X2 t03 03 parameters, ellipse (2012)

Conics & Parametersy

xa-a

a

-a

1) Circle

222 ayx

Conics & Parametersy

xa-a

a

-a

1) Circle

222 ayx

yxP ,

Conics & Parametersy

xa-a

a

-a

1) Circle

yxP ,a

x

222 ayx

cosax

Conics & Parametersy

xa-a

a

-a

1) Circle

yxP ,a

x

222 ayx

cosax

cosax

Conics & Parametersy

xa-a

a

-a

1) Circle

yxP ,a

x

cosax

cosax

y

sinay

sinay

222 ayx

Conics & Parametersy

xa-a

a

-a

1) Circle

yxP ,a

x

cosax

cosax

y

sinay

sinay

222 ayx

Proof:

2

222

222222

sincossincos

aa

aayx

2) Ellipse y

xa-a

b

-b

yxP ,

2) Ellipse y

xa-a

b

-b222 ayx

yxP ,

2) Ellipse y

xa-a

b

-b222 ayx

yxP ,

x

2) Ellipse y

xa-a

b

-b222 ayx

yxP ,

cosax

cosax

x

2) Ellipse y

xa-a

b

-b222 ayx

yxP ,

cosax

cosax

xy

2) Ellipse y

xa-a

b

-b222 ayx

yxP ,

cosax

cosax

xy

sinby

sinby

2) Ellipse y

xa-a

b

-b

yxP ,

xy

222 ayx

222 byx

cosax

cosax

sinby

sinby

Proof:

1sincos

sincos

22

2

22

2

22

2

2

2

2

b

ba

aby

ax

Equation of Tangent and Normaly

xa-a

b

-b

11, yxP

12

2

2

2

by

ax

Equation of Tangent and Normaly

xa-a

b

-b

11, yxP

12

2

2

2

by

ax

02222

dxdy

by

ax

dxdy

dydf

dxdf

Equation of Tangent and Normaly

xa-a

b

-b

11, yxP

12

2

2

2

by

ax

02222

dxdy

by

ax

dxdy

dydf

dxdf

yaxb

dxdy

ax

dxdy

by

2

2

2222

Equation of Tangent and Normaly

xa-a

b

-b

11, yxP

12

2

2

2

by

ax

02222

dxdy

by

ax

dxdy

dydf

dxdf

yaxb

dxdy

ax

dxdy

by

2

2

2222

11,at yxP

12

12

yaxb

dxdy

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

21

221

21

21

2

21

21

221

21

2

yaxbyyaxxb

xbxxbyayya

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

21

221

21

21

2

21

21

221

21

2

yaxbyyaxxb

xbxxbyayya

2

21

2

21

21

21

by

ax

byy

axx

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

21

221

21

21

2

21

21

221

21

2

yaxbyyaxxb

xbxxbyayya

2

21

2

21

21

21

by

ax

byy

axx

121

21

byy

axx

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

21

221

21

21

2

21

21

221

21

2

yaxbyyaxxb

xbxxbyayya

2

21

2

21

21

21

by

ax

byy

axx

121

21

byy

axx

normal:

11

21

2

1 xxxbyayy

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

21

221

21

21

2

21

21

221

21

2

yaxbyyaxxb

xbxxbyayya

2

21

2

21

21

21

by

ax

byy

axx

121

21

byy

axx

normal:

11

21

2

1 xxxbyayy

112

112

12

12

112

12

112

12

yxbyxayxbxya

yxaxyayxbyxb

y

xa-a

b

-b

11, yxP

tangent:

11

21

2

1 xxyaxbyy

21

221

21

21

2

21

21

221

21

2

yaxbyyaxxb

xbxxbyayya

2

21

2

21

21

21

by

ax

byy

axx

121

21

byy

axx

normal:

11

21

2

1 xxxbyayy

112

112

12

12

112

12

112

12

yxbyxayxbxya

yxaxyayxbyxb

22

1

2

1

2

bay

ybx

xa

Using Parametric Coordinates; sin,cosat baP

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

22

22

cossinsincoscoscossinsin

ababaybxabbxabay

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

22

22

cossinsincoscoscossinsin

ababaybxabbxabay

22 cossinsincos

by

ax

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

22

22

cossinsincoscoscossinsin

ababaybxabbxabay

22 cossinsincos

by

ax

1sincos

by

ax

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

22

22

cossinsincoscoscossinsin

ababaybxabbxabay

22 cossinsincos

by

ax

1sincos

by

ax

normal:

cos

cossinsin ax

baby

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

22

22

cossinsincoscoscossinsin

ababaybxabbxabay

22 cossinsincos

by

ax

1sincos

by

ax

normal:

cos

cossinsin ax

baby

cossin cossin

cossinsin cossincos

22

2

2

babyax

aaxbby

Using Parametric Coordinates; sin,cosat baP

sincos

sincos

2

2

ab

baab

dxdy

tangent:

cos

sincossin ax

abby

22

22

cossinsincoscoscossinsin

ababaybxabbxabay

22 cossinsincos

by

ax

1sincos

by

ax

normal:

cos

cossinsin ax

baby

cossin cossin

cossinsin cossincos

22

2

2

babyax

aaxbby

22

sincosbabyax

For ellipse 12

2

2

2

by

ax

11,at tangent yx

121

21

byy

axx

For ellipse 12

2

2

2

by

ax

11,at tangent yx

121

21

byy

axx

11,at normal yx

22

1

2

1

2

bay

ybx

xa

For ellipse 12

2

2

2

by

ax

11,at tangent yx

121

21

byy

axx

11,at normal yx

22

1

2

1

2

bay

ybx

xa

sin,cosat tangent ba

1sincos

by

ax

For ellipse 12

2

2

2

by

ax

11,at tangent yx

121

21

byy

axx

11,at normal yx

22

1

2

1

2

bay

ybx

xa

sin,cosat tangent ba sin,cosat normal ba

1sincos

by

ax 22

sincosbabyax

e.g. (i)

23,2point at the

116 ellipse theo tangent t theofequation theFind 2

2

yx

e.g. (i)

23,2point at the

116 ellipse theo tangent t theofequation theFind 2

2

yx

yx

dxdydxdyyx

16

028

e.g. (i)

23,2point at the

116 ellipse theo tangent t theofequation theFind 2

2

yx

yx

dxdydxdyyx

16

028

341

2316

2 ,2

3,2at

dxdy

dxdy

e.g. (i)

23,2point at the

116 ellipse theo tangent t theofequation theFind 2

2

yx

yx

dxdydxdyyx

16

028

341

2316

2 ,2

3,2at

dxdy

dxdy

234

123 xy

e.g. (i)

23,2point at the

116 ellipse theo tangent t theofequation theFind 2

2

yx

yx

dxdydxdyyx

16

028

341

2316

2 ,2

3,2at

dxdy

dxdy

234

123 xy

08342634

yxxy

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

cos2x

sin2ddx

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

cos2x

sin2ddx

siny

cosddy

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

cos2x

sin2ddx

siny

cosddy

sin2cos

dxd

ddy

dxdy

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

cos2x

sin2ddx

siny

cosddy

sin2cos

dxd

ddy

dxdy

23

123

,6

at

dxdy

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

cos2x

sin2ddx

siny

cosddy

sin2cos

dxd

ddy

dxdy

23

123

,6

at

dxdy 3

32

21

xy

(ii)

6point at the

sin,2cosellipsethetonormaltheofequation theFind

yx

xfy sin

xfxfdxdy cos

xfy cos

xfxfdxdy sin

cos2x

sin2ddx

siny

cosddy

sin2cos

dxd

ddy

dxdy

23

123

,6

at

dxdy 3

32

21

xy

033324344332

yxxy

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

cosax

sinaddx

sinby

cosbddy

sincos

ab

dxdy

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

cosax

sinaddx

sinby

cosbddy

sincos

ab

dxdy

mdxdykmxy hen tangent ta is If

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

cosax

sinaddx

sinby

cosbddy

sincos

ab

dxdy

mdxdykmxy hen tangent ta is If

sincos i.e.

abm

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

cosax

sinaddx

sinby

cosbddy

sincos

ab

dxdy

mdxdykmxy hen tangent ta is If

sincos i.e.

abm

)1(0cossincossin

bam

bam

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

cosax

sinaddx

sinby

cosbddy

sincos

ab

dxdy

mdxdykmxy hen tangent ta is If

sincos i.e.

abm

)1(0cossincossin

bam

bam

then;sin,cosat ellipsemeets tangent If bakamb cossin

(iii)

22222

2

2

2

then ,1

ellipsetheotangent taisif that Show

kbmaby

ax

kmxy

cosax

sinaddx

sinby

cosbddy

sincos

ab

dxdy

mdxdykmxy hen tangent ta is If

sincos i.e.

abm

)1(0cossincossin

bam

bam

then;sin,cosat ellipsemeets tangent If bakamb cossin)2(sincos kbam

0coscossin2sin:1 222222 babmma

2222222 sincossin2cos:2 kbabmma

(+)

0coscossin2sin:1 222222 babmma

2222222 sincossin2cos:2 kbabmma

(+)

22222222 cossincossin kbma

0coscossin2sin:1 222222 babmma

2222222 sincossin2cos:2 kbabmma

(+)

22222222 cossincossin kbma 2222 kbma

Exercise 6C; 1, 3, 11, 15, 18a

Komaroni Unit 1