A14 LesterPadilla 20191003860

 

929?

Auão ed Dstuejm

RYJCDY ROYKJOF UDMYJO EDF KMCRMYUOCJDLUM EDF KML[TCJEMY

CJKYMDKMLMCÃO CJKYMDKMLM CÃO | Dkmlmcão - TLOG

 

J.

Dl df kuoe kuoerm rm kmfmq kmfmqud ud df lþcd lþcdrm rm qu qud d kmrrds kmrrdspml pmled ed o fo edskr edskrjpk jpkjûl jûl kmrr kmrrdkt dkto. o. Ofaulms kmlkdptms qudeorïl vokãms. Lm.

? 9 ; 5 4

8

1 7 : ?2 ?? ?9 ?; ?5 ?4

?8

Edskrjpkjûl edf Kmlkdptm

Kmlkdptm

Lm.

Fãcjtds ed fos pmsjijfjeoeds ed kmlsucm ed ul jlejvjeum. Jlejko Jlej ko fos pmsjijfj pmsjijfjeoed eoedss ed kmlsu kmlsucm cm ed ul jlejvjeum osdqujifds poro ul jlejvjeum, eoems df prdkjm ed fms ijdlds y df jlardsm. Jlardsm dxprdsoem lm kmcm ejldrm sjlm kmcm fo koltjeoe ed ijdlds qud puded kmcpror. Fo rdstrjkkjûl prdsupudstof sd edspfozo gokjo hudro y sd gokd cïs pfolo. Fãldo qud cudstro kmcijlokjmlds ed ijdlds qud fd rdsuftol jlejhdrdltds o ul kmlsucjemr. Uoso o fo qud ulo pdrsmlo rdlulkjorï of ijdl y (df ijdl cdejem smird df dnd y) poro mitdldr cïs edf ijdl x (df ijdl cdejem smird df dnd x) y of cj cjscm scm tjd tjdcp cpm m pdr pdrco cold ldkdr kdr jle jlejhd jhdrd rdltd ltd (pdrcoldkdr smird fo cjsco kurvo ed jlejhdrdlkjo).

Fãldo ed Ydstrjkkjûl Rrdsupudstorjo Rrdkjm Ydfotjvm Cjdltros ionm sdo df prdkjm edf ijdl ‖x– Cjdltros cdlmr sdo df Uoso Corajlof ed [ustjtukjûl Kurvos ed jlej jlejhd hdrd rdlk lkjo jo po poro ro ems sustjtutms pdrhdktms

:

7

1

Ds ulo dstï tdledlkjo adldrof, sdaþl ofocdlms kuof ulo pdrsmlo ejspudsto o rdlulkjor edf Uoso Corajlof ed [ustjtukjûl ijdl y poro mitdldr ulo uljeoe oejkjmlof edf ijdl   Kurvo ed [d rd rdpr prds dsdl dlto to pm pmrr kkur urvo voss p por orof ofdf dfos os Jlejhdrdlkjo

4

Rrdkjm ed ul ijdl ejvjejem dltrd df prdkjm ed mtrm ijdl. Fo fãldo ed rdstrjkkjûl prdsupudstof dstorï cïs go gokj kjo o fo jjzz uj ujdr dreo eo.. Df idldhjkjm m sotjshokkjûl dl df kmlsucm ed ul ijdl m sdrvjkjm. Ds ulo sjtuo sjtuokjûl kjûl dl fo qud ul kmlsucje kmlsucjemr mr go osjaloem fo tmtofjeoe ed sus jlardsms ejspmljifds

Jlardsm Ydof

9

Jlardsm y Rrdkjms

?

Rmsj sji ijfjfje jeo oeds Kmlsucm

ed

?8

Rrdhdrdlkjos

ed fo coldro qud, eoems fms prdkjms sdrvjkjms, coxjcjzo su utjfjeoe tmtof. ed ijdlds y Ds fo utjfjeoe corajlof ed ul ijdl ejvjejem dltrd Ttjfjeoe su rdkjm. Ds df dhdktm ed ul kocijm dl df jlardsm smird fo Ttjfjeoe Umtof koltjeoe ed ijdlds kmlsucjems. Dstï iosoem dl fo jedo qud fos pdrsmlos pudedl mredlor tmeos fos pmsjifds kmcijlokjml kmcijlokjmlds ds ed Ttjfjeoe Corajlof kmlsucm ed ijdlds dl trds arupms> Rrdhdrjems, Lm Rrdhdrjems d Jlejhdrdltds. Ds df idldh jkjm t mtof qud ulo pdrsmlo rdkjid Dqujfjirjm pmr df kml suc m e d i jdld s. Adl Adldr drof ofcd cdlt ltd d cï cïss Kmlsucjemr  kmlsucm prmpmrkjmlo cïs idldhjkjm.

?;

?1 92 ?9

edf

?5

 

?1

Ds df dhdktm ed ul kocijm ed prdkjm dl fo Ttjfjeoe Corajlof koltjeoe qud sd kmcpro kuolem df kmlsucjemr  pmr Tljeoe (gjpmtêtjkocdltd) (gjpmtêtjkocd ltd) pdrcoldkd jlejhdrdltd dltrd Cmldtorjo Aostoeo fo sjtuokjûl mrjajlof y fo fo ludvo sjtuokjûl.

?7

?7

Ds df kocijm dl fo utjfjeoe tmtof qud rdsufto ed ul ou oucdl cdltm tm ed ulo ulje uljeoe oe dl fo kol koltje tjeoe oe Dhdktm [ustjtukjûl kmlsucjeo ed ul ijdl.

?4

?:

Ku Kuol olem em ul ij ijdl dl ds ul ij ijdl dl jl jlhd hdrj rjmr mr..

92 9?

JJ.

Dhdktm Jlardsm

[d vdl fjcjtoeos pmr su prdsupudstm y fms prdkjms prdk jms ed fms prmeuk prmeuktms tms y sdrvj sdrvjkjms kjms qud ul Dhdktm Rrdkjm kmlsucjemr kmcpro. [d dfoimro rdfokjmlolem df prdkjm ed fms ijdlds y Kurvo ed Edcoleo fos koltjeoeds kmlsucjeos.

[dfdkkjûl [dfdkkjûl cþftjpfd. cþftjpfd. - [dfdk [dfdkkjmld kjmld fos rdspudst rdspudstos os kmrrdkto kmrrdktos s

?. Fos mpkjmlds edf kmlsucm dl ul gmaor dstïl fjcjtoeos pmr> o) Df jlardsm edf gmaor. i) Fms prdkjms ed fms prmeuktms y sdrvjkjms qud kmcpro. k) Lj fms jlardsms ed fms gmaords lj fms prdkjms ed fms prmeuktms y sdrvjkjms qud

kmcpro. 9. [upmlaocms qud soidcms qud o Hrdeey fd austo kmlsucjr sûfm ems prmeuktms>

smeos y idijeos ed hrutos. Fos smeos y idijeos ed hrutos tjdldl ul prdkjm ed T[$?.22 koeo ulm. Kml dso jlhmrcokjûl pmedcms> o) Edtdrcjlor fo fãldo ed rdstrjkkjûl prdsupudsto prdsupudstorjo rjo ed Hrdeey. i) Edtdrcjlor fo utjfjeoe ed Hrdeey toltm poro smeos kmcm poro idijeos ed hrutos. k) Edtdrcjlor fo kmcijlokjûl ed smeos y idijeos ed hrutos qud coxjcjzol fo

utjfjeoe ed Hrdeey. e) Lm pmedcms edtdrcjlor fo fãldo ed rdstrjkkjûl prdsupudstorjo ed Hrdeey m su utjfjeoe. ;. Fo utjfjeoe tmtof edskrjid> o) i) k) e)

Df idldhjkjm mitdljem ed tmem df kmlsucm. Tl oucdltm dl df lþcdrm ed vdkds qud sd kmlsucd ul ijdl. Df lþcdrm ed vdkds ed fo utjfjeoe corajlof. Df kmlsucm tmtof ejvjejem dltrd fo utjfjeoe corajlof.

5. ½Kuïf ed fos sjaujdltds ohjrcokjmlds ds vdreoedro= o) Cdejeo qud sd kmlsucd cïs ed ul ijdl, oucdlto su utjfjeoe tmtof, o cdlms qud

df ijdl dstê sundtm o fo utjfjeoe corajlof edkrdkjdltd. i)  O cdejeo qud sd kmlsucd cïs e ed d ul ijdl, oucd oucdlto lto su utjfje utjfjeoe oe tmtof, jlkfusm sj d dff ijdl dstï sundtm o fo utjfjeoe corajlof edkrdkjdl edkrdkjdltd. td. k) Fo utjfjeoe corajlof krdkd o rjtcms edkrdkjdltds. e)  Ocims O y K.

 

4. Fo utjfjeoe corajlof cjed> o) i) k) e)

Fo utjfjeoe tmtof edf kmlsucm ed ulo pdrsmlo. Fo utjfjeoe ed ulo uljeoe oejkjmlof ed kmlsucm. Fo utjfjeoe prmcdejm ed ulo uljeoe oejkjmlof ed kmlsucm. Df kmstm ed ulo uljeoe oejkjmlof ed kmlsucm.

8. [upmlaocms qud df prdkjm edf kohê ds ed T[$;.22, df prdkjm ed ul pol ds ed

T[$9.22 y df prdsupudstm ed ulo pdrsmlo ds ed T[$52.22 T[$52.22.. Fo dkuokjûl ed fo fãldo prdsupudstorjo ds> o)

 

;

9 +

_ kohê  kohê  _ pol

3 52

i) 9 (_pol) + ;(_kohê) 3 52 k)

  _kohê  _pol ;

+

9

3 52

e) _pol x _kohê 3 52 / (; + 9). 1. ½Kuïf ed fms sjaujdltds dvdltms, eol fuaor o ul kocijm dl fos pdlejdltds ed sus

kurvos ed jlejhdrdlkjo dltrd fo aosmfjlo y df ofqujfdr ed pdfãkufos= o) i) k) e)

Tl kocijm dl sus prdhdrdlkjos pmr ofaulm ed fms ems ijdlds. [ûfm ul kocijm dl df prdkjm ed kuofqujdro ed fms ems ijdlds. [ûfm ul kocijm dl sus jlardsms. Uoltm ul kocijm dl df prdkjm ed kuofqujdro ed fms ijdlds y dl df jlardsm

7. Qofem tjdld ul jlardsm ed F95.22 y kmlsucd smfocdltd ems prmeuktms> Kmlhjtds y

Amcos ed Coskor kuyms prdkjms sml F2.82 y F9.52, dl su mredl. ½Kuïf ed fos sjaujdltdss ohjrcokjml sjaujdltd ohjrcokjmlds ds ds kmrrdkto= o) i) k) e)

Df jlardsm rdof ed Qofem ds ed 52 kmlhjtds. Df jlardsm rdof ed Qofem ds ed ?2 amcos ed coskor. Df prdkjm rdfotjvm ed fos amcos ed coskor ds ed 5 kmlhjtds. Umeos fos rdspudstos oltdrjmrds sml kmrrdktos.

:. Fo Ttjfjeoe Corajlof Edkrdkjdltd sjaljhjko> o) i) k) e)

Fo utjfjeoe tmtof edf kmlsucm ed ulo uljeoe dxkded fo utjfjeoe tmtof ed ems uljeoeds. Df prdkjm ed ul ijdl ds df emifd edf prdkjm ed mtrm ijdl. Tlo pdrsmlo ejshruto su sdauleo uljeoe ed kmlsucm cdlms qud fo prjcdro. Gosto kjdrtm pultm dl df kmlsucm, fo utjfjeoe tmtof edkrdkd y df Jlardsm oucdlto.

?2. Kuïf ed fms sjaujdltds hoktmrds LM sd puded mitdldr o portjr ed fo dkuokjûl ed

Fãldo ed Ydstrjkkjû Ydstrjkkjûl l Rrdsupudsto Rrdsupudstorjo. rjo. o) i) k) e)

Rrdkjms Ydfotjvms Jlardsm Ydof Koltjeoe ed Ijdlds Rrdhdrdlkjos edf Kmlsucjemr 

 

JJJ. JJJ. Olïf Olïfjs jsjs js Arïh Arïhjk jkm m o) Ejiund ulo arïhjko qud cudstrd df dqujfjirjm edf kmlsucjemr kml oucdltm dl df

Rrdkjm ed Z, smfm kml lmcdlkfoturos, lm tjdld qud kmfmkor lþcdrms. _y J/Ry? J/Ry9

D? D9

F Y   R  

F Y   R  

_x

Y/3  Tl oucdltm edf prdkjm ed Z, df jlardsm rdof ejscjluyd y kmlsjam ulo fãldo ed

rdstrjkkjûl prdsupudstorjm. i) Ejiund ulo arïhjko qud cudstrd df dqujfjirjm edf kmlsucjemr kml ejscjlukjûl dl

df jlardsm, soijdlem qud ocims ijdlds sml jlhdrjmrds, lm tjdld qud kmfmkor  lþcdrms smfm lmcdlkfoturo. J?/Ry

J9/Ry

F  Y   R  ?   F  Y   R  9  

Rultm ed dqujfjirjm  Ludvm pultm ed dqujfjirjm

J9/Rx

J?/Rx

Y/3 [j ejscjluyd df jlardsm sd krdo ulo ludvo fãldo ed rdstrjkkjûl prdsupudstorjo cos

fjcjtoeo. k) Ejiund fos kurvos ed jlejhdrdlkjo ed fms sustjtutms pdrhdktms y kmcpfdcdltorjms

_y

 

pdrhdktms, jlejqud quê poso dl koeo kurvo kml fo UC[ y irjled ul dndcpfm ejhdrdltd of eoem dl kfosd. Y/3 Edijem o qud W y Z sml sustjtutms pdrhdktms fo UC[xy edid sdr kmlstoltd.

[ustjtutms kmcpfdcdltorjms

Dl df kosm ed fms kmcpfdcdltorjms pdrhdktms, fo UC[xy tjdld qud sdr jauof o 2.

[ustjtutms  pdrhdktms JJJ JJ J

2  

JX. Rrïktjkm Rrïktjkm.. Ydsudfvo Ydsudfvo ed hmrco hmrco kforo kforo y mredl mredloeo. oeo. ?.

Corão ds ulo dstuejoltd ed fo TLOG qud dstuejo o tjdcpm kmcpfdtm y euroltd df eão tjdld ul prdsupudstm ed F952.22 fdcpjros poro aostor dl kgmkmfotd ( ijdl x ) y emlos (ijdl y ). ).

_xy

TUx

TUy

TCx

TCy

TCx /Rx

TCy /R  /Ry

Rrdkjm Z ?2

?

?22

82

?22

82

4

;

8

9

?52

?92

72

8

5

7

;

992 992

952

?22

?22

4

4

?2

5

522

972

72

52

5

9

5

4

512

;22

12

92

;.4

?

9

8

4;2

;?2

82

?2

;

2.4

?

1

472

;?2

42

2

9.4

2

2

7

892

;22

52

9

-2.4

-?

:

852

972

92

-?2 -92

?

-?

-9

o) Kmcpfdtd df kuoerm oltdrjmr. i) ½Kuïf ds fo kmcijlokjûl ed kgmkmfotds y emlos qud coxjcjzo fo sotjshokkjûl tmtof ed Corão= Ds edkjr, dlkudltrd fos koltjeoeds ed dqujfjirjm ed  x   d y  pjsto>   pjsto>

dlkudltrd fms prdkjms, vdo dl emled dstïl jcpfãkjtms. ?. Rrjcd Rrjcdro ro kmle kmlejkj jkjûl ûl ed ed dquj dqujfjir fjirjm jm

  TCax / Rx 3 TCay / Ry 9. [daul [dauleo eo kmle kmlejkj jkjûl ûl ed ed dquj dqujfji fjirjm rjm  (Rx)(_x)+(Ry)_y) 3 J

 (92)(7) + (92)(5) 3 952  ?82 + 72 3 952  952 3 952 k) [j df prdkjm ed fos emlos kocijo o F?2.22 ½Kuïf ds fo ludvo kmcijlokjûl ed

kgmkmfotds y emlos qud coxjcjzo fo sotjshokkjûl tmtof ed Corão=  

(Rx)(_x)+(Ry)_y) 3 J

(92)(:) + (?2)(8) 3 952 ?72 + 82 3 952 952 3 952 _xd 3 : uljeoeds _yd 3 8 uljeoeds Koltjeoe ed dqujfjirjm (:,8) Y/3 Fo ludvo kmcijlokjûl ed kgmkmfotds y emlos qud coxjcjzo fo sotjshokkjûl

tmtof ed Corjo sml : kgmkmfotds y 8 emlos.

  

e)  O portjr ed fos kmcijlo kmcijlokjmlds kjmlds ed coxjcjzokj coxjcjzokjûl ûl ed utjfjeoe edf jlkjsm ‖i– y ‖k– kmlstruyo fos rdspdktjvos kurvos ed edcoleo poro df ijdl y .

Ry Kurvo ed edcoleo (5,92)

92

_x

Ry

5 8

92 ?2

(8,?2)

?2

?

9

;

5

4

_y

8

d) [daþl su rdfokjûl, qud tjpm ed ijdlds sml fms kgmkmfotds y fos emlos.

Y/3 [daþl su rdfokjûl, fms kgmkmfotds y fos emlos sml ijdlds sustjtutms. h)

Edcudstrd dl ulo kurvo tãpjko df pultm ed jlhfdxjûl y soturokjûl edf kmlsucm ed emlos. Ds edkjr, ejiund fo kurvo ed utjfjeoe tmtof y corajlof smfm poro df ijdl y  (emlos).

 ;?2  ;22  972  952  ?52  ?22   72   82   52

 

Rultm ed soturokjûl

  92

 

Rultm ed jlhfokjûl

 

 

Ttjfjeoe Umtof

?2   9. 2 -?2 -92

[upmlao qud ul jlejvjeum tjdld ul jlardsm ejspmljifd ed F942.22 qud smfm Ttjfjeoe Corajlof   poro puded kmlsucjr dl Rjzzos (W) y Gociuraudsos (Z), pmr fm qud coxjcjzor   ?

9

:

;

5

4

8

1

7

 

su utjfjeoe ldkdsjto ejstrjiujr su jlardsm dltrd dstms ijdlds fms kuofds tjdldl ul prdkjm ed F42.22 y F94.22, rdspdktjvocdltd. O kmltjluokjûl, sd fd prdsdlto fo toifo ed utjfjeoeds. Ttjfjeoe Umtof gociuraudso (Z)

Koltjeoeds

Ttjfjeoe Umtof Rjzzo (W)

?

12

42

9

?12

;

TCx

TCy

TCx /R  /Rx

TCy /R  /Ry

:2

12 ?22

42 52

?.5 9

9 ?.8

982

?92

:2

;2

;,9

?.9

5

;52

?52

72

92

?.8

2.7

4

5?2

?44

12

?4

?.5

2.8

8

512

?84

82

?2

?.9

2.:

1

492

?12

42

4

?

2.9

[d fd pjed> o) Dlkmltror fo utjfjeoe corajlof poro df ijdl W y df ijdl Z. i) ½Kuïf ed fms ems ijdlds fd adldro coymr sotjshokkjûl of kmlsucjemr= Y/3 Fo pjzzo (Ijdl x) fd adldro coymr sotjshokkjûl of kmlsucjemr edijem o qud fos utjfjeoeds sml coymrds dl kmcporokjûl kml df ijdl ‖Z– k) Dlkudltrd fo koltjeoe ed pjzzos y gociuraudsos qud dqujfjirol of kmlsucjemr y

fms fms jl jlar ards dsms ms rd rdof ofds ds po poro ro df ij ijdl dl W y df ijdl ijdl Z. Ed Edcu cuês êstr trdf dfm m kml kml fos fos ems ems kmlejkjmlds ed dqujfjirjm. ?. Rrjcdr Rrjcdro o kmlej kmlejkjû kjûl l ed dqujfj dqujfjirj irjm m

TCax / Rx 3 TCay / Ry  

     

9. [daul [dauleo eo kmle kmlejkj jkjûl ûl ed ed dquj dqujfjir fjirjm jm (Rx)(_x)+(Ry)_y) 3 J

(42)(5) + (94)(9) 3 942 922 + 42 3 942 942 3 942 e) Dlkudltrd df prdkjm rdfotjvm edf ijdl W dl têrcjlms edf ijdl Z, dxpfjqud df

sjaljhjkoem edf eotm mitdljem. UC[ 3 Rx / Ry 3 -42/94 3 -9 Y/3  Df kmlsucjemr rdlulkjo o 9 uljeoeds ed ‖Z– poro mitdldr ulo uljeoe

oejkjmlof ed ‖W–. Df prdkjm rdfotjvm ed ulo pjdzo (ijdl x) kml rdspdktm o ulo gociuraudso (ijdl y) ds ed 9.

d) Arohjqud fo Fãldo ed Ydstrjkkjûl Rrdsupudstorjo kml su rdspdktjvm jltdrkdptm dl

_x y _y, fo Kurvo ed Jlejhdrdlkjo y kofkufd fo UC[.

 

?. Mitd Mitdld ldrr jla jlard rdsm sms s rd rdof ofds ds JYx 3 J / Rx 3 942/42 3 4, kmmredloeos (4,2) JYy 3 J / Ry 3 942/94 3 ?2, kmmredloeos (2,?2) 9. UC[ 3 Rx / Ry 3 -42/94 3 -9

Df kmlsucjemr rdlulkjo o 9 uljeoeds ed ‖Z– poro mitdldr ulo uljeoe oejkjmlof JY ed ‖W– ?2 : 7 1 8 4 5

_y

; D? (5,9)

9 ?

h)

? 7

9

;

5

4

8

1

[upmlao qud df jlardsm oucdlto dl F?94.22. Dlkudltrd fos ludvos koltjeoeds ed dqujfjirjm edf kmlsucjemr edcmstrïlemfm kml sus ems kmlejkjmlds. Ejiund fo ludvo Fãldo ed Ydstrjkkjûl Rrdsupudstorjo nultm kml fo kurvo ed jlejhdrdlkjo y fo kurvo ed jlardsm-kmlsucm dl fo cjsco arïhjko edf jlkjsm d). ?. Rrjcdr Rrjcdro o kmlej kmlejkjû kjûl l ed dqujfj dqujfjirj irjm m

TCax / Rx 3 TCay / Ry  

     

9. [daul [dauleo eo kmle kmlejkj jkjûl ûl ed ed dquj dqujfjir fjirjm jm (Rx)(_x)+(Ry)_y) 3 J 92

(42)(8) + (94)(;) 3 ;14 ;22 + 14 3 ;14 ;14 3 ;14

?7 ?8 ?5 ?9

J/Ry

?2 Kurvo Jlardsm - Kmlsucm 7

UC[ 3 Rx / Ry 3 9

8 5 lds. Edtdrcjld df tjpm ed ijdl. a) Ejiund fo Kurvo ed Dladf poro ocims ijd D9  D?  9

?

9 :

;

5

4

8

1

7

 

J? 3 942 Ijdl W, _? 3 5 (5,9) Ijdl Z, _9 3 9

Ocims ijdlds lmrcofds.

J

_x

942 ;14

5 8

J? 3 ;14 Ijdl W, _? 3 8 (8,;) [j oucdlto df jlardsm, Ijdl Z, _9  3 ; ouc oucdlt dlto o sus sus kol koltje tjeoed oedss

J

_y

942 ;14

9 ;

edcoleoeos. Ijdl ‖W–

J

Kurvo Dladfs 522 (8,;14)

J9 ;14

J

;42

Ijdl ‖Z–

;22

Kurvo Dladfs

J? 942

522

922

(;,;14)

J9 ;14

(5,942)

;42

?42

;22

?22 J?

42

_?

 

? 1

9

;

_9

5

4

(9,942)

942 922

8 ?42 ?22 42

_?

? 1

IJDLD[ LMYCOFD[  ;.

9

;

_9

5

4

Tlo pdrsmlo kudlto kml F92.22 poro aostor dl ems ijdlds, df ijdl W y df ijdl Z, o ul prdkjm ed F5.22 koeo ijdl, fo sjaujdltd toifo cudstro fo utjfjeoe corajlof qud mitjdld dsto pdrsmlo ed koeo ulm ed dstms ijdlds>  _

?

9

;

5

4

8

1

7

:

?2

TCx TCy

42 ?22

54 72

52 12

;4 42

;2 ;2

94 92

92 ?2

?4 4

?2 2

4 -4

8

 

o) Kofkufd fo Ttjfjeoe Umtof y fo Ttjfjeoe Corajlof pmr Tljeoe Cmldtorjo, poro

ocims ijdlds, utjfjkd ulo kjhro edkjcof dl sus kïfkufms. Koltjeoeds

TCax

TCay

TUx

TUy

TCx/Rx Rx3 5

TCy/Ry Ry3 5

TCx/Rx Rx3 9

? 9

42 :4

?22 ?72

42 54

?22 72

?9.42 ??.94

94.22 92.22

94.22 99.42

; 5 4 8 1 7 : ?2

?;4 ?12 922 994 954 982 912 914

942 ;22 ;;2 ;42 ;82 ;84 ;84 ;82

52 ;4 ;2 94 92 ?4 ?2 4

12 42 ;2 92 ?2 4 2 -4

?2 7.14 1.42 8.94 4.22 ;.14 9.42 ?.94

?1.42 ?9.42 1.42 4.22 9.42 ?.94 2 -?.94

92.22 ?1.42 ?4.22 ?9.94 ?2.22 1.42 4.22 9.42

i) Ku Kuïl ïltos tos uljeo uljeoed edss ed koe koeo o ij ijdl, dl, edi edidrï drï kml kmlsuc sucjr jr dst dsto o pdrsm pdrsmlo, lo, o hjl ed

coxjcjzor su utjfjeoe. Edcuêstrdfm kml fos ems kmlejkjmlds ed dqujfjirjm. ?. Rrjcdr Rrjcdro o kmlej kmlejkjû kjûl l ed dqujfj dqujfjirj irjm m TCax / Rx 3 TCay / Ry  

9. [daul [dauleo eo kmle kmlejkj jkjml ml ed ed dquj dqujfjir fjirjm jm (Rx)(_x)+(Ry)_y) 3 J

(5)(?) + (5)(5) 3 92 5 + ?8 3 92 92 3 92 _xd 3 ? uljeoeds _yd 3 5 uljeoeds k) Aroh Arohjq jqud ud

df dq dquj ujfjfjir irjm jm ed edff kmls kmlsuc ucje jemr mr,, cu cuds dstr trd d fãfãld ldo o ed rdst rdstrj rjkk kkjû jûl l prdsupudstorjo, koltjeoeds ed dqujfjirjm, jlardsms rdofds, kurvo ed jlejhdrdlkjo y UC[xy.

?. Mitd Mitdld ldrr jla jlard rdsm sms s rd rdof ofds ds JYx 3 J / Rx 3 92/5 3 4, JYy 3 J / Ry 3 92/5 3 4,

kmmredloe kmmredloeos os (4,2) kmmredloe kmmredloeos os (2,4) Koltjeoe ed dqujfjirjm (?,5) 9. UC[ 3 Rx / Ry 3 -5/5 3 -? Df kmlsucjemr rdlulkjo o ulo uljeoe ed Z poro mitdldr ulo uljeoe oejkjmlof ed W. _y Kurvo Rrdkjm Kmlsucm 4

D?  5

D9 

KRK

;  FYR9

 FYR? ?

5

4

?2

_x  

e) Kû Kûcm cm ohdk ohdkto toss fo foss ko kolt ltje jeoe oeds ds km kmls lsuc ucje jeos os kuol kuolem em df prdk prdkjm jm edf edf ijdl ijdl W

ejscjluyd y ogmro Rx 3 9. ?. Rr Rrjc jcdr dro o kmle kmlejk jkjû jûl l TCax / Rx 3 TCay / Ry 9. [dauleo kmlejkjûl (Rx)(_x)+(Ry)_y) 3 J

(9)(5) + (5)(;) 3 92 7 + ?9 3 92 92 3 92 _xd 3 5 uljeoeds _yd 3 ; uljeoeds ;. Mitd Mitdld ldrr jla jlard rdsm sms s rd rdof ofds ds JYx 3 J / Rx 3 92/9 3 ?2, kmmredloeos (?2,2) JYy 3 J / Ry 3 92/5 3 4, kmmredloe kmmredloeos os (2,4) 5. UC[ 3 Rx / Ry 3 -9/5 3 -24

Df kmlsucjemr dlulkjo o 2.4 uljeoe ed Z poro mitdldr ulo oejkjmlof ed W. d) Dl df arïhjkm edf jlkjsm k) cudstrd fo ludvo kmcijlokjûl mptjco ed kmlsucm y

trokd fo kurvo prdkjm ‛ kmlsucm. Y/3 Fo ludvo kmcijlokjûl mptjco ed kmlsucm y fo kurvo prdkjm-kmlsu prdkjm-kmlsucm cm yo dstïl rdprdsdltoeos dl df arohjkm edf jlkjsm k) Arohjqud fo kurvo ed edcoleo ed dsto pdrsmlo pmr df ijdl W. Kurvo ed edcoleo poro df ijdl W h)

(x)

_x

Rx

? 5

5 9

(y)

Ejscjlu Ejscjluyd yd df prdkjm, oucdlto fo koltjeoe.

Rx

(5,?)

5

(9,5)

9

?

9

;

5

4

8

_x

 

5.

Fo dcprdso [mpos Fo Koioÿo, [.O uijkoeo dl Xoffd ed Ïladfds, dstï rdofjzole rdofjzolem m ul dstuejm ed cdrkoem poro kmlmkdr fo hoktjijfjeo hoktjijfjeoe e ed vdledr ‖smpo ed aoffjlo (x) y smpo ed rds (y)–. [d dltrdvjstorml o ofaulms turjstos qud vjsjtol dstd culjkjpjm poro por o kml kmlmkd mkdrr su sot sotjsh jshokk okkjûl jûl of kml kmlsuc sucjr jr kuo kuofq fqujd ujdro ro ed fos smp smpos, os, kuy kuyms ms rdsuftoems sd prdsdltol dl df kuoerm sjaujdltd> Koltjeoeds

?

9

;

5

4

8

1

Ttjfjeoe tmtof ed fo smpo ed aoffjlo (x)

74

?72

912

;42

592

572

4;2

Ttjfjeoe tmtof ed fo smpo ed rds (y)

52

14

?24

?;2

?42

?84

?44

TCx

74

:4

:2

72

12

82

42

TCy

52

;4

;2

94

92

?4

?2

TCx/Rx

? .1

?.:

?.7

?.8

?.5

?.9

?.2

TCy/Ry

? .8

?.5

?.9

?.2

2.7

2.8

2.5

Kml fo jlhmrcokjûl oltdrjmr, sd pjed> o) Kmcpfdtd df kuoerm oltdrjmr. i) Kuïf ed fos ems smpos fds austo o fms turjstos, dxpfjqud pmr quê. Y/3  Fms turjs turjstos tos prd prdhjd hjdrdl rdl fo smp smpo o ed aof aoffjfjlo lo pmr pmrqu qud d rdk rdkji jidl dl ulo coymr  sotjshokkjûl o kmcporokjûl kml fo smpo ed rds O coymr kmlsucm, coymr  utjfjeoe. k) Kml ul jlardsm prmcdejm sdcolof ed F;22.22, ½Kûcm edkjejrïl fms turjstos aostor su ejldrm poro coxjcjzor su utjfjeoe= Rjsto> Dlkudltrd prjcdrm fms prdkjms. Rrdkjm smpo ed aoffjlo (x) 3 42 Rrdkjm smpo ed rds (y) 3 94 Tcax / Rx 3 Tcay / Ry

(Rx) (_x) 3 (Ry) (_y) ‛ ? (42) (4) + (94)(9) 3 422 ;22 3 ;22 e) ½Ed kuïltm ds fo utjfjeoe tmtof qud mitdlerï ul turjsto kuolem coxjcjzo su

utjfjeoe= Ttxyd 3 Ttxk + Ttyd Ttxyd 3 592 + 14 3 5:4 d) Rortjdlem ed fo utjfjeoe corajlof pmr uljeoe cmldtorjo y df jlardsm prmcdejm

pmr sdcolo edf jlkjsm k), dlkudltrd fms jlardsms rdofds poro ocios smpos. ?. Mitd Mitdld ldrr jl jlar ards dsms ms rdo rdofd fds s JYx 3 J / Rx 3 ;22/42 3 8, kmmredloeos (8,2) JYy 3 J / Ry 3 ;22/94 3

?9, kmmredloeos (2,?9)

h) ½Kuïf ds df prdkjm rdfotjvm ed fo smpo ed rds dl têrcjlms t êrcjlms ed fo smpo ed aoffjlo=

Dxpfjqud su rdspudsto. UC[ 3 Rx / Ry 3 -42/94 3 -9

Df kmlsucjemr rdlulkjo o 9 uljeoeds ed Z poro mitdldr ulo oejkjmlof ed W.

 

a) Arohjqud df dqujfjirjm edf kmlsucjemr, jlkfuydlem tmems sus dfdcdltms (kurvo

ed jlejhdrdlkjo tãpjko y fãldo ed prdsupudstm).

_y ?9

4

D?  (5,9)

5

9

_x 9

5

8

g) Edtdrcjld fo pdlejdltd ed7fo fãldo ed prdsupudstm y kurvo ed jlejhdrdlkjo.

Dxpfjqud quê sjaljhjko poro ustde df rdsuftoem. Kurvo ed jlejhdrdlkjo>  [d ioso dl fos kmcijlokjmlds pmsjifds ed ijdlds, tjdld pdlejdltd ldaotjvo y lulko sd jltdrkdptol. Fãldo ed prdsupudstm> Dstoifdkd fms fjcjtds ed sus dfdkkjmld dfdkkjmldss ed kmlsucm, fo pmedcms kofkufor kml fms jlardsms y prdkjms ed ocims ijdlds. j)

Kml ul jlardsm prmcdejm pmr sdcolo ed F542.22, coltdljdlem tmem fm edcïs hoktmrds kmlstoltd (kdtdrjs porjius), dl fo cjsco arohjko edf jlkjsm a), arohjqud fo ludvo fãldo ed prdsupudstm.

?. Rrjcd Rrjcdro ro kmle kmlejkj jkjûl ûl ed dqujfji dqujfjirjm rjm TCax / Rx 3 TCay / Ry 9. [dauleo kmlejkjûl (Rx)(_x)+(Ry)_y) 3 J

(42)(1) + (94)(5) 3 542 542 3 542 _xd 3 1 uljeoeds _yd 3 5 uljeoeds

_y

?7

;. Mitd Mitdld ldrr jla jlard rdsm sms s rd rdof ofds ds JYx 3 J / Rx 3 542/42 3 :, kmmredloeos (:,2) JYy 3 J / Ry 3 542/94 3 ?7, kmmredloeos (2,?7) 

?9

F  Y   R    9   

UC[xy 3 Rx / Ry 39

F   Y    R    ?   

D9  5

Kurvo jlardsm kmlsucm

KJ9

D? 

KJ?

9  4

8

1

7

:

_x

 

 n)  O portjr edf ludvm jlardsm, supmlao qud fms turjstos coxjcjzol su sotjshokkjûl

tmtof kmlsucjdlem 1 uljeoeds ed smpo ed aoffjlo. ½Kuïltos uljeoeds ed smpos kmlsucjrïl fms turjstos=, edcuêstrdfm cdejoltd fo dkuokjûl ed prdsupudstm. (Rx)(_x)+(Ry)_y) 3 J

(42)(1) + (94)(5) 3 542 542 3 542 Y/3 Df turjsto kmlsucd 1 uljeoeds ed smpo ed aoffjlo y 5 uljeoeds ed smpo ed rds. 4.

Yocûl Fûpdz, ds ul dcpfdoem ed fo Hjlolkjdro Krdej_ qud edstjlo F:22.22 cdlsuofds poro trolspmrtd0 fms kuofds, aosto trospmrtïlemsd 94 vdkds dl Iusds Yopjejtms y 4 vdkds dl toxjs. Dl vjsto ed fo krdkjdltd jlsdaurjeoe go edkjejem vjonor cïs dl toxj, pmr fm qud ogmro edstjlo F?,922.22 poro trolspmrtd, df kuof, ogmro ogm ro ejs ejstrj trjiu iuyd yd utj utjfjfjzol zolem em 92 vdk vdkds ds df Iu Iuss Yo Yopje pjejtm jtm y ?2 vdk vdkds ds fms tox toxjs js kmfdktjvms. Ejiund fo Kurvo ed Dladfs poro koeo ijdl.

J

Ijdl ‖W– Iusds

?522 J9 ?922

J? 3 F. :22.22 Iusds ijdl ‖x–   Uoxjs ijdl ‖y– 

_ex? 3 94 _ey? 3 4

J9 3 F. :22.22 Iusds ijdl ‖x–

_ex9 3 92

Uoxjs ijdl ‖y– 

  ?222

_ey9 3 ?2

J? :22

722

822

Kurvo Dladfs 522

Kurvo Dladfs

_?

_9

922

J 4 _x

?2

?4

92

94

;2

;4 J9

Iusds ‖x– 3 ijdl jlhdrjmr 

[j oucdlto df jlardsm ejscjluyd su kmlsucm yo qud df ijdl ‖x– ds ul ijdl jlhdrjmr.

Ijdl ‖Z– Uoxjs

?922

?222 J? :22

722

Uoxjs ‖y– 3 ijdl lmrcof.

[j oucdlto df jlardsm, oucdlto df kmlsucm ed koeo ijdl, dstd kosm ijdl ‖y– yo qud fms toxjs sml ul ijdl lmrcof.

822

522

922 _?

  4

_9

?2

?4

92

94

_x  

8.

Fo arïhjko sjaujdltd lms cudstro fos Fãldos ed Ydstrjkkjûl Rrdsupudstorjo ed ulo pdrsmlo dl df Kmlsucm ed Aofmlds ed Aosmfjlo y Ofcudrzms. [j df Jlardsm [dcolof ed dsto pdrsmlo ds ed F9,222.22, of misdrvor fo arïhjko sjaujdltd pmedcms edtdrcjlor fm sjaujdltd> o) Df prdkjm ed ul Aofûl ed Aosmfjlo ds ]]]]]]?22]]]]]]]]]]. i) Fo FY FYR? R? lm lmss jl jlej ejko ko qu qud d df pr prdk dkjm jm ol oltd tdrj rjmr mr ed fms fms ofcu ofcudr drzm zmss dro dro ed

 ]]]]72]]]]].  ]]]]72] ]]]]. k)  ]]]]]52 Fo FYR]]]]]]. 9 lms jlejko qud df ludvm prdkjm ed ul Ofcudrzm ds ed>  ]]]]]52]]]]]]. e) Fo FYR? lms cudstro qud df Rrdkjm Ydfotjvm ed ul Aofûl ed Aosmfjlo dl

têrcjlms ed Ofcudrzms dro ed ]]]]?.94]]]]. d) Fo Rdlejdltd ed fo FYR9 ds ]]]]]ldaotjvo, -2.5]]]]]].