11 x1 t11 06 tangents & normals ii (2012)
TRANSCRIPT
![Page 1: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/1.jpg)
Tangents & Normals (ii) Using Cartesian
![Page 2: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/2.jpg)
Tangents & Normals (ii) Using Cartesian
(1) Tangent
![Page 3: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/3.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
(1) Tangent
![Page 4: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/4.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
![Page 5: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/5.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
![Page 6: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/6.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
![Page 7: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/7.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
![Page 8: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/8.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
11when ,
2
xdyx x
dx a
![Page 9: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/9.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
11when ,
2
xdyx x
dx a
1 slope of tangent is 2
x
a
![Page 10: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/10.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
11when ,
2
xdyx x
dx a
1 slope of tangent is 2
x
a
11
12
xxa
xyy
![Page 11: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/11.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
11when ,
2
xdyx x
dx a
1 slope of tangent is 2
x
a
11
12
xxa
xyy
2
11122 xxxayay
![Page 12: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/12.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
11when ,
2
xdyx x
dx a
1 slope of tangent is 2
x
a
11
12
xxa
xyy
2
11122 xxxayay
111 422 ayxxayay
![Page 13: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/13.jpg)
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
a
x
dx
dy
2
11when ,
2
xdyx x
dx a
1 slope of tangent is 2
x
a
11
12
xxa
xyy
2
11122 xxxayay
111 422 ayxxayay
11 2 yyaxx
![Page 14: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/14.jpg)
(2) Normal
![Page 15: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/15.jpg)
(2) Normal
y
x
2 4x ay
![Page 16: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/16.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y
![Page 17: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/17.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y
![Page 18: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/18.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2
xP
a
![Page 19: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/19.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2
xP
a
2 1
2slope of normal is
a
x
![Page 20: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/20.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2
xP
a
2 1
2slope of normal is
a
x
1
1
1
2xx
x
ayy
![Page 21: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/21.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2
xP
a
2 1
2slope of normal is
a
x
1
1
1
2xx
x
ayy
1111 22 axaxyxyx
![Page 22: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/22.jpg)
(2) Normal
y
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2
xP
a
2 1
2slope of normal is
a
x
1
1
1
2xx
x
ayy
1111 22 axaxyxyx
1111 22 yxaxyxax
![Page 23: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/23.jpg)
(3) Line cutting/touching/missing parabola
![Page 24: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/24.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
![Page 25: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/25.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
![Page 26: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/26.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
![Page 27: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/html5/thumbnails/27.jpg)
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b
![Page 28: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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
2
4 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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
2
4 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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
2
4 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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
2
4 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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
2
4 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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
2
4 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 3m b
![Page 44: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 3m b
2 3b m
![Page 45: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 3m b
2 3b m
tangents are 2 3y mx m
![Page 46: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 3m b
2 3b m
tangents are 2 3y mx m 2 4x y
![Page 47: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 3m b
2 3b m
tangents are 2 3y mx m 2 4x y
2 4 2 3x mx m
![Page 48: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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
2
4 4 1 12 8 0m m
![Page 51: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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
2
4 4 1 12 8 0m m 216 48 32 0m m
![Page 52: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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
2
4 4 1 12 8 0m m 216 48 32 0m m
2 3 2 0
1 2 0
m m
m m
![Page 53: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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
2
4 4 1 12 8 0m m 216 48 32 0m m
2 3 2 0
1 2 0
m m
m m
1 or 2m m
![Page 54: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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
2
4 4 1 12 8 0m m 216 48 32 0m m
2 3 2 0
1 2 0
m m
m m
1 or 2m m
tangents are 1 and 2 4y x y x
![Page 55: 11 x1 t11 06 tangents & normals ii (2012)](https://reader033.vdocument.in/reader033/viewer/2022042607/55622c0cd8b42ac6588b5503/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 + b
2 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
2
4 4 1 12 8 0m m 216 48 32 0m m
2 3 2 0
1 2 0
m m
m 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