ganit (3) - निर्देशांक ज्यामिति

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File Name : C:\Computer Station\Math-IX (Hindi)\Chapter\Chap-3\Chap-3.PM65

v;k;3

funsZ'kk ad T;kfefr

Whats the good of Mercators North Poles and Equators, Tropics, Zones andMeridian Lines? So the Bellman would cry; and crew would reply They aremerely conventional signs!

(ejosQVj ds mkjh /zqoksa vkSj fo"kqor o`kksa] m".k dfVca/ksa] eaMyksavkSj ;kekskj js[kkvksa esaD;k vPNkbZgS\ blfy, csyeSu us'kksj epk;k gksxk vkSj ukfod ny us mkj fn;k gksxk]^^;sosQoy ijaijkxr fp gSa**A)

LEWIS CARROLL, The Hunting of the Snark

3.1 Hkwfedk

vki ;g i

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funsZ'kkad T;kfefr 61


izR;sd lM+d (ekxZ) ij cus edkuksaij la[;k, vafdr dj nh xbZgSaA vkidks;gk viuhlgsyh osQ edku dk irk yxkuk gSA D;k blosQ fy, osQoy ,d funsZ'k&fcUn q dk Kkr gksuki;kZIr gksxk\ mnkgj.k osQ fy,] ;fn gesa osQoy ;g Kkr gks fd og lM+d 2 ij jgrhgSrksD;k ge mlosQ ?kj dk irk ljyrk lsyxk ldrs gSa\ mruh ljyrk ls ugha ftruhljyrk lsrc tcfd gesanks tkudkfj;kvFkkZr~lM+d dh og la[;k ftl ij mldkedku gS vkSj edku dh la[;k Kkr gksus ij gksrh gSA ;fn vki ml edku ij tkuk pkgrsgSa tkslM+d 2 ij fLFkr gS vkSj ftldh la[;k 5 gS] rkslcls igysvkidks;g irkyxkuk gksxk fd lM+d 2 dkSu&lh gSvkSj rc ml edku dk irk yxkuk gksrk gS ftldhla[;k 5 gSA vko`Qfr 3-1 esaHblh edku dk LFkku n'kkZrk gSA blh izdkj] Pml edku

dksn'kkZrk gS tkslM+d la[;k 7 ij gSvkSj ftldh la[;k 4 gSAII.eku yhft, vki ,d dkxt dh 'khV ij ,d fcUnq yxk nsrs gSa

[vko`Qfr 3-2 (a)]A ;fn ge vkilsdkxt ij yxs fcUnqdh fLFkfr osQ ckjs esaiwNsa] rksvki blsoSQls crk,xs\ laHkor% vki bl iz'u dk mkj bl izdkj nsa % fcUnq dkxt osQvk/sosQ ijh Hkkx esafLFkr gS ;k ^;g Hkh dg ldrsgSafd ;g fcUnqdkxt dh ck;hadksj osQ fudV fLFkr gS** ;k ^^;g fcUnqdkxt dh ck;havksj osQ ijh dksus osQ dkiQhfudV fLFkr gSA D;k ij fn, x, dFkuksa esa ls fdlh Hkh dFku osQ vk/kj ij vki fcUnqdh Bhd&Bhd fLFkfr crk ldrs gSa\ Li"V gS fd mkj ^^ugha** gSA ijUrq] ;fn vki ;gdgsafd fcUnq dkxt dh ck;hadksj ls yxHkx 5 cmnwj gS] rksbllsvkidksfcUnqdh

fLFkfr dk vkHkkl rksgkstkrk gSfiQj Hkh Bhd&Bkd fLFkfr dk irk ughapyrkA FkksM+k cgqrlksp&fopkj osQ ckn vki ;g dg ldrs gSa fd lclsuhps okyh js[kk lsfcUnq9 cmdhnwjh ij gSA vc ge fcUnqdh fLFkfr Bhd&Bkd crk ldrs gSaA

vko`Qfr 3.2
http://schools.aglasem.com/

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blosQ fy, ge nksfu;r js[kkvksa vFkkZr~dkxt dh ck;ha dksj vkSj dkxt dh lclsuhpsokyh j s[kk l s fcUn qdh fLFkfr fu;r djrs gSa [vkoQfr 3-2 (b)]A nwljs'kCnksa esa] ge ;gdg ldrs gSa fd fcUn q dh fLFkfr Kkr djusosQ fy, nks Lora=k lwpukvksadk gksuk vko';dgksrk gSA

vc vki d{kk esa cSBusdh ;kstuk uked fuEufyf[kr f;kdyki dhft,%

f;kdyki1(cSBus dh ;kstuk):lHkh estksadks ,d lkFk [khapdj viuh d{kk esacSBus dh ,d ;kstuk cukb,A izR;sd est dks ,d oxZlsfu:fir dhft,A izR;sd oxZ esaml fo|kFkhZ dk uke fyf[k, ftl ij og cSBrk gSvkSj ftls og oxZ fu:fir djrk gSA

d{kk esa izR;sd fo|kFkhZdh fLFkfr dk Bhd&Bhd fu/kZj.k fuEufyf[kr nks lwpukvksa dhlgk;rk ls fd;k tkrk gSA(i) og LraHk ftlesaog cSBrk @cSBrh gSA(ii)og iafDr ftlesaog cSBrk @cSBrh gSA

;fn vki ml est ij cSBrsgSa tks 5osaLraHk vkSj rhljh iafDr esagS] ftlsvko`Qfr 3-3 esaNkf;r oxZ lsfn[kk;k x;k gS] rks vkidh fLFkfr dks (5] 3) osQ :i esaO;Dr fd;k tk ldrk gS] tgkigyh la[;k LraHk la[;k dks izdV djrh gSvkSj nwljhla[;k iafDr la[;k dksizdV djrh gSA D;k ;g ogh gStks fd (3] 5) gS\ vki viuhd{kk osQ vU; fo|kfFkZ;ksa osQ uke vkSj muosQ cSBus dh fLFkfr;k fy[ksaA mnkgj.k osQ fy,]

;fn lksfu;k pkSFks LraHk vkSj igyh iafDr esacSBrh gS] rksmlosQ fy, S(4] 1) fyf[k,Af'k{kd dh est vkiosQ cSBus dh ;kstuk osQ varxZr ugha vkrh gSA ;gk ge f'k{kd dksosQoy ,d izs{kd gh ekursgSaA

vko`Qfr 3.3

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ij dh ppkZesa vkius;g ns[kk gS fd ,d ry ij j[kh gqbZfdlh oLrq dh fLFkfrnksyac js[kkvksadh lgk;rk ls fu:fir dh tk ldrh gSA ;fn oLrq,d fcUnqgS] rks gesalclsuhps okyh js[kk ls vkSj dkxt dh ck;ha dksj ls fcUnq dh nwjh Kkr gksuk vko';dgksrk gSA ^cSBus dh ;kstuk** osQ laca/ esa gesaLraHk dh la[;k vkSj iafDr dh la[;k dk tkuukvko';d gksrk gSA bl ljy fopkj/kjk dsnwjxkeh ifj.kke gksrs gSavkSj bllsxf.kr dh

funsZ'kkad T;kfefr (Coordinate Geometry)uked ,d vfr egRoiw.kZ'kk[kk dh O;qRifkgqbZA bl v;k; esa] gekjk y{; funsZ'kkad T;kfefr dh oqQN vk/kjHkwr ladYiukvksa lsvkidksifjfpr djkuk gSA blosQ ckjs esa vki foLrkj ls v;;u mPp d{kkvksa esadjsaxsA izkjaHk esaizQkalhlh nk'kZfud v kSj xf.krKjsusndkrsZus bl v;;u dks fodflr fd;k FkkA

oqQN yksx izkr%dky esa fcLrj ij ysVsjguk ilan djrs gSaA;gh vknr l=kgoha 'krkCnh osQ egku izQkalhlh xf.krKjsus ndkrsZdh FkhA ijUrq og vkylh O;fDr ugha Fkk] og ;gle>rk Fkk fd fcLrj ij iM+s&iM+sgh vf/d pru fd;ktk ldrk gSA ,d fnu tcfd og viusfcLrj ij ysVs&ysVsvkjke dj jgk Fkk] mlus ,d ry esa ,d fcUnq dh fLFkfr

dk fu/kZj.k djus ls lacaf/r leL;k dk gy

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vkiosQ ekWMy esa ,d&nwljs dks dkVrh gqbZvusd kWl&LVhV (pkSjkgs) gksldrh gSaA,d fo'ks"k kWl&LVhV nkslM+dksalscuh gS] ftuesals ,dmkj&nf{k.k fn'kk e satkrh gSvkSj nwljh iwoZ&if'pe dh fn'kk esaA izR;sd kWl&LVhV dk funsZ'ku bl izdkj fd;k tkrk

gS% ;fn nwljh lM+d mkj&nf{k.k fn'kk esa tkrh gS vkSj ikpoha lM+d iwoZ&if'pe fn'kkesatkrh gSvkSj ;s ,d kWlx ij feyrh gSa] rc bls ge kWl&LVhV (2] 5) dgsaxsA blhijaijk ls;g Kkr dhft, fd

(i) fdruh kWl&LVhVksadks (4] 3) ekuk tk ldrk gSA

(ii) fdruh kWl&LVhVksadks (3] 4) ekuk tk ldrk gSA

3.2dkrhZ; ifr^la[;k ifr* osQ v;k; esa vkila[;k js[kkosQ ckjs esa i

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vko`Qfr 3.6

okLro esa;sjs[kk, bl izdkj izkIr dh tkrh gSa % nksla[;k js[kk,

yhft, vkSj mUg saXX vkSjY

Ydk uke n snhft,A X

Xdks {kSfrt jf[k,[tSlk fd vko`Qfr3.7(a)esafn[kk;k x;k gS]vkSj bl ij Bhd mlh

izdkj la[;k, fyf[k, t Slk fd la[;k js[kk ij fy[kh tkrh gSaA ;sgh lHkhf;k, vkiY YosQ lkFk dhft,A varj osQoy ;gh gSfd Y Y {kSfrtugha gS] vfirq okZ/j gS[nsf[k, vkfr 3.7 (b)]A

vko`Qfr 3.7

nksuksa js[kkvksa dk la;kstu bl izdkj dhft, fd ;snksjs[kk, ,d&nwljsdksewyfcUnq

ij dkVrh gksa (vko`Qfr 3-8)A {kSfrt js[kk X Xdks x - v{k dgk tkrk gS vkSj okZkjjs[kk YYdksy - v{k dgk tkrk gSA og fcUnq] tgk XX vkSj YY,d&nwljs dks dkVrhgSa] mlsewyfcUn q (origin)dgk tkrk gS vkSj bls Ols izdV fd;k tkrk gSA D;ksafdkukRed la[;k, OX vkSj OYdh fn'kkvksa esafLFkr gSa] blfy,OX vkSjOYdkse'k%

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x-v{k vkSj y-v{k dh /ukRed fn'kk, dgk tkrk gSA blh izdkj] OX vkSjOYdksx-v{k vkSj y-v{k dh e'k% .kkRed fn'kk,dgk tkrk gSA

;gk vki ;g ns[krs gSafd ;snksuksa v{k ry dks pkj Hkkxksaesa foHkkftr djrh gSaA bupkj Hkkxksa dksprqFkk'k(quadrants) (,d&pkSFkkbZ) dgk tkrk gSA OXls okekorZfn'kk esabUgsa I, II, III vkSjIVprqFkk'k dgk tkrk gS (nsf[k, vko`Qfr 3-9)A bl izdkj] bl ry esanksuksav{k vkSj pkjksa prqFkk'k lfEefyr gSaA ge bl ry dks dkrhZ; ry (Cartesianplane);k funsZ'kkad ry(Coordinate plane );k xy-ry (xy-plane )dgrs gSaA v{kksadksfunsZ'kkad v{k(coordinate axes )dgk t krk gSA

vko`Qfr 3.8 vko`Qfr 3.9

vkb, vc ge ;g ns[ksafd xf.kr esa bl ifr dk bruk egRo D;ksagS vkSj ;g fdlizdkj mi;ksxh gksrh gSA vkxsfn;k x;k vkjs[k yhft,] tgk v{kksa dksvkys[k dkxt

(graph paper)ij [khapk x;k gSA vkb, ge v{kksalsfcUnqvksa P vkSj Qdh n wfj;k KkrdjsaA blosQ fy,x-v{k ij yac PMvkSjy-v{k ij yacPNMkfy,Ablh i zdkj] ge yacQR vkSjQSMkyrsgSa] tSlk fd vko`Qfr 3-10 esa fn[kk;k x;k gSA

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vko`Qfr 3.10

vki ikrs gSafd(i) y-v{k ls fcUnqPdh yk afcd n wjh] ftls x-v{k dh /ukRed fn'kk esaekik x;k

gS] PN = OM = 4,dd gSA

(ii) x-v{k ls fcUnqPdh ykafcd nwjh] ftlsy-v{k dh /ukRed fn'kk esaekik x;kgS]PM = ON = 3,dd gSA

(iii) y-v{k lsfcUnq Qdh ykafcd nwjh] ftls x-v{k dh .kkRed fn'kk esaekik x;k

gS]OR = SQ = 6,dd gSA(iv)x-v{k lsfcUnqQdh ykafcd nwjh] ftl sy-v{k dh .kkRed fn'kk eas ekik x;k

gS] OS = RQ = 2,dd gSA

bu nwfj;ksadh lgk;rk ls ge fcUnqvksadk fu/kZj.k fdl izdkj djsa fd dksbZHkze ujg tk,\

ge fuEufyf[kr ijaijkvksa dks ;ku esa j[kdj ,d fcUnq osQ funsZ'kkad fy[krs gSa%

(i) ,d fcUn qdkx -funsZ'kkad (x-coordinate), y-v{k ls bl fcUnqdh ykafcd nwjhgS] ftlsx-v{k ij ekik tkrk gS (tks fdx-v{k dh /ukRed fn'kk esa/ukRed

vkSj x-v{k dh .kkRed fn'kk esa .kkRed gksrh gS)AfcUnq PosQ fy, ;g+ 4gS vkSj QosQ fy, ;g 6gSAx -funsZ'kkad dks Hkqt (abscissa) Hkh dgk tkrk

gSA

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(ii),d fcUnqdk y-funsZ'kkad]x-v{k ls mldh ykafcd n wjh gksrh gS ftlsy-v{k ijekik tkrk gS (tks y-v{kdh /ukRed fn'kk esa/ukRed vkSj y-v{k dh.kkRed fn'kk esa.kkRed gksrh gS)AfcUnq PosQ fy, ;g + 3gSvkSj QosQfy, 2 gSA y-funsZ'kkad dksdksfV (ordinate) Hkh dgk tkrk gSA

(iii)funsZ'kkad ry esa ,d fcUn qosQ funsZ'kkad fy[kr s le; igys x-funsZ'kkad fy[krsgSavk Sj mlosQ ckn y-funsZ'kkad fy[krs gSaA ge funsZ'kkadk sa dksdks"Bd osQ vanj fy[krs

gSaA

vr%PosQ funsZ'kkad(4, 3)gSavkSjQosQ funsZ'kkad( 6, 2)gSaA

;ku nhft, fd ry esa,d fcUnqosQ funsZ'kkad vf}rh; gksrs gSaA blosQ vuqlkjfunsZ'kkad(3, 4) vk Sj funsZ'kkad(4, 3)leku ughagSaA

mnkgj.k1 :uhpsnh xbZ vko`Qfr 3-11 dks ns[kdj fuEufyf[kr dFkuksa dksiwjk dhft,%

(i) fcUnq Bdk Hkqt vkSj dksfV e'k% _ _ _ _ vkSj_ _ _ _gSaA vr%BosQ funsZ'kkad(_ _ _, _ _ _)gSaA

(ii) fcUnq MosQ x-funsZ'kkad vkSjy-funsZ'kkad e'k% _ _ _ vkSj_ _ _gSaA vr%MosQfunsZ'kkad(_ _, _ _)gSaA

(iii) fcUnq L osQx-funsZ'kkad vkSjy-funsZ'kkad e'k%_ _ _ vkSj_ _ _gSaA vr% LosQfunsZ'kkad(_ _, _ _)gSaA

(iv) fcUnq SosQ x-funsZ'kkad vkSj y-funsZ'kkad e'k%_ _ _ vkSj_ _ _gSaA vr% SosQfunsZ'kkad (_ _, _ _)gSaA

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vkfr 3.11

gy:(i)D;ksafd y-

v{k ls fcUnqBdh n wjh 4 ,dd gS] blfy, fcUnq Bdk x-funsZ'kkad

;k Hkqt 4 gksxkA x-v{k ls fcUnqBdh nwjh 3 ,dd gS] blfy, fcUnq B dky-funsZ'kkadvFkkZr~dksfV 3 gksxhAvr% fcUnq BosQ funsZ'kkad (4] 3) gSaA

ij (i)dh Hkk afr%

(ii) fcUnq MosQx-funsZ'kkad vkSj y-funsZ'kkad e'k%3 vkSj 4gSaA vr% fcUnq MosQfunsZ'kkad(3, 4)gSaA

(iii) fcUnq LosQ x-funsZ'kkad vkSjy -funsZ'kkad e'k% 5 vkSj 4gSaA vr% fcUnq LosQfunsZ'kkad(5, 4)gSaA

(iv) fcUnq SosQx-funsZ'kkad vkSj y-funsZ'kkad e'k% 3 vkSj 4gSA vr% fcUnq SosQfunsZ'kkad(3, 4)gSaA

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mnkgj.k2 :vko`Qfr3.12esa v{kksa ij vafdr fcUnqvksaosQ funsZ'kkad fyf[k,%gy:vki ;gk ns[k ldrs gSafd %

(i) fcUnq A, y-v{k ls + 4,dd dhnwjh ij gSvkSj x-v{k ls nwjh 0ijgSA vr% fcUnq Adk x-funsZ'kkad 4gSvkSj y-funsZ'kkad 0gSA blfy, AosQfunsZ'kkad(4, 0)gSaA

(ii) BosQ fun sZ'kkad(0, 3)gSaA D;ksa\

(iii) CosQ fun sZ'kkad( 5, 0)gSaA D;k sa\(iv) DosQ funsZ'kkad(0, 4)gSaA D;ksa\

(v) EosQ funsZ'kkad2

, 03

gSaA D;ksa\

D;ksafdx-v{k dk izR;sd fcUnqx-v{k ls 'kwU; nwjh ij gS] blfy, x-v{k ij fLFkrizR;sd fcUn q dk y-funsZ'kkad lnk gh 'kwU; gksxkA bl rjg] x-v{k ij fLFkr fdlh Hkh fcUnqosQ funsZ'kkad (x, 0)osQ :i osQ gksaxs] tgky-v{k ls fcUnq dh nwjh x gSA blh izdkj]y-v{k ij fLFkr fdlh Hkh fcUnqosQ funsZ'kkad (0,y)osQ :i osQ gksaxs] tgkx-v{k lsfcUnq

dh nwjh ygSA D;ksa\ewyfcUnqOosQ funsZ'kkad D;k gSa\ D;ksafd nksuksa v{kksals bldh nwjh 'kwU; gS] blfy, blosQ

Hkqt vkSj dksfV nksuksagh 'kwU; gksaxsA vr% ewyfcUnqosQ funsZ'kkad (0, 0)gksrsgSaA

ij osQ mnkgj.kksa esa] vkius,d fcUnq osQ funsZ'kkadksaesayxs fpksavkSj ml fcUnqosQprqFkk'k] ftlesaog fLFkr gS] osQ chp osQ fuEufyf[kr laca/ksadh vksj vo'; ;ku fn;kgksxk%

(i) ;fn fcUnq igys prqFkk'k esagS] rks fcUnq (+, +)osQ :i dk gksxk] D;ksafd igykprqFkk'k /ukRedx-v{k vkSj /ukRed y-v{k lsifjc gSA

(ii) ;fn fcUnq nwljsprqFkk'k esa gS] rks fcUnq (, +)osQ :i dk gksxk] D;ksafd nwljk prqFkk'k.kkRedx-v{k vkSj /ukRed y-v{k ls ifjc gSA

(iii) ;fn fcUn q rhljs prqFkk 'k esagS] rks fcUnq (, )osQ :i esagksxk] D;ksafd rhljk prqFkk'k.kkRedx-v{k vkSj .kkRed y-v{k ls ifjc gSA

vkfr 3-12

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(iv) ;fn fcUn qpkSFks prqFkk'k esagS] rksfcUnq (+, )osQ :i esa gksxk] D;ksafd pkSFkk prqFkk'k/ukRed x-v{k vkSj .kkRed y-v{k ls ifjc gS (nsf[k, vko`Qfr 3-13)A

vko`Qfr 3.13

fVIi.kh:,d ry e sa fLFkr ,d fcUnqdh O;k[;k djusosQ laca/ esaij geus ftlifr osQ ckjsesa ppkZ dh gS] og osQoy ,d ijaijk gS ftldksiwjs fo'o esa Lohdkj fd;ktkrk gSA mnkgj.k osQ fy,] ifr esa,slk Hkh gks ldrk gSfd igys dksfV fy[kh tk, vkSjmlosQ ckn Hkqt fy[kk tk,A fiQj Hkh] ftl ifr dk mYys[ k geusfd;k gS mlsiwjk fo'ofcuk fdlh Hkze osQ Lohdkj djrk gSA

iz'ukoyh3.2

1. fuEufyf[kr iz'uksaesalsizR;sd iz'u dk mkj nhft,%

(i) dkrh Z; ry esafdlh fcUnq dh fLFkfr fu/kZfjr djus okyh {kSfrt vk Sj okZ/j js[kkvksaosQ D;k uke gSa\

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(ii) bu nks js[kkvksa lscusry osQ izR;sd Hkkx osQ uke crkb,A(iii)ml fcUnqdk uke crkb, tgk ;s nksjs[kk, izfrPNsfnr gksrh gSaA

2. vko`Qfr 3-14 ns[kdj fuEufyf[kr dksfyf[k,%

(i) BosQ funsZ'kkad

(ii) CosQ funsZ'kkad

(iii)funsZ'kkad (3,5)}kjk igpkuk x;k fcUnq

(iv)funsZ'kkad (2, 4)}kjk igpkuk x;k fcUnq

(v) Ddk Hkqt

(vi)fcUnqHoQh dksfV

(vii)fcUnq LosQ funsZ'kkad

(viii)fcUnq M osQ funsZ'kkad

vko`Qfr 3.14

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3.3ry esa,d fcUnq vkysf[kr djuk tcfd blosQ funsZ'kkad fn, gq, gksa

vHkh rd geusvkiosQ fy, fcUnq [khapsgSa vk Sj vkilsmuosQ funsZ'kkad crkus osQ fy, dgkgSA vc ge vkidks ;g n'kkZ,xs fd ry esabu fcUnqvksa dks fdl izdkj vafdr djrsgSatcfd buosQ funsZ'kkad gesaKkr gksaA bl ize dksge fcUnqdk vkys[ku (plotting thepoint)dgrsgSaA

eku yhft, ,d fcUnq osQ funsZ'kkad (3] 5) gSaA ge bl fcUnq dksfunsZ'kkad ry esavkysf[kr djuk pkgrs gSaA ge funsZ'kkad v{k [khaprsgSa vkSj vius,ddksadk p;u blizdkj djrs gSa fd nksuksa v{kksa ij ,d lsaVhehVj ,d ,dd dks fu:fir djrk gksA fcUnq

(3] 5) osQ funsZ'kkad lsgesa ;g irk pyrk gSfd /ukRed x-v{k ij bl fcUnqdhy-v{k ls nwjh 3 ,dd gSvkSj /ukRedy-v{k ij bl fcUnq dh x-v{k lsnwjh 5 ,ddgSA ewyfcUnqOls izkjaH k djosQ ge /ukRedx-v{k ij 3 ,dd dh fxurh djrsgSa vkSjlaxr fcUnqdks AosQ :i esa vafdr djrsgSaA vc AlsizkjaHk djosQ ge y-v{k

vko`Qfr 3.15

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dh /ukRed fn'kk esapyrsgSavkSj 5 ,dd dh fxurh djrsgSarFkk laxr fcUnqdks PosQ :i esa vafdr djrsgSa(nsf[k, vko`Qfr 3-15)A vki ;gk;g ns[krsgSafd y-v{kls Pdh nwjh 3 ,dd gSvkSj x - v{k ls 5 ,dd gSA vr% Pfopkjk/hu fcUnq dh fLFkfrgSA ;ku nhft, fd Pigys prqFkk'k esa fLFkr gS] D;kasafd PosQ nksuksafunsZ'kkad /ukRed gSaAblh izdkj vki funsZ'kkad ry esa fcUnq Q (5, 4) vkysf[kr dj ldrs gSaA .kkRedy-v{k osQ vuqfn'k x-v{k ls Qdh nwjh 4 ,dd gSftllsfd bldk y-funsZ'kkad 4gS(nsf[k, vko`Qfr 3-15)AfcUnq QpkSFks prqFkk'k esafLFkr gSA D;ksa\

mnkgj.k3 :dkrhZ; ry esa fcUnqvksa(5, 0), (0, 5), (2, 5), (5, 2), (3, 5), (3, 5), (5, 3)

vkSj(6, 1)dk LFkku fu/kZj.k dhft,Agy:1cm = 1,dd ysdj gex-v{k vkSj y-v{k [khaprs gSaA fcUnqvksadh fLFkfr;ksadksvko`Qfr 3-16 esaxgjs fcUnqvksals fn[kk;k x;k gSA

vko`Qfr 3.16

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uksV:ij fn, x, mnkgj.k e sage ;g ns[kr s gSafd(5, 0) vkSj(0, 5)leku ugh agSaA blhizdkj](5, 2) vkSj (2, 5) vyx&vyx fLFkfr;k gSa vkSj (3, 5) vkSj (5, 3)fHkUu&fHkUufLFkfr;kgSaA blh izdkj osQ vusd mnkgj.k ysus ij vki ;g ns[ksaxs fd ;fn xygks] rkssdkrhZ; ry esa (x,y)dh fLFkfr (y,x)dh fLFkfr lsfHkUu gksrh gSA vr% ;fn gefunsZ'kkadksa x vkSj y esavnyk&cnyh djsa] rks (y,x)dh fLFkfr (x,y)dh fLFkfr l sfHkUugkstk,xhA blls;g vFkZfudyrk gSfd (x,y)esax vkSj yosQ e dk dkiQh egRogksrk g SA vr% (x,y)dks fer ; qXe (ordered pair)dgk tkrk gSA fer ;qXe (x,y) fer ;qXe (y,x),;fnxygSvkSj (x,y) = (y,x),;fnx=ygSA

mnkgj.k4 :fuEufyf[kr la[;k&;qXeksa dksdkrhZ; ry ds fcUnqvksa osQ :i esa vkysf[krdhft,A v{kksa ij iSekuk 1 lsaVhehVj =1 ,dd yhft,A

x 3 0 1 4 2

y 7 3 .5 3 4 3

gy:lkj.kh esafn, x, la[;k&;qXeksadksfcUnqvk sa (3, 7), (0, 3.5), ( 1, 3), (4, 4) vkSj(2, 3)ls fu:fir fd;k tk ldrk gSA fcUnqvksa osQ LFkkuksadks vko`Qfr 3.17esaxgjsfcUnqvksalsfn[kk;k x;k gSA

vko`Qfr 3.17

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76 Xkf.kr


f;kdyki2 :nks O;fDr;ksadk ,d [ksyA(vko';d oLrq,& nks dkmaVj ;k flDosQ]xzkiQ isij] vyx&vyx jaxksaosQ nks ikls] eku yhft, yky vkSj gjs jax osQ nks ikls)A

izR;sd dkmaVj dks(0, 0)ij jf[k,A izR;sd f[kykM +h ,d lkFk nksikls isaQdrh gSaA tcigyh f[kykM+h ,slk djrh gS] rks eku yhft, yky ikls ij 3 vkrk gSvkSj gjs iklsij1 vkrk gSA vr% og viuk dkmaVj (3, 1)dh vksj ystkrh gSA blh izdkj] ;fn nwljhf[kykM+h yky ij 2 vkSj gjsij 4 isaQdrh gS] rks og viuk dkmaVj (2, 4)dh vksj ystkrh gSA nwljh ckj isaQdus ij ;fn igyh f[kykM+h yky ij 1 vkSj gjs ij 4 isaQdrh gS]rksog viuk dkmaVj(3, 1)ls(3 + 1, 1 + 4)dh vksj ys tkrh gSA vFkkZr~(3] 1) osQ

x-funsZ'kkad esa 1 tksM+nsrh gSvkSj y-funsZ'kkad esa4 tksM+nsrh gSA

bl [ksy dk ms'; lhek yk?ks fcuk igys (10] 10) ij igqpuk gS] vFkkZr~u rks HkqtvkSj u gh dksfV 10 ls vf/d gks ldrh gSA ;ku jgsfd ,d dkmaVj dh fLFkfr nwljs

dkmaVj dh fLFkfr osQ lkFk laikrh ugha gksuh pkfg,A mnkgj.k osQ fy,] ;fn igyh f[kykM+hdk dkmaVj ml fcUn qdh vksj xfreku gksrk gStgkigys ls gh nwljh f[kykM +h dk dkm aVjfLFkr gS] rks nwljh f[kykM+h dk dkmaVj (0] 0) ij vk tk,xkA ;fn lhek yk?k s fcuk pkypyuk laHko ughagS] rksf[kykM+h lsog ckjh NwV tkrh gSA bl [ksy dks vf/d la[;kesaviuh lgsfy;ksa dks ysdj vki [ksy ldrh gSaA

fVIi.kh:dkrhZ; ry esa fcUnqvksa osQ vkys[ku dh rqyuk oqQN lhek rd le;&nwjh xzkiQ]

Hkqtk&ifjeki xzkiQ] vkfn ftudk v;;u vki fiNyh d{kkvksa esadj pqosQ gSa] tSlhfofHkUu fLFkfr;ksalsdh tk ldrh gSA ,slh fLFkfr esa] x-v{k vkSjy-v{k osQ LFkku ijv{kksa dks t-v{k] d-v{k, s-v{k ;k p-v{k vkfn dg ldrsgSaA

iz'ukoyh 3.3

1. fdl prqFkk'k esa;k fdl v{k ij fcUnq (2, 4), (3,1), ( 1, 0), (1, 2) vkSj (3, 5 )fLFkrgSa\ dkrhZ; ry ij budk LFkku fu/kZj.k djosQ viusmkj lR;kfir dhft,A

2. v{kksaij nwjh dk mi;qDr ,dd ysdj uhpslkj.kh esa fn, x, fcUnqvksadksry ijvkysf[kr dhft,%

x 2 1 0 1 3

y 8 7 1 .25 3 1

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3.4 lkjka'kbl v;k; esa] vkiusfuEufyf[kr fcUnqvksa dk v;;u fd;k gS%

1. ,d ry esa ,d oLrq;k ,d fcUnqdk LFkku fu/kZj.k djus osQ fy, nks ykafcd js[kkvksadh vko';drk gksrh gS ftlesa ,d {kSfrt gksrh gS vkSj nwljh okZ/j gksrh gSA

2. ry dks dkrhZ; ;k funsZ'kkad ry dgk tkrk gSvkSj js[kkvksa dksfunsZ'kkad v{k dgk tkrkgSA

3. {kSfrt js[kk dks x-v{k vkSj okZ/j js[kk dks y-v{k dgk tkrk gSA

4. funsZ'kkad v{k ry dkspkj Hkkxksa esackV nsrsgSa] ftUgsaprqFkk'k dgk tkrk gSA

5. v{kksadsizfrPNsn fcUnq dks ewyfcUnq dgk tkrk gSA6. y- v{k lsfdlh fcUnqdh nwjh dks mldk x-funsZ'kkad ;k Hkqt dgk tkrk gSA lkFk gh]

x-v{k ls fcUnq dh nwjh dks y-funsZ'kkad ;k dksfV dgk tkrk gSA

7. ;fn ,d fcUnqdk Hkqt xgks vkSj dksfV y gks] rks (x,y)dksfcUnq osQ fun sZ'kkad dgk tkrkgSA

8. x-v{k ij ,d fcUnqosQ funsZ'kkad (x, 0)osQ :i osQ gksrsgSavkSj y-v{k ij fcUnqdsfunsZ'kkad(0,y)osQ :i osQ gksrs gSaA

9. ewyfcUnqosQ funsZ'kkad(0, 0)gksrsgSaA

10.,d fcUnq osQ fun sZ'kkad igysprqFkk'k easa (+ , +)osQ :i osQ n wljsprqFkk'k esa (, +)osQ :iosQ] rhljs prqFkk'k esa (, )osQ :i osQ vkSj pkSFksprqFkk'k esa (+, )osQ :i osQ gksrs g Sa]tgk +,d /ukRed okLrfod la[;k dksvkSj ,d .kkRed okLrfod la[;k dksizdVdjr sgSaA

11.;fn xygks] rks (x,y) (y,x)gksxk vkSj ;fn x= ygks] rks(x, y)= (y,x)gksxkA

ganit (3) - निर्देशांक ज्यामिति

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