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