unidad 5 - analítica plana - problemas resueltos

8/11/2019 Unidad 5 - Analtica Plana - Problemas resueltos

1/16

1 BUP - Matemticas - Unidad 5 - Analtica plana

Pg.115 I Punto medio = [ ] [ ]( )=++ 2115282 ( )83

Pg.115 II Baricentro (G= !ntersecci"n medianas#ircuncentro (T = !ntersecci"n mediatrices$rtocentro (H = !ntersecci"n alturas

%ecta de Euler = (GTH&' = altura) mediana) mediatri* (lado desigual

Pg.115 III Baricentro (G= !ntersecci"n medianas#ircuncentro (T = !ntersecci"n mediatrices$rtocentro (H = !ntersecci"n alturas%ecta de Euler = (GTH&' = altura) mediana (+ipotenusa

Pg.115 IV a =

++

=3

333

3

034G

13

1

, ( ) ( ) ( )02023321.. =LadosMP Baricentro = G= ( )c ( ) ( ) ( )=sPiesAltura $rtocentro = H= ( )d #ircuncentro = T= ( )e %ecta uler = GTH= ( )

Pg.11 1 /a recta en param0tricas es += 02x 36 =y 2=x'odo punto de a,scisa "2" pertenece a la recta A, B y C s pertenecen

os puntos ms de la recta son) por e2emplo ( )02 3 ( )12

Pg.11 2 a ( ) ( ) ( )+= 2225 yx ( ) ( ) ( ) 2225 += yx, ( ) ( ) 2225 +r 25 =x 22 +=y

Pg.11 3 a ( ) ( ) =+= 5321v ( )21 +

=

2

5

1

2 yx 012 =++ yx

, ( ) ( ) =+= 3333v ( )06 63 =x 03 =+y 3=yc ( ) ( ) =+= 3411v ( )70 01 =x 74=y 1=x

d ( ) ( )=++= 4223v ( )25 +=+

2

4

5

2 yx 01652 = yx

Pg.11 4 /os respecti4os puntos medios de los lados ABCABC ,, sern( ) =+= 2' CBA ( )2321 ( ) =+= 2' ACB ( )01 ( ) =+= 2 BAC ( )2321

'AA

=

323

3

221

2 yx

=

29

3

25

2 yx 0359 = yx

'BB 00

0

11

1

=

++ yx

0=y

'CC ++

=

323

3

021

0 yx

+=

29

3

21

yx 039 = yx

Pg.11 5 Paralela a OX 5=y Paralela a OY 3=x

161


2/16


Pg.11 6 7ector director recta OA = =OA ( )312 7ector cual8uiera de la recta OA = ( ) = 3123 ( )167ector normal (ortogonal a la recta OA) por e2emplo) es ( )61

Pg.11 7 7ector director = ( )23 cuaci"n general 2

2

3

1 =+ yx 0832 =+ yx

cuaci"n normal ( ) yxP = 0nAP ( ) ( ) =+ 03221 yx( ) ( )( ) 02312 =++ yx 0832 =+ yx

cuaci"n normal can"nica 032

8

32

3

32

2

222222

=+

++

+

yx

cuaci"n coordenadas origen 13121 =+yx

Pg.11 8 7ector normal = ( )23=n 7ector director = ( )32=u

Pg.11 9 7ector director = ( ) ( ) =+== 3231ABAB ( )54 7ector normal = =n ( )45cuaci"n normal An ( ) ( ) 3345 ( ) ( ) 03435 =++ yx

cuaci"n general =++ 0124155 yx 0345 =+ yx

cuaci"n normal can"nica 045

3

45

4

45

5

222242=

+

++

+yx

cuaci"n coordenadas origen 14353 =+ yx

Pg.11 10 7ector normal a recta ( ) ( ) =+== 2401ABAB ( )61 cuaci"n ( ) ( ) 0361 C ( ) ( ) =+ 00631 yx 036 =+ yx

Pg.119 11 cuaci"n ( )+= 22

14 xy 5

2

1 += xy 0102 =+ yx 1510=+

yx

Pg.119 12 Pendiente x

y =

+12

52

3

7 cuaci"n ( )1

3

75 += xy

$rdenada en origen = 0x ( )+= 103

75y

3

8=y

Pg.1:1 13 a5

5

3

2

secantes ,

5

5

15

5

9

3 = paralelas

Pg.1:1 14 !ntersecci"n ( ) 05=+ yxs ( ) 01 =++ yxt( ) ( )+ ts =+ 062x 3=x ( ) ( ) st = 042y 2=y

Paralela a ( ) 012 =++ yxr ) por ( )23 ( ) ( ) =++ 0232 yx 042 =++ yx

Pg.1:; 15 a ( ) ( ) =++= 22 1320AB 22 ( ) ( ) =+= 22 3703BC 5( ) ( ) =+= 22 7132CA 61 CABCAB Escalen

, =

+

=

22

2

1

2

1

2

1

2

3AB 1 =

++

=

22

2

1

2

31

2

31BC 1

=

++

=22

2

31

2

11

2

1CA 1 CABCAB == E!u"l#ter

Pg.1:; 16 a %ecta por ( )32 3 ( )22

=++

32

3

22

2 yx 0104 =+ yx

:61


3/16


, istancia de ( )010 a recta ==+

+

17

0

41

1040110

220

c Por tanto) los puntos CBA ,, estn al"nea$s

Pg.1:; 17 a ( ) 043 = yxr ( )43 =rn ( ) 0322 =++ yxs ( )22=sn

( ) ( ) ( )

=

++

+=

=2222

2243

2423,cos

sr

sr

nn

nnsr

25

1 ( ) '5281, srng

, ( ) 5= xyr 1=rm ( ) += 22xys 2=sm

( ) =+

=+

=211

21

1,

sr

sr

mmmm

srtg3

1 ( ) '2618,. srng

Pg.1:; 18 ( ) ( ) == 0641AB ( )65 ( ) ( ) == 0046AC ( )010 ( )

=++

+=

=

222201065

050cos

ACAB

ACABA

61

5 '1250A

=BA ( )65 ( ) ( ) =+= 6016BC ( )65

6111

6565

3625.cos

2222

=

++

+=

=

BCBA

BCBAB

'3679B=CB ( )65 =CA ( )010

( ) ( )

=++

+=

=

222201065

06105cos

CACB

CACBC

61

5 '1250C

l tringulo es "s%sceles 3 acut#n&ul

Pg.1:; 19 ( ) 2bam + amb = 2 1=m 2=a 4=b 4=m 3=a 11=b ( )114

Pg.1:; 20 7ector normal recta ( ) =+ 03yxr ( )11 =n

%ecta ( ) ( )= 15larPorPerpendicus +=1

1

1

5 yx ( ) 04=+ yxs

!ntersecci"n srM ( ) ( )+ sr 2

1=x ( ) ( ) rs 2

7=y

2

7

2

1M

== 52

12

'Px 4 ( )== 12

72

'Py 8 ( )84' P

Pg.1:; 21 ( ) 0= xOY ( ) 01 =+ yxr !ntersecci"n ( ) ( )OYrQ ( )10Q Un punto particular del e2e es ( )00O

7ector perpendicular ( )r ( )11 Perpendicular por O ( )1

0

1

0

= yxs

!ntersecci"n ( ) ( )srM ( ) 01 =+ yxr ( ) 0=+ yxs ( ) ( )+ sr 21=x ( ) ( ) rs 21=y ( )2121 M


4/16


5/16


6/16


ormal ( ) 0142

32 =++

+ yx &eneral 0742 =++ yx

#an"nica +

++

++ 222222 42

7

42

4

42

2yx 0

52

7

5

1

52

1 =++ yx

Pg.1:> 34 a == 12m 2 , +=3

5

3

2xy 32=m

c += 252

15xy 215=m d =+=11

23m2

1

e ===

0

2

11

3 aaam ( )0a C =

=

3

5m

3

5

g ( )= 27u 7

2=m + ( )= 25u

5

2=m

Pg.1:> 35 ( )r += 332 xy 32=m 3=n ( )s = 331 xy 31=m 3=n ( )t = 5x =m =n ( )u = 4y 0=m 4=n

Pg.1:> 36 = 306090 33

30 =tg ( )+=+ 233

5 xy 31532

3

3 += xy

Pg.1;1 37 a ( ) ( )+= 122 xy 42 += xy ,

=++

34

3

12

1 yx

3

10

3

1+= xy

c ==3

330tgm ( ) ( )=+ 2

3

33 xy

3

329

3

3 = xy

d == 360tgm ( ) ( )+= 235 xy ( )3253 ++= xy

Pg.1;1 38 a ( )= 32n ( ) ( ) ( )+= 2363 yx 33+=x 26+=y 02432 =+ yx 18

3

2 += xy

, ( ) ( )=+= 313001PQ ( )= 31u ( ) ( ) ( )3130 += yx

=x 33+=y ++

=

30

3

01

0 yx

3

3

1

+=

yx

033 =++ yx 33 = xy

Pg.1;1 39 ( ) ( )7121

=11

27m

2

9=m

Pg.1;1 40 +=1

1

2

2 yx =

2

1m ( ) ( )= 2

2

16 xy 0142 =+ yx

Pg.1;1 41 a ( ) 02 =+ yxr ( ) 04=+ yxs =4

2

1

1

1

1paralelas

, ( ) 0723 = yxr ( ) 0832 = yxs

3

2

2

3secantes

c ( ) 052 = yxr ( ) 053

1

3

2 =+ yxs

= 5

5

31

1

32

2paralelas

d ( ) 07=+ yxr ( ) 02

7

2

1

2

1=+ yxs

=

=

277

21

1

21

1c"nc"$entes

e ( ) 02 =+ yxr ( ) 084 =+ yxs 1

1

4

2secantes

61


7/16


C ( ) 032 =+ yxr ( ) 02 = yxs 2

1

1

2secantes

Pg.1;1 42 a ( ) 53 =+ yxr ( ) 52 =+ yxs ( ) ( ) sr 0=y 5=x ( )05 , ( ) ( ) sr 54 = 147x 2=x 23=y ( )232 c ( ) ( )+ sr 246 = 3018y 35=y 928=x ( )35928 d ( ) 232 =+ yxr ( ) ( ) sr = 2y 2=x ( )22

Pg.1;1 43 a 52=m ( ) ( )+= 2526 xy 02652 =+ yx

, 0=m 4=y c =m 1=x d 2=m = xy 2 02 = yx e 21=m ( ) ( )+= 2214 xy 0102 =+ yx C 0=m = 2y 02=+y g 1=m ( ) ( )= 015 xy 05=+ yx

Pg.1;1 44 a 2=m ( )=+ 221 xy 032 =+ yx, =m D 04=+xc 0=m D = 3y 03 =y

d 1=m D ( )= 010 xy 0=+ yx e 2=m ( )+= 120 xy 022 =++ yx

C ( ) ( ) =++= 321235ABm = 23m ( )+= 3232 xy 01323 =+ yx

Pg.1;1 45 a

=3

1

4

1 &

& = 42& 2=&

, == 44

32

1&& 32=&

c

=

2

15

3

2

&

& = 152 2& = imaginario& "(ps")le paralelas

Pg.1;1 462

1

6

3

28

03 ==

=CAm 2

5

64

14 =+=BDm

4

5= BDCA mm 0

CA 3 BD no so perpendiculares la Cigura n es un r()

Pg.1;1 47 ( ) 41P identidadbb = 1441 ( ) 32Q = 1432 bb 3=b ( ) ( ) = 1343 yx 0113 =+ yx

Pg.1;1 48 +

=

1

1

1

22

&&

& =+ 032 2 && 1=& 23=&

Pg.1;1 49 2

1=&

mr 1

43 = &ms =+ 073 2 && 2=& 3

1=&

= 2& ( ) 042 =+ yxr ( ) 042 =++ yxs ( ) ( )+ sr 2 512=x 54=y ( )54512

= 31& ( ) 032232 =+ yxr ( ) 0913 =++ yxs ( ) ( )+ sr 2 152=x 4513=y ( )513152

Pg.1;1 50 ( ) ( )+= 23 xmy 032 =++ mymx = 21m 042 =+ yx

Pg.1;1 51 ( ) 032 = yxr ( ) 053 = yxs 12 == yx ( )21 = xmy ( ) 22 ( )= 2212 m = 41m ( )= 2411 xy 064 =+ yx

>61


8/16


$ ,ien ( ) ( ) 05332 =+ yxyx'a$ ( ) 22 = 139 064 =+ yx

Pg.1;1 52 ( ) ( ) sr ( )00 mxy= = 32m = xy 32 032 =+ yx

Pg.1;1 53 ( )r 032 =++ &yx ( ) 31 = 11& 01132 =+ yx

Pg.1;1 54 &xy += 2 ( )00 = 0& 02 =+ yx

Pg.1;1 55 ( )r = 23m = 32m &xy += 32 ( ) 11 = 31& 0132 =+ yx

Pg.1;1 56 ( )t = 25m 52 =m ( ) ( ) sr ( )26 ( )= 6522 xy 02252 =+ yx

=+

+

++

0

52

22

52

5

52

2

222222

yx 029

22

29

5

29

2=+ yx

Pg.1;1 57 a ( ) ( ) ==+++= 802235 22D 54

, ==

+

= 8

3

1

5

3

2

1

2

5 22

D 22

c =

++

=

22

3

53

2

1

5

3D

30

1609

d =

+

=

22

2

3

2

3

2

2

2

2D 3

Pg.1;1 58 ( ) ( ) sr ( )11 ( ) ( )[ ] ( ) ( ) =++= 22 11211132D 17

Pg.1;1 59 a( ) ( )

=+

+=

2232

54332D

13

1,

( ) ( )=

+

+=

2212

32102D

5

7

c ( ) ( ) =++= 22

22332212D

25 d ( ) ( ) ==+

++=20

1112111

22D 0

e ++

=

342

3

212

21 yx=++ 0231014 yx

( ) ( ) =+

++=

221014

23010114D

742

9

C = 1m = 2xy = 02yx ( ) ( ) =

+

=22

11

22131D

2

3

Pg.1;1 60 a

=8

7

1

1

2

2


9/16


=

+

=

22

12

30

2

3CA 1 'riangulo e!u"l#ter

Pg.1;1 62 ( ) ( ) =++= 22

2125AB 58 ( ) ( ) =++= 22

1453BC 29

( ) ( ) =++= 22 2423CA 29 Permetro = =++ 292958 2229 +

[ ] ( ) ( )

=+

+=

2273

84733ABCD

2

29 ==

2

2958

2

1(rea

2

29

Pg.1;1 63 ( ) ( ) rs 3 = 077y =1y 2=x ( ) A12 ( ) ( )+ ts 2 =07x = 0x 2=y ( ) B 20 ( ) ( ) rt 2 = 007y = 0y 1=x ( ) C 01 ( ) ( ) =++= 22 2001BC 5 ( ) ( ) =++= 22 0112CA 10

( ) ( ) =+= 22 1220AB 13

Permetro =13105 ++

Pg.1;1 64 a 3=rm 1=sm ( ) =+

=131

13tg 2 '2663

,2

1=rm 1=sm

( )( ) ( )

=+

=1211

121tg 3 '3471

c 1=rm 3=sm ( )

( ) ( )=

+

=311

31tg

2

1 '3426

d 1=rm 2

1=sm ( )

( ) ( )=

+

=2111

211tg

3

1 '2618

Pg.1;1 65 ( ) 134=+ yxr ( ) 1

56=+ yxs ( )04A ( )06B ( )50C ( )30 D

== 46AB 2 == 35CD 2 =+= 22 34AD 5 =+= 22 56AC 61

Permetro = =+++ 61522 619 + Frea = ( ) ( )[ ]== 34562

1OADOBC 9

Pg.1;; 66 a( ) ( )

=

++

=

=

1317

5

3214

3214cos

2222ACB

ACABA

A '40109A )tus#n&ul

,( ) ( )

04010

0

2631

2631cos

2222==

++

=

=

CBCA

CBCAC 90=C rect#n&ul

c( ) ( ) 0

4520

0

3642

3642cos

2222==

++

=

=

CBCA

CBCAC 90=C rect#n&ul

Pg.1;; 67 ( ) ( ) sr 2 = 077x 1=x 0=y ( )01A ( ) ( ) tr = 0122x 6=x 215=y ( )2156 B ( ) ( ) ts 2 = 0205y 4=y 1=x ( )41 C

( )2155 =AB ( )42 =AC ( )2155=BA ( )277=BC ( )42=CA ( )277 =CB

( ) ( ) ( )

( )=

++

+=

2222422155

421525cosA

65

44961,0 "18'1560A

( ) ( )

( ) ( )=

++

+=

22222772155

2721575cosB

65

78682,0 "18'4429B

961


10/16


( )

( )=

++

+=

222227742

27472cosC =

35

00 90=C

l tringulo es rect#n&ul 3 escalen

Pg.1;; 68 ( ) ( ) ( )APA yxyxyx = 2' == 232'Ax 4 == 352'Ay 7 ( )74' A ( ) ( ) ( )BPB yxyxyx = 2' == 432'Bx 2 == 152'By 9 ( )92' B

Pg.1;; 69 = 1rm ( ) == 111'AAm ( ) ( )== 113' xyAA 02=+ yx( ) ( ) rAA' ( ) ( )+ rAA' = 12x 21=x 25=y ( )2521 a

== 1212'Ax 0 == 3252'Ay 2 ( )20' A = 1rm ( ) == 111'BBm ( ) ( )== 3125' xyBB 021 = yx

( ) ( ) rBB' ( ) ( )+ rBB' = 272x = 47x 45=y ( )4547 b== 3472'Bx 21 == 25452'By 0 ( )021' B

Pg.1;; 70 a ( ) ( ) sr2 0=y 2=x Punto corte = ( )02Q , ( ) ( ) ( ) ( ) =+ 4231221 sP 0 P pertenece a (s)

= 41rm ( ) == 4411'PPm ( ) ( )= 142' xyPP 064 =+ yx( ) ( ) rPP' ( ) ( ) rPP 4' = 01417y 1714=y 1722=x== 117222'Px 1727 == 217142'Py 176

( )1761727' P

c ( ) '' QPs

=+

+0176

0

21727

2 yx

=

+661

2 yx 012616 =++ yx

m

Pg.1;; 71 a ( )AB

=

40

4

13

1 yx 03=+ yx

( ) ( ) +2

' BA

C ( )21' C

=1ABm 1' =Cm Mediatri*6AB ( ) ( )+= 112 xy 01=+ yx

( )BC =++ 02 033 3 yx 033 =++ yx ( ) ( ) += 2' CBA ( )

10' A

= 31BCm 3' =Am Mediatri*6B#( ) ( )=+ 031 xy 013 = yx

( )CA ++

=

24

2

31

3 yx 073 =+ yx

( ) ( ) +2

' AC

B ( )12' B

= 3CAm 31' =Bm Mediatri*6#A( ) ( )= 2311 xy 013 =+ yx

, ( ) ( ) ( )01301301 =+=+=+ yxyxyx T ( )2121

Pg.1;; 72 ( ) 0223 =+ yxr ( ) 022 =+ yxs Bisectrices 222221

22

23

223

+

+=

+

+ yxyx

Beatri* 1G 0513251321353 =+ yx ( )51321353

1

=m

Beatri* :G ( ) ( ) ( ) 0513251321353 =++++ yx ( )51321353

2

+

+=m

( ) ( ) ( )=

+

+

=

5134

1345

5132

1353

5132

135321

mm 1


11/16


12/16


Pg.1;; 77 a ( ) =+= 0855AC ( )810 ==+= 164810 22AC 412 81,12

( ) == 2637BD ( )810 ==+= 164810 22BD 412 81,12

, =

+

+=

2

20

2

35L ( )11 =

+=

2

82

2

53M ( )34

=

=

2

68

2

75+ ( )71 =

=

2

60

2

57P ( )36

=

=

2

71

2

11.. L+MedioP ( )31 =

=

233

264.. MPMedioP

( )31

/a identidad de estos puntos demuestra 8ue LM+Pes paralel&ra(.Adems) al ser ( )80L+ perpendicular a ( )010MP ) es r().

c ( ) ( ) =++== 22 131444 LMPer,metro 414

d


13/16


( )ABCD , ( )( )

=+

+=

2213

15163 &D

10

3 &+ 20

10

3102

2

1=

+=

&rea

103 =+ & 103 =+& 7=& 13=&

Pg.1; 84 ( )AC 134=+ yx ( ) 01243 =+ yxAC ==+= 2534 22AC 5

( ) ( )2322324.. == MACMedioP

43=ACm

( ) ( )43 BD ( ) ( ) ==+= 222 2543 MD 5 255 = 21= ( ) ( )++ 42332B ( ) ( )=++ 223232B ( )2727

[ ] =

2

7

2

32

2

722D

2

1

2

1 ( ) ===

222

2

12

2

1ACLLrea

2

25

Pg.1; 85 =

+

+

2

31

2

06M ( )23 ( )= 022132D ( )45

( ) == 3106AC ( )26 =+= 22 26AC 102 ( ) == 0415BD ( )44 =+= 22 44BD 24

( ) =

=

=

24102

4246,cosBDAC

BDACBDAC5

1 ( ) '2663,. BDACng

Pg.1; 86 1=+&y

&x

== &&rea2

1

2

2&

52

2

=&

10=&

Pg.1; 87 ( )12 = xmy = 0x my = 2 = 0ym

mx

2= ( )m

mmA

22

2

15,4

=

=++ 0452 mm ( )( ) 041 =++ mm 1=m ( )= 112 xy 03 =+ yx

4=m

( )= 142 xy 064 =+ yx

Pg.1; 88


14/16


( ) AC 0233 = yx ( ) ( )0233072 ==++ yxyxC ( )65 C

( ) baB Punto medio de AB

+

2

7

2

2 baM

l punto M est en la mediana =+++ 072

72

2

2 ba 022 =++ ba

l punto B est en la altura 0113 =++ ba 4=a 1=b ( )14B

( )AB =

++

17

1

42

4 yx 01334 =++ yx ( )BC =

++

16

1

45

4 yx 01997 =++ yx

Pg.1; 93


15/16


( ) ( ) r = 03111y = 1131y 1140=x ( )11311140 Q

Pg.1;5 99 ( )22 A ( )0 aB ( )13C

= BCAB ( ) ( ) ( ) ( ) +=++ 2222 013202 aa 51=a ( )051 B

MedioACP

=

+

2

3

2

1

2

12

2

23O MedioBDP ODBO =

54

5

1

2

12 ==Dx 302

32 ==Dy ( )354 D

( ) ( ) =++= 22 2123AC 26 ( ) =+

= 2

2

035

1

5

4BD

5

263

===5

26326

2

1

2

1BDACrea

5

39

Pg.1;5 100


16/16


=CD ( )cdcd + 7 cd=4 71 += cd 1=c 5=d ( )61C ( )55D

=CD ( )dcdc 7 dc =4 cd= 71 6=c 2=d ( )16C ( )22D

Pg.1;5 105 ( )AB12

1

21

2

=

yx

( )BC23

2

11

1

++=

yx

( )CA31

3

12

1

=

++ yx

( )AB 053 = yx ( )BC 0125 =+ yx ( )CA 0732 =+ yx

+ +=+ + 2222 32732

25125 yxyx ( ) ( ) ( )013297132293292135 =+ yx

53 = xy ( ) 132921329 = x 16

27745=Px

16

27755=Py

( )( ) CB

CA

x

x

PB

PA

P

P +

+=

=

=

=22

22

25

32

29

13

132929

291313

291329

291313

1

2

Pg.1;5 106 = 0& 1=y = 1& 024 =++ yx 2=x V.rt"ce= ( )12 %ecta 8ue 'alta & ( ) 0222 =+++ yyx& 02 =+ yx

Pg.1;5 107( )27

P

( )34

Q

( ) oPuntoDePasaM = 0

( )34'

Q

eO*im/tricoD= l camino "#$es id0ntico al "#$%) 8ue ser mnimo si es la recta "$%.

( )'PQ

=++

23

2

74

7 yx 013115 =++ yx ( ) ( ) =++ 0130115a

5

13=a

Por tanto) el punto de paso es

0

5

13M

Pg.1;5 108 2===== )AD)CDBCAB ( )01C ( )01D ( ) ( ) == 108525180. DCBng ( ) == 108180. BC-ng 72

72cos

+= BCxx

CB 72

senBCyC = ( ) ( )7272cos1 senB

+ #omo &es sim0trico de ') respecto O(, es ( ) ( )7272cos1 sen) 72cos72cos44

222 ++== OADAOA ( ) ( 72cos72cos440 2++A

Pg.1;5 109 a abmAB = ( )AB =+ 1byax 0=+ abaybx abm

O' = ( )O' = xaby 0=byax

22

2

ba

abx

+=

22

2

ba

bay

+=

+

+

22

2

22

2

ba

ba

ba

ab'

,( )

( )

=

+

+=

+

+

+

=2

22

22222

22

22

22

2

ba

baba

ba

ba

ba

abO'

22 ba

ab

+

( )

=+

+=

+

+

+=

222

2462

22

22

22

2

ba

baa

ba

baa

ba

ab'A

22

2

ba

a

+

( )

=+

+=

++

+

=222

6422

22

22

22

2

ba

bbab

ba

ba

ba

ab'B

22

2

ba

b

+

c 2O' =+

+

+

=22

2

22

2

22

22

ba

b

ba

a

ba

ba'B'A

d 2OA =

+

+=22

2

222

ba

abaa 'AAB

2OB =+

+=22

2

222

ba

bbab 'BAB

161

unidad 5 - analítica plana - problemas resueltos

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