11 x1 t11 05 tangents & normals i (2013)
TRANSCRIPT
![Page 1: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/1.jpg)
Tangents & Normals (i) Using Parametrics
![Page 2: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/2.jpg)
Tangents & Normals (i) Using Parametrics
(1) Tangent
![Page 3: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/3.jpg)
Tangents & Normals (i) Using Parametrics
y
x
2 4x ay
(1) Tangent
![Page 4: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/4.jpg)
Tangents & Normals (i) Using Parametrics
y
x
2 4x ay
2(2 , )P ap ap
(1) Tangent
![Page 5: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/5.jpg)
Tangents & Normals (i) Using Parametrics
y
x
2 4x ay
2(2 , )P ap ap
(1) Tangent
![Page 6: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/6.jpg)
Tangents & Normals (i) Using Parametrics
y
x
2 4x ay
2(2 , )P ap ap
(1) Tangent
a
xy
4
2
![Page 7: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/7.jpg)
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
![Page 8: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/8.jpg)
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
![Page 9: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/9.jpg)
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
![Page 10: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/10.jpg)
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
![Page 11: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/11.jpg)
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
![Page 12: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/12.jpg)
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
![Page 13: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/13.jpg)
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
![Page 14: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/14.jpg)
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
![Page 15: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/15.jpg)
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
![Page 16: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/16.jpg)
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
![Page 17: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/17.jpg)
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
![Page 18: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/18.jpg)
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
![Page 19: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/19.jpg)
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
![Page 20: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/20.jpg)
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
![Page 21: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/21.jpg)
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
![Page 22: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/22.jpg)
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
![Page 23: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/23.jpg)
(2) Tangents from an external point
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(2) Tangents from an external point 2 4x y
y
x
![Page 25: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/25.jpg)
(2) Tangents from an external point 2 4x y
y
x
(3,2)
![Page 26: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/26.jpg)
(2) Tangents from an external point 2 4x y
y
x
(3,2)
![Page 27: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/27.jpg)
(2) Tangents from an external point 2 4x y
2(2 , )P p p
y
x
(3,2)
![Page 28: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/28.jpg)
(2) Tangents from an external point 2 4x y
2(2 , )P p p
y
x
(3,2)
![Page 29: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/29.jpg)
(2) Tangents from an external point 2 4x y
2(2 , )P p p
y
x
(3,2)
2(2 , )Q q q
![Page 30: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/30.jpg)
(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
![Page 31: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/31.jpg)
(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
![Page 32: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/32.jpg)
(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
![Page 33: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/33.jpg)
(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
![Page 34: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/34.jpg)
(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
![Page 35: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/35.jpg)
(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
![Page 36: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/36.jpg)
(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)
![Page 37: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/37.jpg)
(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
![Page 38: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/38.jpg)
(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
![Page 39: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/39.jpg)
(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
![Page 40: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/40.jpg)
(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
![Page 41: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/41.jpg)
(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
![Page 42: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/42.jpg)
(3) Intersection of tangents
![Page 43: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/43.jpg)
(3) Intersection of tangents
y
x
2 4x ay
![Page 44: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/44.jpg)
(3) Intersection of tangents
y
x
2 4x ay
2(2 , )P ap ap
![Page 45: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/45.jpg)
(3) Intersection of tangents
y
x
2 4x ay
2(2 , )P ap ap
![Page 46: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/46.jpg)
(3) Intersection of tangents
y
x
2 4x ay
2(2 , )P ap ap
1 2Show tangent at is P y px ap
![Page 47: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/47.jpg)
(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
![Page 48: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/48.jpg)
(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
![Page 49: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/49.jpg)
(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
![Page 50: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/50.jpg)
(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
![Page 51: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/51.jpg)
(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
![Page 52: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/52.jpg)
(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
![Page 53: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/53.jpg)
(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
![Page 54: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/54.jpg)
(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
![Page 55: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/55.jpg)
(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
![Page 56: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/56.jpg)
(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
![Page 57: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/57.jpg)
(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 ,
![Page 58: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/58.jpg)
(4) Normal
![Page 59: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/59.jpg)
(4) Normal
y
x
2 4x ay
![Page 60: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/60.jpg)
(4) Normal
y
x
2 4x ay
2(2 , )P ap ap
![Page 61: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/61.jpg)
(4) Normal
y
x
2 4x ay
2(2 , )P ap ap
![Page 62: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/62.jpg)
(4) Normal
y
x
2 4x ay
2(2 , )P ap ap 1 Show the slope of tangent at is P p
![Page 63: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/63.jpg)
(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
![Page 64: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/64.jpg)
(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
![Page 65: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/65.jpg)
(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
![Page 66: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/66.jpg)
(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
![Page 67: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/67.jpg)
(5) Intersection of normals
![Page 68: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/68.jpg)
(5) Intersection of normals
y
x
2 4x ay
![Page 69: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/69.jpg)
(5) Intersection of normals
y
x
2 4x ay
2(2 , )P ap ap
![Page 70: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/70.jpg)
(5) Intersection of normals
y
x
2 4x ay
2(2 , )P ap ap
![Page 71: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/71.jpg)
(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
![Page 72: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/72.jpg)
(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
![Page 73: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/73.jpg)
(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
![Page 74: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/74.jpg)
(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
![Page 75: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/75.jpg)
(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
![Page 76: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/76.jpg)
(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
![Page 77: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/77.jpg)
(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
![Page 78: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/78.jpg)
(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
![Page 79: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/79.jpg)
(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
![Page 80: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/80.jpg)
(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
![Page 81: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/81.jpg)
(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
![Page 82: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/82.jpg)
(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
![Page 83: 11 x1 t11 05 tangents & normals i (2013)](https://reader034.vdocuments.net/reader034/viewer/2022051315/55a9294f1a28abae768b47d2/html5/thumbnails/83.jpg)
Exercise 9F; 1ac, 2ac, 3, 6, 8, 9ac, 11, 12, 13