11 x1 t11 05 tangents & normals i (2012)

Tangents & Normals (i) Using Parametrics

Tangents & Normals (i) Using Parametrics

(1) Tangent

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

(1) Tangent

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

2 4OR x ay

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

2 4OR x ay

2 4dy

x adx

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

2dx

adt

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

2dx

adt

2y at

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

2dx

adt

2dy

atdt

2y at

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

12

2

dyat

dx a

t

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

12

2

dyat

dx a

t

when ,dy

t p pdx

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

slope of tangent is p

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

12

2

dyat

dx a

t

when ,dy

t p pdx

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

slope of tangent is p

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

12

2

dyat

dx a

t

when ,dy

t p pdx

apxpapy 22

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

slope of tangent is p

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

12

2

dyat

dx a

t

when ,dy

t p pdx

apxpapy 22

22 2appxapy

Tangents & Normals (i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

a

xy

4

2

a

x

dx

dy

2

pdx

dyapx ,2when

2 4OR x ay

2 4dy

x adx

a

x

dx

dy

2

slope of tangent is p

2

OR

x at

2dx

adt

2dy

atdt

2y at

dy dy dt

dx dt dx

12

2

dyat

dx a

t

when ,dy

t p pdx

apxpapy 22

22 2appxapy

2appxy

(2) Tangents from an external point

(2) Tangents from an external point 2 4x y

y

x

(2) Tangents from an external point 2 4x y

y

x

(3,2)

(2) Tangents from an external point 2 4x y

y

x

(3,2)

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

however tangent passes through (3,2)

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

however tangent passes through (3,2) 22 3p p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

however tangent passes through (3,2) 22 3p p

2 3 2 0p p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

however tangent passes through (3,2) 22 3p p

2 3 2 0p p

2 1 0p p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

however tangent passes through (3,2) 22 3p p

2 3 2 0p p

2 1 0p p

1 or 2p p

(2) Tangents from an external point 2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4

xy

2

dy x

dx

when 2 ,dy

x p pdx

2 2y p p x p

2 22y p px p

2y px p

however tangent passes through (3,2) 22 3p p

2 3 2 0p p

2 1 0p p

1 or 2p p

tangents are 1 and 2 4y x y x

(3) Intersection of tangents

(3) Intersection of tangents

y

x

2 4x ay

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap2(2 , )Q aq aq

1 2Show tangent at is P y px ap

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

2

2

aqyqx

apypx

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

2

2

aqyqx

apypx

22 qpaxqp

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

2

2

aqyqx

apypx

22 qpaxqp

qpax

qpqpaxqp

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

2

2

aqyqx

apypx

22 qpaxqp

qpax

qpqpaxqp

2apqpapy

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

2

2

aqyqx

apypx

22 qpaxqp

qpax

qpqpaxqp

2apqpapy

apq

apapqapy

22

(3) Intersection of tangents

y

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

2(2 , )Q aq aq

2 2tangent at is Q y qx aq

T

3 Solve simultaneously

2

2

aqyqx

apypx

22 qpaxqp

qpax

qpqpaxqp

2apqpapy

apq

apapqapy

22

apqqpaT ,

(4) Normal

(4) Normal

y

x

2 4x ay

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap 1 Show the slope of tangent at is P p

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap 1 Show the slope of tangent at is P p

2 1

slope of normal is p

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap 1 Show the slope of tangent at is P p

2 1

slope of normal is p

apxp

apy 212

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap 1 Show the slope of tangent at is P p

2 1

slope of normal is p

apxp

apy 212

apxappy 23

(4) Normal

y

x

2 4x ay

2(2 , )P ap ap 1 Show the slope of tangent at is P p

2 1

slope of normal is p

apxp

apy 212

apxappy 23

apappyx 23

(5) Intersection of normals

(5) Intersection of normals

y

x

2 4x ay

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2(2 , )Q aq aq

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2(2 , )Q aq aq

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

aqaqqyx

apappyx

2

2

3

3

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

aqaqqyx

apappyx

2

2

3

3

qpaqpayqp 233

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

aqaqqyx

apappyx

2

2

3

3

qpaqpayqp 233

2

2

22

22

qpqpay

qpaqpqpqpayqp

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

aqaqqyx

apappyx

2

2

3

3

qpaqpayqp 233

2

2

22

22

qpqpay

qpaqpqpqpayqp

apapqpqpapx 22 322

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

aqaqqyx

apappyx

2

2

3

3

qpaqpayqp 233

2

2

22

22

qpqpay

qpaqpqpqpayqp

apapqpqpapx 22 322

qpapq

apapqqapapapapx

22 2233

N

(5) Intersection of normals

y

x

2 4x ay

2(2 , )P ap ap

1 3Show normal at is p 2P x y ap ap

2 3normal at is 2Q x qy aq aq

2(2 , )Q aq aq 3 Solve simultaneously

aqaqqyx

apappyx

2

2

3

3

qpaqpayqp 233

2

2

22

22

qpqpay

qpaqpqpqpayqp

apapqpqpapx 22 322

qpapq

apapqqapapapapx

22 2233

2, 22 qpqpaqpapqN

N

Exercise 9F; 1ac, 2ac, 3, 6, 8, 9ac, 11, 12, 13