19.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2xPa
21
2slope of normal is ax
![Page 20: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/20.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2xPa
21
2slope of normal is ax
11
12 xxx
ayy
![Page 21: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/21.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2xPa
21
2slope of normal is ax
11
12 xxx
ayy
1111 22 axaxyxyx
![Page 22: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/22.jpg)
(2) Normaly
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2xPa
21
2slope of normal is ax
11
12 xxx
ayy
1111 22 axaxyxyx
1111 22 yxaxyxax
![Page 23: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/23.jpg)
(3) Line cutting/touching/missing parabola
![Page 24: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/24.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
![Page 25: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/25.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
![Page 26: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/26.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
![Page 27: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/27.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b
![Page 28: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/28.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b
parabola and tangent meet when;
![Page 29: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/29.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b parabola and tangent meet when;
![Page 30: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/30.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b parabola and tangent meet when;
2 4 4 0x amx ab
![Page 31: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/31.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
![Page 32: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/32.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0
![Page 33: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/33.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
![Page 34: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/34.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
![Page 35: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/35.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac 24 4 1 4am ab
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
![Page 36: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/36.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac 24 4 1 4am ab
2 2
2
16 16
16
a m ab
a am b
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
![Page 37: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/37.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac 24 4 1 4am ab
2 2
2
16 16
16
a m ab
a am b
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
2two solutions (cuts) when 0am b
![Page 38: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/38.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac 24 4 1 4am ab
2 2
2
16 16
16
a m ab
a am b
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
2two solutions (cuts) when 0am b 2one solution (touches) when 0am b
![Page 39: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/39.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac 24 4 1 4am ab
2 2
2
16 16
16
a m ab
a am b
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
2two solutions (cuts) when 0am b 2one solution (touches) when 0am b (common idea)
![Page 40: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/40.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
2 4b ac 24 4 1 4am ab
2 2
2
16 16
16
a m ab
a am b
parabola and tangent meet when;
2 4 4 0x amx ab two solutions (cuts) when 0
one solution (touches) when 0 no solutions (misses) when 0
2two solutions (cuts) when 0am b 2one solution (touches) when 0am b
2no solutions (misses) when 0am b (common idea)
![Page 41: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/41.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
![Page 42: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/42.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b
![Page 43: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/43.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
![Page 44: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/44.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m
![Page 45: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/45.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
![Page 46: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/46.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y
![Page 47: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/47.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
![Page 48: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/48.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m
![Page 49: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/49.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
![Page 50: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/50.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
24 4 1 12 8 0m m
![Page 51: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/51.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
24 4 1 12 8 0m m 216 48 32 0m m
![Page 52: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/52.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
24 4 1 12 8 0m m 216 48 32 0m m
2 3 2 01 2 0
m mm m
![Page 53: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/53.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
24 4 1 12 8 0m m 216 48 32 0m m
2 3 2 01 2 0
m mm m
1 or 2m m
![Page 54: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/54.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
24 4 1 12 8 0m m 216 48 32 0m m
2 3 2 01 2 0
m mm m
1 or 2m m
tangents are 1 and 2 4y x y x
![Page 55: 11X1 T12 06 tangents & normals II](https://reader034.fdocuments.net/reader034/viewer/2022052215/55622c09d8b42ac6588b5501/html5/thumbnails/55.jpg)
e.g. Find the equation of the two tangents to the parabola passing through the point (3,2).
2 4x y
tangent will be of the form y = mx + b2 3m b
2 3b m tangents are 2 3y mx m
2 4x y 2 4 2 3x mx m
2 4 12 8 0x mx m line is a tangent if 0
24 4 1 12 8 0m m 216 48 32 0m m
2 3 2 01 2 0
m mm m
1 or 2m m
tangents are 1 and 2 4y x y x
Exercise 9G; 1ac, 2ac, 3a, 4, 7, 9, 11, 12,
13, 15, 17, 18