11 x1 t11 06 tangents & normals ii (2012)
TRANSCRIPT
Tangents & Normals (ii) Using Cartesian
Tangents & Normals (ii) Using Cartesian
(1) Tangent
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
(1) Tangent
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
Tangents & Normals (ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
a
xy
4
2
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
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
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
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
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
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
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
(2) Normal
(2) Normal
y
x
2 4x ay
(2) Normal
y
x
2 4x ay
1 1( , )P x y
(2) Normal
y
x
2 4x ay
1 1( , )P x y
(2) Normal
y
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2
xP
a
(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
(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
(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
(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
(3) Line cutting/touching/missing parabola
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b
parabola and tangent meet when;
(3) Line cutting/touching/missing parabola
y
x
2 4x ay
y mx b 2 4x a mx b
parabola and tangent meet when;
(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
(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
(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
(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
(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
(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
(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
(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
(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
(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)
(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)
e.g. Find the equation of the two tangents to the parabola
passing through the point (3,2).
2 4x y
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
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
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
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
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
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
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
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
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
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
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
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
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
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