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

axy4

2

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

2 4OR x ay

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

2 4OR x ay

2 4 dyx adx

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

2dx adt

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

2dx adt

2y at

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

2dx adt

2dy atdt

2y at

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

122

dy atdx a

t

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

122

dy atdx a

t

when , dyt p pdx

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

slope of tangent is p

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

122

dy atdx a

t

when , dyt p pdx

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

slope of tangent is p

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

122

dy atdx a

t

when , dyt p pdx

apxpapy 22

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

slope of tangent is p

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

122

dy atdx a

t

when , dyt p pdx

apxpapy 22

22 2appxapy

Tangents & Normals(i) Using Parametrics

y

x

2 4x ay

2(2 , )P ap ap

(1) Tangent

axy4

2

ax

dxdy

2

pdxdyapx ,2when

2 4OR x ay

2 4 dyx adx

ax

dxdy

2

slope of tangent is p

2

ORx at

2dx adt

2dy atdt

2y at

dy dy dtdx dt dx

122

dy atdx a

t

when , dyt p pdx

apxpapy 22

22 2appxapy

2appxy

(2) Tangents from an external point

(2) Tangents from an external point2 4x y

y

x

(2) Tangents from an external point2 4x y

y

x

(3,2)

(2) Tangents from an external point2 4x y

y

x

(3,2)

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx p pdx

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx p pdx

2 2y p p x p

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx p pdx

2 2y p p x p 2 22y p px p

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx p pdx

2 2y p p x p 2 22y p px p

2y px p

(2) Tangents from an external point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx 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 point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx 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 point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx 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 point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx 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 point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx 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 point2 4x y

2(2 , )P p p

y

x

(3,2)

2(2 , )Q q q

2

4xy

2dy xdx

when 2 , dyx 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 tangentsy

x

2 4x ay

(3) Intersection of tangentsy

x

2 4x ay

2(2 , )P ap ap

(3) Intersection of tangentsy

x

2 4x ay

2(2 , )P ap ap

(3) Intersection of tangentsy

x

2 4x ay

2(2 , )P ap ap

1 2Show tangent at is P y px ap

(3) Intersection of tangentsy

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 tangentsy

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 tangentsy

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 tangentsy

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 tangentsy

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 tangentsy

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

aqyqxapypx

(3) Intersection of tangentsy

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

aqyqxapypx

22 qpaxqp

(3) Intersection of tangentsy

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

aqyqxapypx

22 qpaxqp

qpaxqpqpaxqp

(3) Intersection of tangentsy

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

aqyqxapypx

22 qpaxqp

qpaxqpqpaxqp

2apqpapy

(3) Intersection of tangentsy

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

aqyqxapypx

22 qpaxqp

qpaxqpqpaxqp

2apqpapy

apqapapqapy

22

(3) Intersection of tangentsy

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

aqyqxapypx

22 qpaxqp

qpaxqpqpaxqp

2apqpapy

apqapapqapy

22

apqqpaT ,

(4) Normal

(4) Normaly

x

2 4x ay

(4) Normaly

x

2 4x ay

2(2 , )P ap ap

(4) Normaly

x

2 4x ay

2(2 , )P ap ap

(4) Normaly

x

2 4x ay

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

(4) Normaly

x

2 4x ay

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

2 1slope of normal is p

(4) Normaly

x

2 4x ay

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

2 1slope of normal is p

apxp

apy 212

(4) Normaly

x

2 4x ay

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

2 1slope of normal is p

apxp

apy 212

apxappy 23

(4) Normaly

x

2 4x ay

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

2 1slope 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 aqN

(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 simultaneouslyN

(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

aqaqqyxapappyx

22

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

aqaqqyxapappyx

22

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

aqaqqyxapappyx

22

3

3

qpaqpayqp 233

2

222

22

qpqpayqpaqpqpqpayqp

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

aqaqqyxapappyx

22

3

3

qpaqpayqp 233

2

222

22

qpqpayqpaqpqpqpayqp

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

aqaqqyxapappyx

22

3

3

qpaqpayqp 233

2

222

22

qpqpayqpaqpqpqpayqp

apapqpqpapx 22 322

qpapqapapqqapapapapx

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

aqaqqyxapappyx

22

3

3

qpaqpayqp 233

2

222

22

qpqpayqpaqpqpqpayqp

apapqpqpapx 22 322

qpapqapapqqapapapapx

22 2233

2, 22 qpqpaqpapqN

N

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