11 x1 t11 05 tangents & normals i (2012)
TRANSCRIPT
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