11x1 t12 06 tangents & normals ii
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
axy4
2
Tangents & Normals(ii) Using Cartesian
y
x
2 4x ay
1 1( , )P x y
(1) Tangent
axy4
2
ax
dxdy
2
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
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
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
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
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
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
(2) Normal
(2) Normaly
x
2 4x ay
(2) Normaly
x
2 4x ay
1 1( , )P x y
(2) Normaly
x
2 4x ay
1 1( , )P x y
(2) Normaly
x
2 4x ay
1 1( , )P x y 1 1Show the slope of tangent at is 2xPa
(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
(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
(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
(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
(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 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
(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
(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
(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
(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)
(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)
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 + b2 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 + b2 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 + b2 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 + b2 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 + b2 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 + 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
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
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
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
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
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
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
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