b{ih$ : àm.s>m°.gxmoed x{dvar x{irb ‘m [wpævh{$À‘m e{dq>À‘m xm{z od^mjmv oÌh$m{u...

$: b{IH$ : àm.S>m°.gXmoed X{dVar X{Irb ‘m [wpæVH{$À‘m e{dQ>À‘m Xm{Z od^mJmV oÌH$m{U odf‘mÀ‘m [brH$S>{ OmD$Z ^yo_VrMm Om{ odæVma AmVm Pmbm Amh{ d hm{V Amh{ ¸‘mMr Wm{S>•‘mV$
bIH$ : ‡m.S>m∞.gXmoed Xd

2

AZwH«$_oUH$m

‡ÒVmdZm ...................................................................... 3

1. emb` ^yo_VrVrb oÃH$mUmMm A‰`mg .................................... 4

2. oÃH$mUm¿`m A‰`mgmgmR>r H$mhr gyÃ .................................... 10

3. oÃH$mUmMr [nao_Vr AmoU jÃ\$i ...................................... 16

4. H$mQ>H$mZ oÃH$mU ........................................................... 19

5. EH$ ^yo_Vr` ‡˚Z : AßXmO ]mßYUr ....................................... 22

6. oÃH$mU ZJarVrb EH$ Hw$Qw>ß] .............................................. 26

7. H$mR>rM VwH$S> AmoU oÃH$mU oZo_©Vr ..................................... 30

8. Q>∞∑gr - ^yo_Vr ............................................................. 35

9. CÀH´$mßV ^yo_Vr¿`m H$mhr emIm ........................................... 50

àñVmdZmyo_Vr odf`m¿`m emYmMm _mZ J´rH$ XemVrb JoUVk wp∑bS>bm oXbm OmVm. À`mMm H$mi B.g. [yd© 5

d eVH$ Amh. [U wp∑bS>¿`m AJmXa H$mhr JoUVk hmD$Z Jb d À`mßZr yo_VrMr H$mhr ‡_ , gyÃ emYyZH$mT>br hmVr. ^maVmV B.g.[yd© 2800 `m H$mimV eyÎ]gyÃ hm gßÒH•$V ^mfVrb JßW obohbm Jbm, À`mV

yo_Vr odf`mMm A‰`mg H$bbm AmT>iVm. Jrg$XemVM wS>m∑gg (Eudoxus) m, B.g.[yd© 1000 H$mimVPmbÎ`m JoUVkmZ yo_VrMr H$mhr gyÃ emYbr hmVr. wp∑bS>Z À`mH$mir _mhrV Agbbr yo_VrMr gyÃgß[moXV H$Í$Z "X Eb_|Q>g≤' (The Elements) `m ZmdmM yo_VrM [wÒVH$ obohb, d V 2000 dfm©[jmOmÒV H$mi H´$o_H$ [wÒVH$ ÂhUyZ bmH$o‡` Pmb. À`mZ yo_VrV H$mhr âhr KmVbr hmVr.

" yo_Vr' `m gßÒH•$V ^mfVrb e„XmMr \$mS>, " y' ÂhUO O_rZ, [•œdr AmoU "o_Vr' ÂhUO _mO_m[Aer Amh. BßJÓrVrb "Geometry'`m e„XmMr \$mS> Geo AmoU metry Aer d À`m e„XmßMm AW©hr VmMAmh. OJmVrb ododY dÒVyßM AmH$ma_mZ, Kamß¿`m, aÒÀ`mß¿`m aMZm, ZJaoZ`mOZ, S>m|Ja, [d©V, Z⁄m,dZÒ[Vr g•ÓQ>r d À`mVrb AmH$ma, AmH•$À`m, Oo_ZrM jÃ\$i, dÒVyßM KZ\$i, À`mßMr bmß]r, ÈßXr, CßMrAmoU OmS>r Aem ododY ‡H$mamßMm A‰`mg yo_VrV g_modÓQ> hmVm.

yo_VrMm odMma q]Xy AmoU afm m XmZ _yb yV gßH$Î[Zmß[mgyZ gwÍ$ hmVm. emiV q]Xy AmoU afm mßMrAmiI H$Í$Z oXbr OmV. EdT>r _mohVr hr [wpÒVH$m dmM `mg [waer Amh.

q]Xy AmoU afm `mßMr AmiI Pmbr H$s ZßVa yo_VrVrb gdm©V gm[r ]h˛ wOmH•$Vr ÂhUO "oÃH$mU'hm . Ia Va, oÃH$mUmMm A‰`mg hr yo_Vr odf`mMr gwadmVM. VrZ wOm d VrZ H$mZ Agbbr hr AmH•$VrH$mhr jÃ\$i ]ßoXÒV H$aV. hr AmH•$Vr oeH$VmZmM Vr ]h˛JwUr AgÎ`mM bjmV V. emb A‰`mgH´$_mVoÃH$mUmMr Or _mohVr Am[Î`mbm hmV d ZßVa hr [wpÒVH$m dmM `mgmR>r C[ w∑V hmV Vr EH$m od^mJmVgßojflV[U oXbr Amh. `m AmH•$VrMr ]arM AmiI H´$o_H$ [wÒVH$mVyZ oXbbr ZgV. Aer _mohVr"oÃH$mUZJargh yo_VrMr ododYVm' `m [wpÒVH$V H$Í$Z X `mMm ‡`ÀZ H$bm Amh. oÃH$mU ZJarVrbH$mhr _mR>Ám aÒÀ`mßZr OmD$Z `m ZJarMr AmiI W H$Í$Z oXbr Amh. `m ZJarVrb bhmZ aÒV, JÎÎ`m,]mi `mß¿`m_YyZ o\$ÈZ _mohVr H$ÈZ KVbr AgVr Va `m [wpÒVH$Mm JßWM Pmbm AgVm, V W Q>mibAmh.

oÃH$mUmgmaª`mM A›` ]h˛ wOmH•$Vr, dVw©i, bß]dVw©i, Aem AZH$ AmH•$À`m Am[Î`m AmiIr¿`mAgVrbM [U À`mßMm A‰`mg W A[ojV Zmhr. Var XIrb m [wpÒVH$¿`m edQ>¿`m XmZ od^mJmV oÃH$mUodf`m¿`m [brH$S> OmD$Z yo_VrMm Om odÒVma AmVm Pmbm Amh d hmV Amh À`mMr WmS>∑`mV AmiIH$ÈZ oXbr Amh. oejH$ AmoU od⁄mœ`m™Zm hr _mohVr AmYr Agb Va À`mß¿`mhr H$Î[ZmßZm [ßI \w$Q>Vrb.EH$ _mÃ bjmV KVb [mohO. H$s, yo_VrMm A‰`mg gd© d°kmoZH$ emImßZm Ï`m[yZ Cabm Amh, À`m_wiemb [mVirdaM Vm [∑H$m ÈOm`bm hdm.

Jmdm od⁄m[rR>mVrb S>m∞.èdßV dmbmdbrH$a AmoU _S>Jmd Wrb Mm°Jwb _hmod⁄mb`mM C[‡mMm ©‡m.am_H•$ÓU IßS>[maH$a `mßZr _bm hdr Agbbr _mohVr [wadbr `m]‘b À`mßM _r Am^ma _mZVm.

Aer [wpÒVH$m oboh `mMm AmJh lr. ZmJe eßH$a _mZ mßZr H$bm. moZo_ŒmmZ _r obohbb hÒVoboIV‡m.dgßV JOmZZ oQ>H$H$a `mßZr dmMyZ À`mV gwYmaUm gwModÎ`m. `m XmKmßM _r _Z:[yd©H$ Am^ma _mZVm.

gXmoed Xd12, ‡gr o]pÎS>®J, _im,

[UOr, (403001), Jmdm

4

1. emb yo_VrVrb oÌH$mUmMm Aä`mg

emiV oVgË`m, Mm°œ`m B`ŒmV AgVmZm od⁄mœ`m™Zm yo_VrMr AmiI H$ÈZ oXbr OmV. q]Xy, afm, oÃH$mU, Mm°H$mZ AmoUdVw©i `m AmH•$À`m H´$_mZ ZßVa oeH$dÎ`m OmVmV. yo_VrMm A‰`mg _ybV: AmH•$À`mß¿`m _m‹`_mVyZ H$bm OmVm. q]XyM dU©Z¡`mbm bmß]r Zmhr, ÈßXr Zmhr AmoU OmS>r Zmhr Ag gd©gmYmaU[U H$b OmV. Var[U AoVe` ]marH$ Q>mH$ Agbbr [p›gbdm[Í$Z Am[U q]XyM oMÃ ‡Vbmda (H$mJX, [mQ>r) H$mT>Vm. h oMÃ Vg AßXmO H$mT>bb AgV H$maU H$mJXmda H$mT>bÎ`mq]Xybm XIrb AJXr gy _ H$m hmB©Zm bmß]r, ÈßXr AmoU OmS>r AgVM. _ZmV Agbb q]XyM oMÃ ‡À`j H$mJXmda aImQ>b H$s VAßXmOM ]am]a AgV. [U Ï`dhmam¿`m [mVirda Am[U Ag oMÃ qH$dm AmH•$Vr q]Xy ÂhUyZ _m›` H$aVm. yo_Vr¿`m A‰`mgmMrgwadmV q]Xy[mgyZ hmV. Am[Î`mbm C[ w∑V AgUmar yo_Vr H´$_mZ Imbr oXbr Amh.

(1) AßVa A AmoU B h XmZ q]Xy EH$mM ‡Vbmda KVb Va À`mß¿`m_Yrb gdm©V H$_r AßVa À`mßZm OmS>UmË`m gai afZ Ï`∑V H$b

OmV. Imbr H$mT>bÎ`m VrZ AmH•$À`m [hm.

A `moR>H$mUr Am[b Ka Amh AmoU B [mer emim Amh Ag g_Oy. Kam[mgyZ emiV Om `mgmR>r dJdJi _mJ©[napÒWVrZwÈ[ Agy eH$VmV.

AmH•$Vr -1 _‹ Ka AmoU emim `mVrb AßVa gdm©V H$_r Amh. AB hm W gai _mJ© Amh. AmH$memV Ogm H$mdim CS>VOmB©b (as the crow flies) Vm hm _mJ© Amh.

AmH•$Vr -2 Ka AmoU emim `mV EImXm Vbmd ]mßYbbm Agb Va _J digm ø`mdm bmJVm, hm _mJ© OmÒV bmß]rMm AgVm.

AmH•$Vr-3 emim AmoU Ka `m_Yrb _mJ© ImßMIiΩ`mßZr Ï`m[bbm Agb Va hm AB _mJ© ø`mdm bmJVm. `Whr A AmoU B_Yrb AßVa gai afdarb AßVam[jm OmÒV AgV.

A

B

A B

A B

AmH•$Vr - 1

AmH•$Vr - 2

AmH•$Vr - 3

5

A [mgyZ B [ ™V Om `mgmR>r Ag oH$Vr _mJ© Agy eH$Vrb ? Ag odMmab Va m ‡˚ZmM CŒma Amh, AZßV ! W VrZM _mJ©XmIdb AmhV. `m _mohVrMm C[`mJ Am[U ZßVa [wT> H$aUma AmhmV.

(2) gai afmgai afgß]ßYr H$mhr _mohVr Am[Î`mbm Amh.

AmH•$Vr-4 ‡Vbmda Agbb A AmoU B q]Xy gai afZ OmS>b AmhV. hr AmH•$Vr [mohÎ`mda Am[Î`m bjmV V H$s‡Vbmda KVbÎ`m XmZ q]XyVyZ OmUmar \$∑V EH$M afm AgV,

AmH•$Vr - 5 AB AmoU CD `m XmZ gai afm EH$_H$mßZm 0 `m q]XyV N>XVmV. `mMm AW© XmZ gai d N>XUmË`m afmßda EH$Mg_mZ (gm_mB©H$) q]Xy 0 AgVm. da q]Xy AmoU afm `mgß]Yr XmZ odYmZ H$br AmhV. EH$mV XmZ q]Xy d EH$ afmAmh Va XwgË`mV XmZ afm d EH$ q]Xy Amh. `m gß]Ymbm EH$_H$mßMr ¤°V (Dual) odYmZ ÂhUVmV. da H$bbrodYmZ AmH•$Vr [mohÎ`mda ghO bjmV VmV. [U À`mß¿`mM C[`mJmZ yo_Vr¿`m H$mhr ‡JV emIm oZ_m©UPmÎ`m AmhV. N>mQ>Ámem odf`mV Ame` H$Yr $& _mR>m oH$Vr AmT>i $&&' `m H$mÏ`[ß∑VrMr AmR>dU _bm hr odYmZdmMyZ Pmbr. [mhˇ `m h oH$V[V Ia Amh !

(3) oÃH$mU ï-Imbr H$mT>bbr AmH•$Vr - 6 Am[Î`m gdm™¿`m [naM`mMr Amh.

`m oÃH$mU AmH•$Vrgß]ßYr AmdÌ`H$ À`m _mohVrMr Zm|X H$Í$.

i) `m oÃH$mUmM A, B, C h oeamq]Xy AmhV.

ii) BC, CA, AB `m VrZ wOm AmhV.

iii) `m VrZ wOmßMr bmß]r AZwH´$_ a, b, c `m oM›hmßZr AmiIbr OmV.

iv) oÃH$mUmM VrZ H$mZ ∠ BAC = ∠ A, ∠ CBA = ∠ B AmoU ∠ ACB = ∠ C Ag AmiIb OmVmV.

v) oÃH$mUmß¿`m VrZ wOmßZr AmVrb ]mOyg H$mhr jÃ\$i ]ßoXÒV H$b OmV. À`mbm oÃH$mUmM jÃ\$i ÂhUVmV.

vi) AmH•$Vr - 6 ∆ ABC qH$dm ∆ BCA, AJa ∆ CAB m oM›hmßZr AmiIbr OmV. H$mhr dim AmH•$Vr ∆ ABC = (a, b,c) Aerhr obohbr OmV. AWm©V (a, b, c), (b, c, a) AmoU (c, a, b) hr gd© oÃHy$Q> EH$M oÃH$mU Xe©dVmV. a, b, c`m gßª`m Zh_rM YZ AgVmV.

A

BAmH•$Vr - 4

A

BAmH•$Vr - 5

C

DO

A

b

C

c

Ba

AmH•$Vr - 6

6

vii) Oa A, B AmoU C h q]Xy gai afV AgVrb Va ABC oÃH$mU V`ma hmB©b [U hm oÃH$mU A‰`mgm¿`m —ÓQ>rZoZÍ$[`mJr AgVm. Aem oÃH$mUmßZm "jwÎbH$' oÃH$mU (Trivial Triangle) ÂhUVmV.

viii) oÃH$mUm¿`m VrZ wOm oXbÎ`m AgVrb Va EH$M EH$ oÃH$mU oZo¸V hmVm. Oa oÃH$mUmM VrZ H$mZ oXb Va _mÃ EH$MEH$ oÃH$mU oZo¸V hmV Zmhr. AmH•$Vr 6 (A) [hm. [U oÃH$mUmM EH$ Hw$Qw>ß] _mÃ oZo¸V H$aVm V. oÃH$mU oZn¸VH$a `mgmR>r AmUIrhr H$mhr dJir odYmZ XVm Vrb. Vr gd© odYmZ W oXbbr ZmhrV.

ix) AmH•$Vr - 6 _‹ ∆ ABC H$mT>bm Amh. VW ∆ PQR, ∆ XYZ hr oM›h gw’m dm[aVm Vrb. JoUV jÃmV [aß[abmÒWmZ Zmhr.

(4) oÃH$mUmßM JwUY_© ï-(A) wOmß¿`m bmß]rda AmYmnaV dJ©dmar ï

oÃH$mUm¿`m gd© wOm g_mZ AgVrb Va Vm oÃH$mU "g_ wO oÃH$mU' AgVm.oÃH$mUm¿`m XmZ wOm g_mZ AgVrb d oVgar wOm Ag_mZ Agb Va Vm "g_o¤ wO oÃH$mU' AgVm.oÃH$mUm¿`m gd© wOm Ag_mZ AgVrb Va Vm "odf_ wO oÃH$mU' AgVm.

(Am) H$mZmda AmYmnaV oÃH$mUmßMr od^mJUr ïoÃH$mUmVrb gd© H$mZ bKwH$mZ AgVrb Va Vm "bKwH$mZ oÃH$mU' AgVm.oÃH$mUmVrb EH$ H$mZ 900 Mm Agb Va Vm "H$mQ>H$mZ oÃH$mU' AgVm. ( W ]mH$sM XmZ H$mZ bKwH$mZ AgVmV.)oÃH$mUmVrb EH$ H$mZ 900 [jm _mR>m Agb Va Vm "odembH$mZ oÃH$mU' AgVm. ( W ]mH$sM XmZ H$mZ bKwH$mZAgVmV.)

(B) i) oÃH$mUm¿`m VrZ H$mZmßMr ]arO 1800 AgV.ii) oÃH$mUmMr EH$ ]mOy dmT>dbr Va hmUmam ]m¯H$mZ amohbÎ`m XmZ AßVJ©V H$mZmß¿`m ]aO]am]a AgVm.iii) g_o¤ wO oÃH$mUmV g_mZ ]mOyß¿`m g_marb H$mZ gmaI AgVmV. `m odYmZmMm Ï`À`mghr gÀ` Amh.iv) g_ wO oÃH$mUmV gd© H$mZ g_mZ (600 M) AgVmV. `m odYmZmMm Ï`À`mghr gÀ` Amh.v) oÃH$mUmVrb gdm©V _mR>Ám H$mZmg_marb wOm, gd© wOmßV _mR>r AgV. `m odYmZmMm Ï`À`mg gÀ` Amh.vi) oÃH$mUm¿`m gd© wOm Oa Z°goJ©H$ gßª`m EdT>Ám bmß]r¿`m AgVrb Va À`mbm "Zm∞_©b oÃH$mU' ÂhUVmV.vii) oÃH$mUm¿`m gd© wOm Z°goJ©H$ gßª`m AgVrb d À`mM jÃ\$i Z°goJ©H$ gßª`m Agb Va À`mbm "ham∞Z oÃH$mU' ÂhUVmV.viii) ¡`m ham∞Z oÃH$mUmV [nao_Vr = 2 (jÃ\$i) Agb Va À`mbm "[na[yU© oÃH$mU' ÂhUVmV.

CXmhaUmW© Imbr H$mT>bbm oÃH$mU A

B C

c=4 b=5

a=3

AmH•$Vr - 7

AmH•$Vr - 6A

7

AmH•$Vr - 7 _Yrb (3, 4, 5) hm oÃH$mU Zm∞_©b, ham∞Z AmoU [na[yU© oÃH$mU Amh. À`mgmR>r (vi), (vii) d (viii) mÏ`mª`m bmJy hmVmV H$m ? V V[mgmd bmJb.

ix) oÃH$mUmßMr AmiI AmUIr H$mhr q]XyZr hmV. Imbr oXbÎ`m VrZ AmH•$À`m [hm ï

AmH•$Vr - 8 _‹ AD, BE d CF `m BC, CA d AB `m afdarb bß] afm AmhV. `m VrZhr bß]afm 0 `m q]XyVo_iVmV. 0 `m q]Xybm "oeambß] gß[mV q]Xy' ÂhUVmV.

AmH•$Vr - 9 _‹ AD, BE d CF `m afm AZwH$_ A, B, d C `mß¿`m H$mZXw mOH$ afm ÂhUVmV. `m VrZhr afmg_mZ q]Xy I _‹ o_iVmV. I `mbm H$mZXw mOH$mßMm "o¤^mOH$ gß[mV q]Xy' ÂhUVmV.

AmH•$Vr - 10 _‹ D, E d F h q]Xy AZwH´$_ BC, CA d AB `m wOmßM _‹`q]Xy AmhV. AD, BE d CF `mafmßZm oÃH$mUm¿`m _‹`Jm afm ÂhUVmV d À`m gd© G `m q]XyV o_iVmV. G `m q]Xybm "_‹`Jm gß[mV q]Xy'ÂhUVmV. oÃH$mUmMm JwÈÀd_‹` G AjamZ XmIodbm OmVm.

da oXbÎ`m VrZ q]Xy O, I d G gß]ßYr AZH$ JwUY_© yo_VrV ‡_ ÂhUyZ og’ H$bbr AmhV.

x) g_Í$[ oÃH$mU ï- XmZ oÃH$mUmß¿`m gd© ]mOy AZwH´$_mZ g_mZ bmß]r¿`m AgVrb Va À`mßZm "EH$Í$[ oÃH$mU'ÂhUVmV.

A

AmH•$Vr - 10

F E

BD

C

G

c2

c2

a2

a2

b2

b2

A

B C

bc

a

AmH•$Vr - 11 P

Q R

qr

p

AmH•$Vr - 12

A

AmH•$Vr - 8

FE

BD C

O900

900

900

A

AmH•$Vr - 9

F

E

B D C

I

A2

A2

∠∠

8

AmH•$Vr -11 _‹` Agbbm ∆ ABC d AmH•$Vr - 12 _‹ Agbbm oÃH$mU `mß¿`m_‹ Oa a = p, b = q AmoUc = r Amh. h oÃH$mU EH$Í$[ AmhV.

darb AmH•$VrVrb (AmH•$Vr - 13 d 14)XmZ oÃH$mU gmaI oXgVmV [U À`mß¿`mV wOm _mÃ Ag_mZ AmhV. Oa`m wOm g_‡_mUmV AgVrb Va ∆ ABC d ∆ PQR g_Í$[ oÃH$mU AmhV.

g_Í$[ oÃH$mUmßgß]ßYr H$mhr ‡_` og’ H$bbr AmhV. Aem ‡H$ma¿`m oÃH$mUmß_wi oÃH$mUZJarVrb H$mhrHw$Qw>ß] oZo¸V H$aVm `VmV. Aem Hw$Qw>ß]mV H$mhr g_mZ JwUY_© AgVmV. CXmhaUmW© ∠A = ∠P, ∠B = ∠Q,

∠C = ∠R. WmS>r Ag_mZVmhr AgVM. CXmhaUmW© ∆ ABC AmoU ∆ PQR `mßMr jÃ\$i Ag_mZ AmhV.

xi) oÃH$mU AmoU dVw©i ï- oÃH$mUmMm A‰`mg ‡JV H$a `mgmR>r H$mhr dVw©i _hŒdmMr R>abr AmhV. À`m[°H$s 5

dVw©imßMm CÎbI Imbr H$bm Amh.

A) AmH•$Vr - 15 [hm. ‡À H$ oÃH$mUm¿`m AmVrb ]mOyg, oÃH$mUm¿`m VrZhr wOm Ò[e©afm AgUma dVw©i H$mT>Vm V.Ag dVw©i \$∑V EH$M AgV. À`mbm oÃH$mUmM "AßVd©Vw©i' Ag ÂhUVmV. `m dVw©imMr oÃ¡`m, oÃH$mUm¿`m wOmß¿`mbmß]rda Adbß]yZ AgV. AßVd©Vw©imMr oÃ¡`m r `m oM›hmZ XmIdVmV.

AmH•$Vr - 16 Vrb dVw©i oÃH$mUmM oeamq]Xy A,B,C _YyZ H$mT>b Amh. m_wi oÃH$mUmZ ]ßoXÒV H$bb jÃ\$i dVw©im¿`mAmV AgV. mbm oÃH$mU ABC M "[nadVw©i' ÂhUVmV. m dVw©imMr oÃ¡`m R m oM›hmZ XmIdVmV d ‡À H$ oÃH$mUmbmAg [nadVw©i EH$M AgV.

A

B C

bc

a

AmH•$Vr - 13P

Q R

qr

p

AmH•$Vr - 14

A

B C

bc

a

AmH•$Vr - 15oÃH$mUmM AßVd©Vw©i

r

o

A

BC

bc

aAmH•$Vr - 16

[nadVw©i

D

R

O

9

¡`m AmH•$VrV AB, BC AmoU CA m[°H$s EH$ wOm dVw©imMm Ï`mg AgVm V|Ïhm m Ï`mgmg_marb H$mZ Zh_rM H$mQ>H$mZAgVm. AmH•$Vr - 16 V 0 hm [nadVw©imMm _‹` q]Xy Amh. W ∠BAC = 900 AgVm.

g_ wO oÃH$mUm¿`m ]m]VrV r AmoU R `mV EH$ odoeÓQ> gß]ßY AmT>iVm; 2r = R Agm hm gß]ßY hm .

AmH•$Vr - 17 _‹ ∆ ABC ¿`m jÃ\$im¿`m ]mha H$mT>bbr VrZ dVw©i oXgVmV. hr dVw©i ∆ ABC ¿`m VrZhr wOmßZmÒ[e© H$aVmV. AmdÌ`H$ VW oÃH$mUm¿`m ^wOm dmT>dÎ`m AmhV. ∠A g_marb dVw©i oÃH$mUm¿`m ^wOmßZm A1, A2, A3 `mq]XyV Ò[e© H$aV. `m dVw©imMr oÃ¡`m ra `m oM›hmZ XmIdVmV. AmVm ∠A g_marb dVw©im]‘b O dU©Z H$b VgM ∠B

d ∠C `m H$mZmg_marb dVw©imß]‘b H$aVm B©b. `m dVw©imß¿`m oÃ¡`m AZwH´$_ _J rb d rc AmhV Ag ÂhUVm B©b.‡À H$ oÃH$mUmbm Aer \$∑V VrZ dVw©i AgVmV. `m dVw©imßZm ∆ ABC oÃH$mUmMr ]ohd©Vw©i ÂhUVmV.

AmH•$À`m - 15,16 AmoU 17 _‹` Am[U EHy$U 5 dVw©i H$mT>br AmhV. À`m dVw©imß¿`m oÃ¡`m r, R, ra , rb AmoU rc moM›hmßZr Ï`∑V H$Î`m AmhV. hr dVw©i ∆ ABC er gß]ßYrV AmhV À`m_wi `m [mMhr oÃ¡`m oÃH$mUm¿`m wOm a, b, c

`mß¿`m bmß„`mßda Adbß]yZ AgVmV.

`moedm` ∆ ABC ¿`m AßVd©V©wimMr oÃ¡`m (r) AmoU ]m¯dVw©imß¿`m oÃ¡`m ra , rb, rc `mß¿`mVhr gß]ßY AmT>iVm.

1 =

1 +

1 +

1Agm hm gß]ßY hm .

r ra rb rc

∆ ABC g_ wO AgÎ`mg AmUIr EH$ gß]ßY AgVm. Vm Agm ra = 3r.

g_ wO oÃH$mUmV ra = rb = rc AgV.

AmH•$Vr - 17-oÃH$mUmMr VrZ ]ohd©V©wi

A

cC1

B1a

A1C

A3B

A2

C2

B2

b

10

2. oÌH$mUmÀ`m Aä`mgmgmR>r H$mhr gyÌ

oÃH$mUmMr [na[yU© AmiI À`mß¿`mgß]ßYr¿`m _mohVrVyZ hmV. AmH•$Vr - 18 [hm.

oÃH$mUm¿`m _mohVrV À`m¿`m wOmß¿`m bmß„`m (a, b, c) H$mZmßM _m[ (∠A, ∠B, ∠C), jÃ\$i (K), oÃH$mU gß]ßYrV 5dVw©imß¿`m oÃ¡`m ( r, R, ra, rb, rc ) `m gßª`mßMm g_mde hmVm. `m gd© gßª`m EH$_H$mßer ZmV ]miJyZ AgVmV. CXmhaUmW©,oÃH$mUm¿`m wOmß¿`m bmß„`m _mhrV AgVrb Va gyÃmß¿`m ghmÊ`mZ A›` gßª`m H$mT>Vm VmV. Aem gyÃmßMr `mXr Imbr oXbbrAmh. hr gyÃ emb A‰`mgH´$_mV oeH$dbr OmV AgVrb qH$dm ZgVrb, À`mß¿`m og’Vm _mhrV AgVrb qH$dm ZgVrb,Am[U hr gyÃ og’ AmhV Ag J•hrV YÍ$. sine, cosine, tangent dJ°a oÃH$mUo_Vr` \$b AmhV.

A) Law of Sines : Sin A=

Sin B=

Sin C

a b c

Am) Law of Cosines :

c2 = a2 + b2 - 2ab cosC,

b2 = a2 + c2 - 2ac cosB,

a2 = b2 + c2 - 2bc cosA,

ÂhUOMcos A =

b2 + c2 - a2,

2 bc

cos B =c2 + a2 - b2

, 2 ca

cos C =a2 + b2 - c2

2ab

B) oÃH$mUm¿`m wOmß¿`m bmß„`m a, b, c AmhV. oÃH$mUmMr [nao_Vr a + b + c = 2s Agb Va AY©[nao_Vr

A

a

bc

CB

∠A

∠B ∠C

AmH•$Vr - 18

11

a + b+ c = s AgV. oÃH$mUmM jÃ\$i K `mgmR>r K = √ s ( s - a ) ( s - b ) ( s - c ) 2

h gyÃ og‹X Amh. `mbmM ham∞ZM gyÃ ÂhUVmV.

(i) AßVd©Vw©imMr oÃ¡`m r = K ; s

(ii) ]m¯ dVw©imß¿`m oÃ¡`m

ra = K , rb =

K , rc =

K

s - a s - b s - c

`m gyÃmßZr o_iVmV.

VgM, Imbrb gyÃ og‹X AmhV.

1 = 1 + 1 + 1 ;r ra rb rc

ra + rb + rc = r + 4R ;

R = abc ;4K

K = 1 bc sin A = 1 ca sin B = 1 ab sin C;2 2 2

a = b cos C + c cos B;

b = c cos A + a cos C;

c = a cos B + b cos A;

tan A = r , tan B = r , tan C = r2 s - a 2 s - b 2 s - c

`m gyÃmßMm C[`mJ H$gm H$bm OmVm À`mgmR>r Imbr H$mhr CXmhaU gmS>dyZ XmIdbr AmhV.

∆ ABC _‹ VrZ wOmß¿`m bmß„`m a, b, c AZwH´$_ m2 - n2, m2 + n2 d 2mn oXbÎ`m AmhV. W m AmoU n `mZ°goJ©H$ YZ gßª`m d m > n Amh. [nao_Vr 2 s = (m2 - n2) + (m2 + n2) + 2mn = 2m (m + n) Amh. gyÃmMm C[`mJH$Í$Z jÃ\$i

K = mn (m2 - n2) og’ hmV. VgM

A

B C

c=2mn

b = m 2 + n2

a = m2 - n2

12

r = K = mn (m2 - n2) = n (m - n) ;s m (m + n)

ra = K

= mn (m2 - n2)

= m (m - n) ;s - a m ( m + n) - (m2 - n2)

rb = K = m (m + n) ;s - b

rc = K = n (m + n) ; hm`.s - c

r, ra, rb, rc `m gd© Z°goJ©H$ gßª`m AmhV. AmUIr

r + ra + rb + rc = 2s = a + b + c, hm`.

da oXbbm oÃH$mU H$mQ>H$mZ oÃH$mU Amh h ghO bjmV V; H$maU W b2 = a2 + c2 Amh. `m CXmhaUm¿`m oZo_ŒmmZAZßV oÃH$mUmßMm gßM Am[Î`mbm o_imbm.

CXmhaU - 2 ïCXmhaU 1 _‹ Oa m = 2 d n = 1 `m gßª`m KVÎ`m Va Am[Î`mbm a = 3, c = 4 d b = 5 o_iVmV d (3,4,5) hm

H$mQ>H$mZ oÃH$mU Amh h Am[Î`mbm _mhrV Amh. `m oÃH$mUmV AmVm r = 1, ra = 2, rb = 6, rc = 3 d R = 5 / 2 h _J ghOog’ H$aVm V d r + ra + rb + rc = 1 + 2 + 6 + 3 = 12 = a + b + c hr g_mZVm og’ hmV.

`m gyÃmß¿`m AmYmamZ `m oÃH$mUmM H$mZ oZo¸V H$aVm VmV. W

K= 1 bc Sin A h gyÃ `mOyZ2

6 = 1 x 5 x 4 Sin A = 10 Sin A ÂhUyZ Sin A = 3 Amh.2 5

VgM Sin C = 4 Ag XmIdVm B©b.5

`m oÃH$mUm¿`m Vr›hr bß]afmßMr, Vr›hr _‹`JmßMr d Vr›hr H$mZo¤^mOH$mßMr bmß]r oXbÎ`m gyÃmß¿`m C[`mJmZ H$mT>Vm`B©b.

CXmhaU - 3

EH$m oÃH$mUm¿`m VrZ ]mOy 2a, 2b AmoU 2c bmß]r¿`m AmhV d À`mß¿`m _‹`Jm f, g, h `m bmß]r¿`m AmhV Va

2 b2 + 2 c2 - a2 = f2 ,

(1) 2 c2 + 2 a2 - b2 = g2 ,

2 a2 + 2 b2 - c2 = h2 ,

hr g_mZVm og’ H$am.

13

`m AmH•$VrV AD hr ∆ ABC Mr _‹`Jm Amh. ÂhUyZ BD = DC = a Amh ∆ ABC gmR>r cosine gyÃ dm[Í$Z

AD2 = AB2 + BD2 - 2 AB. BD cos B = 4c2 + a2 - 4ac cos B = f2

VgMAD2 = AC2 + DC2 - 2 AC . DC cos C = 4b2 + a2 - 4ab cos C = f2

]aOZ2 AD2 = 2f2 = 4c2 + a2 + 4b2 + a2 - 4a (c cos B + b cos C)

= 4c2 + 4b2 + 2a2 - 4a (a)

= 4c2 + 4b2 - 2a2 hm`.ÂhUyZ

f2 = 2b2 + 2c2 -a2 Amh.

`mM [’VrZ amohbÎ`m Xm›hr amer og’ hmVmV. VgM `m (1) _Yrb ametMm C[`mJ H$ÈZ

2f2 + 2g2 - h2 = 9c2

(2) 2h2 + 2f2 - g2 = 9b2

2g2 + 2h2 - f2 = 9a2 ;

`m amer og’ hmVmV. (V[mgyZ [hm). AmVm (1) d (2) m ametMm EH$oÃV odMma H$È. m Xm›hr gyÃmß_‹ H$mhr gmaI[UmbjmV Vm. hm JwUY_© AmVm bjmV KD$. ∆ ABC _‹` a, b, c, m wOmß¿`m bmß„`m _mhrV AgVrb Va m oÃH$mUm¿`m _‹`Jmf, g, h (1) _Yrb gyÃmZ H$mT>Vm VmV. AmVm 2f, 2g, 2h `m wOm Agbbm oÃH$mU H$mT>m. g_mZVm VÀd (Principle of

Symmetry) dm[Í$Z Ag ÂhUVm B©b H$s `m ZdrZ oÃH$mUm¿`m _‹`JmßMr bmß]r 3a, 3b, 3c Agbr [mohO. JoUVmV Agg_mZVm VÀd AZw dmg V. _J V `mOyZ, og’Vm Z XVm H$mhr ZdrZ gß]ßY og’ H$aVm VmV.

g_mZVm VÀdmMm C[`mJ XmId `mgmR>r h CXmhaU oXb Amh. (1) d (2) `m gyÃmßMm C[`mJ H$Í$Z Imbr oXbÎ`moÃH$mUmß¿`m _‹`JmßMr bmß]r H$mT>m.

(i) ( a, b, c, ) = ( 158, 127, 131 )

(ii) ( a, b, c, ) = ( 68, 87, 85 )

`m [´ ZmßMr CŒma Imbr oXbr AmhV.(i)* f = 204, g = 261, h = 255;(ii)* f = 158, g = 127, k = 131;

A

B CD

F E

c

c

b

b

a a

f

g h

14

`m CXmhaUmVrb (1) d (2) hr _‹`JmßMr bmß]r H$mT> `mgmR>r gyÃ ÂhUyZ dm[aVm VmV.

CXmhaU 3, A[mbmoZ`g¿`m ‡_`mAmYma XIrb ghO gwQ>V H$g V [hm.

2a2 + 2b2 = 2d2 + 2 ( c )2 Ag gyÃ dm[am.2

CXmhaU - 4∆ ABC _‹ $AB = 70, BC = 200 d CA = 150 Amh. `m oÃH$mUmß¿`m oeambß]mßMr bmß]r H$mT>m.

`m AmH•$VrV AD hm oeambß] Amh. g_Om x = BD d y = DC ÂhUOM x + y = BD + DC = BC = 200 hm . AmVm

AD2 + y2 = 1502 d

AD2 + x2 = 702 ÂhUyZ dOm]mH$sZ

y2 - x2 = (y + x) ( y - x ) = 200 ( y - x ) = 200 ( 200 - x - x )

= 200 ( 200 - 2x ) = 1502 - 702 = 17600 ÂhUyZ

x = 56 Amh, AmVm

AD2 = AB2 - BD2 = 702 - 562 = 1764 ÂhUyZ AD = 42 hm`.

AD hr bß] afm W YZ[yUm™H$ Amh. `mM [’VrZ amohbÎ`m Xm›hr bß]afmßMr bmß]r H$mT>Vm B©b. d `m Xm›hr gßª`mYZ[yUm™H$ AgVrb. (`m gßª`m H$mT>m.)

CXmhaU - 5Oa m hr Z°goJ©H$ gßª`m Agb Va 2m2 + 1, 2m2 + 2 AmoU 4m2 + 1 m ]mOy AgUmË`m oÃH$mUmM jÃ\$i Z°goJ©H$

gßª`m AgV h og’ H$am.

W oXbÎ`m oÃH$mUmMr [nao_Vr2 s = ( 2 m2 + 1 ) + ( 2 m2 + 2 ) + ( 4 m2 + 1) = 8 m2 + 4 d AY©[nao_Vr.

s = 4 m2 + 2 Amh.

AmVm jÃ\$i H$mT> `mgmR>r Am[U ham∞ZM gyÃ dm[È :

K2 = s ( s - a) ( s - b ) ( s - c )

A

B CD

150

200

70

900x

y

15

= ( 4m2 + 2) ( 2m2) ( 2m2 + 1 ) (1)

= 4m2 (2m2 + 1)2 ÂhUyZ

K = 2m (2 m2 + 1) hm`.

`W K hr gßª`m Z°goJ©H$ Amh. m gmR>r 1, 2, 3,..... Aem AZßV qH$_Vr KD$Z AZßV oÃH$mUmßM jÃ\$i H$mT>Vm B©b.

CXmhaU - 6( a, b, c, ) = ( 13, 14, 15) `m oÃH$mUmM dJi[U XmId `mgmR>r `m oÃH$mUm¿`m oeambß]mß¿`m bmß]r H$mTy>.

oÃH$mUm¿`m bß] afmß¿`m bmß„`m H$mT> `mMr arV CXmhaU - 5 _‹` oXbr Amh. da oXbÎ`m oÃH$mUmV BE hr bß]afm Amh.

`W AE = 5 d EC = 9 Amh h og’ H$aVm B©b. À`m_wi BE = 12 Amh. `m oÃH$mUmV BE, AB, AC, BC `m Mmaafmß¿`m bmß„`m AZwH´$_ (12, 13, 14, 15 ) AmhV. `m Mma YZ[yUm™H$ gbJ (Cousecutive) gßª`m AmhV. oÃH$mU ZJarVhm JwUY_© AgUmam EH$M oÃH$mU Amh.

da oXbÎ`m CXmhaUmdÍ$Z bjmV V H$s oÃH$mUm¿`m VrZ wOm _mhrV AgVrb Va gyÃmß¿`m ghmÊ`mZ oÃH$mUmMr [yU©AmiI H$Í$Z KVm V. oÃH$mUmM VrZ H$mZ oXb Va _mÃ Ag hmV Zmhr. [U g_È[ oÃH$mUmßM Hw$Qw>ß] oZp˚MV H$aVm V.

A

BC

5

12

E14

9

15

13

16

3. oÌH$mUmMr [nao_Vr AmoU jÌ\$i

‡À H$ oÃH$mUmer, [nao_Vr AmoU jÃ\$i `m XmZ gßª`m oZJS>rV AgVmV. `m Xm›hr gßª`m EH$_H$mßer gß]ßYrV AgVmVh À`mgß]ßYr _mhrV AgbÎ`m gyÃmdÍ$Z ghO bjmV `V. oÃH$mUmgß]ßYr kmV Agbbr gyÃ bjmV amhmdrV d gwb^[Udm[aVm `mdrV ÂhUyZ gd©gmYmaU[U gdm™Zr EH$gmaIr oM›h C[`mJmV AmUU JaOM Amh. ÂhUyZ ∆ ABC, yOm a, b, c,

jÃ\$i K [nao_Vr 2s Aer oM›h odf`m¿`m odÌbfUmgmR>r g_mZ Agmdr bmJVmV. AmVm [nao_Vr AmoU jÃ\$i `mßMm _iKmb `mgmR>r Am[U µ (C¿Mma Â y, J´rH$ ^mfVrb EH$ Aja) h oM›h [wT>rb bIZmV `mOy. W

µ = [nao_Vr = 2 s

jÃ\$i K

ÂhUOM

2s = µ K ( µ hr gßª`m YZ Amh. )

`W µ hr gßª`m oÃH$mUm¿`m [nao_VrM jÃ\$imer ‡_mU Xe©dV µ gß]ßYr AmVm VrZ [`m© odMmamV KD$ :

A) µ < 1 Am) µ = 1 AmoU B) µ > 1

CXmhaUmW© : ( 3, 4, 5 ) hm oÃH$mU ham∞Z OmVrMm Amh Ag AJmXa Am[U ÂhQ>b Amh. W

2s = 3 + 4 + 5 = 12, [nao_Vr d s = 6 Amh.

d jÃ\$i K = √ s ( s - a) ( s - b ) ( s - c )

K = √ 6 ( 6 - 3) ( 6 - 4 ) ( 6 - 5)

= √ 36 = 6 Amh.ÂhUyZ

µ =2s

=12

= 2 > 1 hm`.K 6

Va (13, 14, 15 ) `m oÃH$mUmV [nao_Vr 2s = 42 d jÃ\$i 84 AgÎ`mZ

µ =2s

=1

< 1 hm`.K 2

ham∞Z OmVrMm oÃH$mU ÂhUO jÃ\$i d [nao_Vr d ]mOy Z°goJ©H$ gßª`m AgmÏ`mV.

Am[U AmUIr EH$ CXmhaU KD$.

17

g_Om m hr Z°goJ©H$ gßª`m Amh. d oÃH$mUm¿`m wOm 2m2 + 1, 2m2 + 2, 4m2 + 1 `m AmhV. AmVm ham∞ZM gyÃC[`mJmV AmUyZ `m oÃH$mUmM jÃ\$i K H$mTy> .

`m oÃH$mUmMr [nao_Vr

2s = (2m2 + 1) + (2m2 + 2) + (4m2 + 1)

= 8m2 + 4

s = 4m2 + 2.

ham∞Z¿`m gyÃm‡_mU -

K2 = s (s-a) (s-b) (s-c)

= (4m2 + 2) (2m2 + 1) (2m2) (1)

= 4m2 (2m2 + 1)2

d K = 2m (2m2 +1).

W m hr Z°goJ©H$ [U H$mUVrhr gßª`m Amh. AmVm m = 1, 2, 3, 4, 5, 6,..... `m gßª`m `m gyÃmgmR>r `mOÎ`m VaoÃH$mUm¿`m ^wOm ( a, b, c,) [nao_Vr 2s, jÃ\$i K d µ `m gßª`m ‡À H$ oÃH$mUmgmR>r H$mT>Vm Vrb d Imbr oXbb H$mÓQ>H$V`ma H$aVm B©b.

m oÃH$mU n©ÃHy$Q> [nao_Vr jÃ\$i u = [nao_Vr

jÃ\$i

(a, b, c) 2s k

1 ( 3, 4, 5,) 12 6 2

2 ( 9, 10, 17) 36 36 1

3 (19, 20, 37) 76 114 0.66

4 (33, 34, 65) 132 264 0.50

5 (51, 52, 101) 204 510 0.40

6 (73, 74, 145) 292 876 0.30

. . . . .

. . . . .

. . . . .

H$mÓQ>H$ - 1 `m H$mÓQ>H$mgß]Yr H$mhr odYmZ H$aVm VmV.

(A) darb H$mÓQ>H$ m = 7, 8, 9,.....`m gßª`m dm[ÈZ hd Vg dmT>dVm B©b.

(Am) µ ¿`m aH$m›`mVrb gßª`m 2 [mgyZ gwÈ hmVmV [U H´$_mZ H$_r hmV OmVmV. µ hr Zh_r YZ gßª`m Amh À`m_wiµ → 0 (µ hr gßª`m H´$_mZ ey›`mH$S> KgaV OmV) Ag ÂhUVm V.

µ hr gßª`m 2 [jm _mR>r AgV H$m ? Agm ‡ÌZ odMmaVm B©b. mgmR>r (2, 2, 2) m oÃHy$Q>mMm odMma H$È. m oÃH$mUmMr[nao_Vr 6 Amh d jÃ\$i √3 Amh. À`m_wi

18

µ = [nao_Vr = 6 = 2 √3 > 2jÃ\$i √3

EH$m ‡_`mZ Ag XmIdyZ XVm `V H$s 2 √3 hr µ Mr gdm™V _mR>r qH$_V Amh. `m gßª`[mgyZM µ gßª`m KaßJiV ey›`mH$S>OmV. `m ‡_ mZ Ag og’ H$aVm V H$s µ > 2 √3 Aer pÒWVr AgUmam oÃH$mU oÃH$mUZJarV ApÒVÀdmV Zmhr.

darb H$mÓQ>H$ - 1 _‹ (9, 10, 17) m oÃHy$Q>mZ hmUmË`m oÃH$mUmMr [nao_Vr 36 Amh d jÃ\$ihr 36 Amh À`m_wi µ = 1 Amh._J ‡ÌZ oZ_m©U Pmbm H$s gd© oÃH$mUZJarV [nao_Vr AmoU jÃ\$i g_mZ AgUma ham∞Z oÃH$mU AmUIr AmhV H$m ? `m‡ÌZmMm [mR>[wamdm ÏhmB©Q>dW© d o]S>b `m JoUVkmßZr H$bm d À`mßZr emYyZ H$mT>b H$s oÃH$mUZJarV Ag 5 ham∞Z oÃH$mU AmhVH$s ¡`mßMr [nao_Vr AmoU jÃ\$i g_mZ Amh. V Ag (5, 12, 13 ) ( 6, 8, 10) (6, 25, 29) (7, 15, 20) (9,10,17).`m oÃH$mUmßMr [nao_Vr d jÃ\$i H$mTy>Z µ = 1 IamIaM Amh H$m ? V V[mgyZ [hm.

Ag ZdZdrZ ‡ÌZ JoUV gßemYZmV gmIir [’VrZ gVV oZ_m©U hmVmV d ZdrZ o[T>rVrb emÒÃk V gmS>dVmV. `m_wiJoUVmMr ]mJ \w$bV amhV. oÃH$mU AmH•$Vr gß]ßYr Ag gßemYZ ]aM ‡JV Amh [U Mm°H$mZ, [ßMH$mZ, fQ>H$mZ....`m AmH•$À`mß]m]V_mÃ AOyZhr \$ma _mohVr C[b„Y Zmhr.

`m od^mJmV µ `m ‡_mUmgß]ßYr _mohVr KD$Z H$mhr oZÓH$f© H$mT>b AmhV. AmUIr H$mhr ZdrZ ‡_mU Ï`mª`m XD$ZoÃH$mUmgß]ßYr AoYH$ A‰`mg H$aVm B©b. JoUVmV gßemYZ gVV ‡JV hmV AgV.

19

4. H$mQ>H$mZ oÌH$mU

H$mQ>H$mZ oÃH$mUmßMm A‰`mg À`mVrb EH$ H$mZ 900 Mm AgÎ`m_wi ]amM gm[m hmVm. VgM oÃH$mUmVrb amohbb XmZH$mZ EH$_H$mßM H$moQ>H$mZ AgVmV ÂhUO `m XmZ H$mZmßMr ]arO 900 EdT>r AgV. Aem oÃH$mUmßM jÃ\$i 1/2 [m`m x CßMr`m gyÃmZ MQ>H$Z H$mT>Vm V. `m gyÃm¿`m C[`mJmgmR>r oÃH$mUmMr AmH•$Vr [wT>rb ‡_mU Agmdr.

AmH•$Vr - 18 _Yrb oÃH$mUmV AC2 + CB2 = AB2 qH$dm a2 + b2 = c2 h gyÃ og’ H$aVm V. `m gyÃmbm[m`Wm∞Jmag og’mßV Ag ÂhUVmV. hm og’mßV ^maVr` JoUVkmßZm d°oXH$ H$mim[mgyZ _mhrV hmVm Ag S>m∞. E. gm`S>Z]J© `mJoUVkmZ "X AmnaOrZ Am∞\$ _∞W_∞oQ>∑g' `m emY oZ]ßYmV ÂhQ>b Amh. hm Pmbm JoUVm¿`m BoVhmgmMm ^mJ !

(a, b, c) h [m`Wm∞Jmar` oÃHy$Q> Amh `mMm AW© a2 + b2 = c2 h gyÃ gÀ` Amh. `W H$mQ>H$mZmg_marb ]mOy c Amh.H$U© Aem oÃHy$Q>mV oVgË`m OmJda AgVm. m oÃH$mUmVrb c hr gdm©V bmß] ]mOy AgV. (5, 12, 13 ) h [m`Wm∞Jmar` oÃHy$Q>Amh `mMm AW© 52 + 122 = 132 hm . `m ‡_ mMm Ï`À`mg : Oa (a, b, c ) `m oÃHy$Q>mV a2 + b2 = c2 Agb Va hmH$mQ>H$mZ oÃH$mU AgVm. hm Ï`À`mg og’ Amh.

Am[U AmVm

a2 + b2 = c2 (3)

h EH$ g_rH$aU Amh Ag J•hrV YÈ. `mMm AW© a, b, AmoU c `m Mbgßª`m AmhV. `m g_rH$aUmMr gd© CŒma o_imbrVa Am[Î`mbm H$mQ>H$mZ oÃH$mUmßM oÃH$mUZJarVrb EH$ Hw$Qw>ß]M o_ib. AWm©V `m g_rH$aUmMr gd©M CŒma o_idU VgAdKS> Amh. oZXmZ `m[°H$s H$mhr CŒma H$er o_iVrb `mMm Am[U AmVm emY KD$. g_Om m AmoU n `m Z°goJ©H$ gßª`mAmhV. d m > n Amh. AmVm Aghr g_Oy H$s

(i) a = m2 - n2, b = 2mn d c = m2 + n2 Amh.

`m (a, b, c) ¿`m oHß$_Vr AmhV Va

a2 + b2 = ( m2 - n2 )2 + (2 mn )2

A

BCa [m`m

cb CßMr

AmH•$Vr - 18

20

= (m2 + n2 )2

= c2 hm`.

`mMm AW© ( a, b, c) = (m2 - n2, 2mn, m2 + n2) h oÃHy$Q> (3) `m g_rH$aUmM CŒma Amh. Ia Va, (3) _‹ AZßVZ°goJ©H$ gßª`m g_modÓQ> AmhV. W Am[Î`mbm m AmoU n `m m > n `m Z°goJ©H$ gßª`m KVm VmV. Aer oZdS> AZßV[’VrZ H$aVm V. Imbr H$mhr CXmhaU oXbr AmhV.

Oam = 3 d n = 2 Agb Va (a, b, c ) = (5, 12, 13),

m = 4 d n = 3 Agb Va (a, b, c ) = (7, 24, 25),

m = 100 d n = 1 Agb Va (a, b, c) = (9999, 200, 10001).

Aer AJoUV oÃHy$Q> o_idVm Vrb. da H$mT>bbr gd© oÃHy$Q> H$mQ>H$mZ oÃH$mU AmhV. Imbr H$mhr ^Ï` H$mQ>H$mZ oÃH$mUoXb AmhV.

AmVm gßJUH$m¿`m ghmÊ`mZ ^Ï` H$mQ>H$mZ oÃH$mU o_idVm VmV. Imbrb H$mÓQ>H$mV Ag 4 H$mQ>H$mZ oÃH$mU oXbAmhV. (i) _‹ oXbÎ`m gyÃmVyZ Imbrb H$mÓQ>H$ V.

CßMr [m`m H$U© jÃ\$i

m n a b c K

(1) 149 58 18837 17284 25565 162789354

(2) 224 153 26767 68544 73585 617358624

(3) 666 5 443531 6660 443581 1476958230

(4) 406 289 86995 226518 262677 9854271630

H$mÓQ>H$ - 2 H$mQ>H$mZ oÃH$mU

(ii) AmVm ( x - d, x, x + d ) m oÃHy$Q>mMm odMma H$È. h oÃHy$Q> H$mQ>H$mZ oÃH$mU hm `mgmR>r x AmoU d da H$mUVr ]ßYZKmbmdrV h edQ>r R>ady. AmVm

( x - d )2 + x2 = ( x + d )2 ÂhUOM

x ( x - 4d ) = 0 ÂhUOM x = 0 d x = 4d

Oa x = 0 Agb Va oXbb oÃHy$Q> (-d, o, d ) hmB©b [U hm jwÎbH$ oÃH$mU Amh. ÂhUyZ x = o hr oZdS> dJiy. [UOa x = 4d Agb Va nXbb oÃHy Q> ( 3d, 4d, 5d ) Agb W d hr gßª`m YZ AgU JaOM Amh. `m ‡À H$qH$_Vrbm AmVm H$mQ>H$mZ oÃH$mU o_iVm.

(iii) n hr H$mR>brhr Z°goJ©H$ gßª`m Amh. Iw‘ [m`Wm∞JmagZ À`m¿`m JßWmV H$mQ>H$mZ oÃH$mU o_id `mgmR>r

( 2n + 1, 2n2 + 2n, 2n2 + 2n + 1)

`m oÃHy$Q>mMm C[`mJ H$bm hmVm. `W n = 3 Agb Va ( 7, 24, 25) h [m`Wm∞Jmar` oÃHy$Q> o_iV.

(iv) S>m`m\ß$Q>g (Diophantus) `m JoUVkmZ

21

( 2mz, ( m2 - 1)z,, z)m2 + 1, m2 + 1

h [m`Wm∞Jmar` oÃHy$Q> H$mQ>H$mZ oÃH$mU o_id `mgmR>r dm[ab hmV. W m AmoU z `m Z°goJ©H$ gßª`m AmhV.

(v) ( 24, 45, 51 ), ( 20, 48, 52 ), ( 30, 40, 50 ) m VrZ [m`Wm∞Jmar` oÃH$mUmßMr [nao_Vr gmaIr ( 2S = 120)

Amh [U À`mßMr jÃ\$i AZwH´$_ 540, 480, 600 AmhV. Vr Ag_mZ AmhV. h VrZhr ham∞Z oÃH$mU AmhV. `mMm AW©[nao_Vr gmaIr Agbr Var jÃ\$i g_mZ AgVM Ag Zmhr.

(vi) Eb≤ H´$mZH$a `m JoUVkmZ

(2pqt), t (p2 - q2 ), t (p2 + q2) h [m`Wm∞Jmar` oÃHy$Q> mOyZ H$mQ>H$mZ oÃH$mU o_idb. W p AmoU q m gßª`mEH$mM dir odf_ ZgmÏ`mV. d `m gßª`m gm[j _yigßª`m (relatively prime ) AgmÏ`mV. p, q, t m Z°goJ©H$ gßª`mAmhV.

`m od^mJmV H$mQ>H$mZ oÃH$mU o_id `mgmR>r EH$yU 6 gyÃ Z_yX H$br AmhV. `m ‡H$ma¿`m emYmVyZ H$mhr ZdrZ[´˚Z oZ_m©U Pmb. `W Am[U a2 + b2 = c2 `m [m`Wm∞Jmar` ng’mßVmMr MMm© H$br Amh. `mMr [wT>rb [m`arÂhUyZ an + bn = cn, n > 2, (n : Z°goJ©H$ gßª`m) `m g_rH$aUmßMm Ï`m[H$ A‰`mg Pmbm Amh. `mMm W \$∑VCÎbI H$bm Amh.

22

5. EH$ yo_Vr` àý : A§XmO ]m§YUr

oÃH$mUmßMr H$mZmß¿`m AmYmaUrda od^mJUr H$br Va Am[Î`mbm bKwH$mZ oÃH$mU, H$mQ>H$mZ oÃH$mU d odembH$mZ oÃH$mUo_iVmV. H$mUVmhr oÃH$mU `m[°H$s Ò[ÓQ>[U EH$mM od^mJmV [S>Vm. ‡À H$ od^mJmV AZßV oÃH$mU AgVmV. _mOVm UmË`m[U A[nao_V gßª bm JUZr` gßª`m ÂhUVmV. (CXmhaUmW© : gd© Z°goJ©H$ gßª`m). A[nao_V [U _mOVm Z UmË`m gßª`mßZmAJUZr` gßª`m ÂhUVmV. (CXmhaUmW© : gd© gV≤ gßª`m )

Am[U EH$m ‡Vbmda Oa oÃH$mU H$mT>V amohbm Va oÃH$mUmßMr gßª`m AJUZr` Amh Ag bjmV V. da Zm|XbÎ`moÃH$mUmß¿`m ‡À H$ od^mJmV Ag AJUZr` oÃH$mU AgVmV.AmVm ‡À H$ oÃH$mU ‡H$mamMm gßM bjmV KD$. h VrZhr gßMAJUZr` AmhV. [U `m[°H$s H$mUVm gßM _mR>m Amh Ag odMmab Va _mÃ Z_H CŒma XU e∑` hmV Zmhr. EH$mM [´VbmdaH$mT>bÎ`m oÃH$mUmV bKwH$mZ oÃH$mU OmÒV AmhV H$m odembH$mZ oÃH$mU OmÒV AmhV `mM CŒma XU Vg AdKS>M Amh.Var[U JoUVk `m ‡˚Zmer PJS>V amohb. Z_H$ CŒma XVm V Zmhr h Ia ! oZXmZ H$mhr AßXmO KVm B©b H$m ? Agm Xwgam‡ÒVmd [wT> Ambm. d [Q>Vrb Aem yo_Vr¿`m H$mhr aMZm gwModÎ`m JÎ`m. Imbr H$mT>bÎ`m VrZ aMZmda Am[U bj H|$o–VH$È.

(1) EH$m _mR>Ám ‡Vbmda Am[U Imbr doU©bÎ`m Mma AmH•$À`m e∑`Vm ‡Vbm¿`m _Yrb ^mJmda H$mT>m.

C

BAO

AmH•$Vr - 19

C

BAO

AmH•$Vr - 20

C

BAO

AmH•$Vr - 21

C

BA O

AmH•$Vr - 22

23

AmH•$À`m 19 V 22 _‹` 0 _‹`q]Xy Agbbr gmaª`mM oÃ¡`Mr Mma dVw©i H$mT>br AmhV. ‡À`H$ dVw©imV A d B q]XyVyZMmahr dVw©imßZm Ò[e©afm H$mT>Î`m AmhV. `m_wi dVw©imßZm gm_mdUmË`m 4 [≈>`m oXgVmV À`m oH$Vrhr dmT>dVm VmV. ABC

oÃH$mUmMm C hm oeamq]Xy dJdJ˘`m oR>H$mUr KVbm Amh Vm Agm :

(A) AmH•$Vr -19 _‹ C q]Xy dVw©im¿`m [naKmda KVbm Amh. À`m_wi ∠ ACB hm 900 Amh. W C q]Xy [naKmdarbH$mR>bmhr q]Xy Amh. ∆ ACB hm H$mQ>H$mZ oÃH$mU Amh. Aem oÃH$mUmßMr gßª`m AJUZr` AgV.

(Am) AmH•$Vr - 20 _‹ C q]Xy dVw©im¿`m AmV H$mR>hr KVbm Amh. `W ∠ ACB hm odembH$mZ Amh h ghO bjmV V.∆ACB odembH$mZ oÃH$mU Amh. Ag AJoUV odembH$mZ ∆ H$mT>Vm Vrb h ghO g_OV.

(B) AmH•$Vr - 21 _‹ C oeamq]Xy dVw©im¿`m ]mha [U VW H$mT>bÎ`m [≈>rV Amh. C Mr oZdS> H$mR>brhr Amh. AmVm∠ACB H$S> [mohb Va bjmV V H$s ∆ ACB hm bKwH$mZ oÃH$mU Amh. [naUm_r Am[Î`mbm Ag AJoUV bKwH$mZoÃH$mU o_iVrb.

(B©) AmH•$Vr - 22 _‹ C hm oeamq]Xy dVw©im¿`m ]mha AmhM [U Vm AmVm VW H$mT>bÎ`m [≈>r¿`mhr ]mha Amh. ∠ ABC

AmVm odembH$mZ Amh. À`m_wi ∆ ABC hm odembH$mZ oÃH$mU Amh. Ag AJoUV odembH$mZ oÃH$mU H$mT>Vm Vrbh ghO bjmV V. Am[U KVbb ‡Vb gd© ]mOyßZr dmT>dVm B©b.

AmVm Aem‡H$ma o_idbÎ`m VrZ od^mJr` gßMmßMm odMma H$Í$. H$mQ>H$mZ oÃH$mU o_id `mgmR>r ‡Vbmda H$mT>bÎ`mdVw©im¿`m \$∑V [naKmdarb q]Xy Am[Î`mbm [wag AmhV. bKwH$mZ oÃH$mU gßM o_id `mgmR>r dVw©im]mharb [U \$∑V H$mT>bÎ`m[≈>r¿`m AmV Uma q]Xy bjmV ø`md bmJVmV Va odembH$mZ oÃH$mUmßMm gßM dVw©im¿`m AmVrb d dVw©im¿`m AmoU [≈>r¿`m]mharb (S>mÏ`m AmoU COÏ`m ]mOygohV) gd© q]Xy KVm VmV.

W V`ma H$bÎ`m ‡À H$ gßMmV AJoUV oÃH$mU AmhV h Z_yX H$b AmhM. [U H$mhrem AßXmOmZ Am[U Ag ÂhUy eH$VmH$s odembH$mZ oÃH$mUmßMm gßM A›` XmZ gßMm[jm _mR>m Agmdm. da H$bb gßMmM dU©Z Ag odYmZ H$a `mg _mÃ [wag Zmhr.`m dU©ZmVyZ H$mhrgm AßXmO _mÃ Ï`∑V H$aVm Vm. EH$m JoUVVkmZ Va Ag ÂhQ>b H$s oÃH$mU ZJarV gd© odembH$mZoÃH$mUM AgVmV. [U EH$mM yo_Vr¿`m aMZVyZ Ag odYmZ H$aUhr w∑V R>aUma Zmhr. Am[U AmUIr EH$m aMZMm odMma H$Í$.

2) AmUIr EH$ yo_Vr` aMZm ï

A

C

D

E

A A

A B

BB

B

C

E

E

E

D

DD

AmH•$Vr - 23 AmH•$Vr - 24


r r

r

r

C

C

24

da H$mT>bÎ`m AmH•$À`mV A AmoU B q]Xy XmZ g_mZ dVw©imßM _‹`q]Xy AgyZ AB AßVa dVw©im¿`m oÃ¡`m r EdT> Amh. AAmoU B q]Xy_YyZ XmZ g_mßVa afm H$mTy>Z [≈>rÒdÍ$[ AmH•$Vr V`ma H$br Amh. oÃH$mU V`ma H$a `mgmR>r oeamq]Xy C dJdJ˘`moR>H$mUr oZdS>bm Amh. AmVm ‡À H$ AmH•$VrVrb oÃH$mUmH$S> bj XD$.

(A) AmH•$Vr - 23 _‹` C q]Xy Xm›hrhr dVw©imß¿`m AmV oZdS>bm Amh. À`m_wi r > AC d r > BC Amh `mMm AW© ∆ ABC

_‹` r = AB hr gdm©V _mR>r ^wOm Amh. C q]XyMr oZdS> Oer H$br Amh À`mdÍ$Z ∆ ACB hm bKwH$mZ oÃH$mU Amh H$modembH$mZ oÃH$mU Amh h oZo¸V H$aU AdKS> Amh. Vm oÃH$mU Xm›hrhr od^mJmV Agy eH$Vm EdT> oZo¸V H$aVm V.EHy$U ‡Vbm¿`m jÃ\$imVyZ ADBE h jÃ\$i Am[U "AoZo¸V' ÒdÍ$[mM _mZy.

(Am) AmH•$Vr - 24 _‹` oeamq]Xy C hm H$mR>bmhr q]Xy EH$m dVw©im¿`m ]mha Va XwgË`m dVw©im¿`m AmV Amh. À`m_wiAC > r > BC Amh. `mMm AW© ∆ ABC _‹ AB wOm _‹`_ AmH$mamMr Amh. hm odembH$mZ oÃH$mU Amh. AgAJoUV odembH$mZ oÃH$mU AmhV.

(B©) AmH•$Vr - 25 _‹` C hm q]Xy Xm›hr dVw©imß¿`m d oXbÎ`m [≈>r¿`m ]mha Amh. À`m_wi AB = r hr ]mOy oÃH$mU ACB Mrgdm©V bhmZ ]mOy Amh. VgM ∠ ABC hm odembH$mZ Amh d ∆ ABC hm odembH$mZ oÃH$mU Amh.

(B) AmH•$Vr - 26 _‹ C hm H$mR>bmhr q]Xy Xm›hr dVw©imß¿`m ]mha [U oXbÎ`m [≈>r¿`m AmV Amh. Oa C q]Xy D d E `moR>H$mUr AgVm Va ∆ ABC hm g_^wO oÃH$mU Pmbm AgVm. d m oÃH$mUmMm ‡À`H$ H$mZ 600 AgVm. oXbÎ`m oMÃmVC [≈>r¿`m AmV [U Xm›hr dVw©imß¿`m ]mha AgÎ`mZ ∆ ACB hm bKwH$mZ oÃH$mU hmVm. `m ‡H$ma [w›hm Am[Î`mbmAJoUV oÃH$mU o_iVrb.

AmVm m gd© AmH•$VrgmR>r KVbÎ`m ‡VbmM jÃ\$i bjmV KD$ d m jÃ\$imMr od^mJUr oÃH$mUmß¿`m ‡H$mamZwgma H$Í$.`m od^mJUrV odembH$mZ oÃH$mUmM jÃ\$i Iy[M _mR> Amh. AßXmOmZ Ag ÂhUVm B©b H$s odembH$mZ oÃH$mU oZo_©VrM‡_mU _mR> Agmd. EH$ bjmV KVb [mohO H$s AJUZr` gßª`mßMr Aer VwbZm H$aU e∑` Zmhr. hr yo_Vr` aMZm ]mY‡X AmhÂhUyZ Vr oXbr Amh, EdT>M !

19 Ï`m eVH$mV O_©Z JoUVk H∞$›Q>a `mZ "AZßV' `m gßkbm Z_H$m AW© ‡mflV H$Í$Z oXbm. JUZr` d AJUZr` AZßVgßª`mßMm A‰`mg ZßVa dmT>bm. da Zm|Xbb oÃH$mUmßM VrZhr gßM AJUZr` AmhV d V g_mZ (Equivalent) AmhV h gÀ`JoUVmV og’ Amh. W \$∑V VwbZmÀ_H$ jÃ\$imVyZ AßXmO ]mßYbm Amh.

(3) AmUIr EH$ yo_Vr` aMZm ï-Imbr XmZ oÃH$mU AmH•$Vr H$mT>Î`m AmhV. mV ∆ ABC Amh bKwH$mZ oÃH$mU AmoU Xwgam Amh odembH$mZ oÃH$mU.

∆ ABC odemb H$mZ oÃH$mU∠A ï odemb H$mZ

A

B C

O

E

F

A

B CD

E

F

O


∆ ABC bKwH$mZ oÃH$mU

25

darb oÃH$mUmV AD, BE AmoU CF h oeambß] AmhV. V bß]gß[mV q]Xy O _‹ o_iVmV. bKwH$mZ oÃH$mUmV O q]XyoÃH$mUm¿`m AmVrb jÃ\$imV AgVm Va odembH$mZ oÃH$mUmV Vm oÃH$mUm¿`m ]mharb jÃmV AgVm. h odYmZ og’ H$aVm

V. AmVm `m Xm›hr AmH•$VrVrb Imbr oXbÎ`m 4 oÃH$mUmßMm odMma H$Í$. ∆ AOB, ∆ BOC, ∆ COA d ∆$ABC

AmVm

∆ AOB Mm bß]gß[mV q]Xy C Amh,∆ BOC Mm bß]gß[mV q]Xy A Amh.∆ COA Mm bß]gß[mV q]Xy B Amh.∆ ABC Mm bß]gß[mV q]Xy O Amh.

h ghO bjmV V. AmH•$Vr - 27 d 28 _‹` h odYmZ V[mgyZ [hm. WmS>∑`mV A, B, C AmoU O m Mma o]ßXy[°H$s H$mUVhr3 q]Xy oZdS>m d À`mßMm oÃH$mU H$mT>m Va amohbbm q]Xy À`m oÃH$mUmMm bß]gß[mV q]Xy AgVm.

AmH•$Vr - 27 _‹ ∆ ABC hm bKwH$mZ oÃH$mU Amh Va ]mH$sM VrZ odembH$mZ oÃH$mU AmhV.

AmH•$Vr - 28 _‹` ∆ BOC hm bKwH$mZ oÃH$mU Amh Va ]mH$sM VrZ odembH$mZ oÃH$mU AmhV. A›` bß]gß[mV q]Xyß¿`mgßX^m©V AgM odYmZ Xm›hr AmH•$VrVyZ oZXe©Zmg B©b. `mMm AW© ‡À H$ oR>H$mUr 3 odembH$mZ oÃH$mU d EH$M bKwH$mZoÃH$mU V`ma hmVmV. [ohÎ`m XmZ yo_Vr` aMZV odembH$mZ oÃH$mUmßMr gßª`m bKwH$mZ oÃH$mUmß[jm OmÒV Amh Agm OmAßXmO Ï`∑V H$bm hmVm. `m oVgË`m yo_Vr` aMZV hm AßXmO AmVm ]iH$Q> Pmbm Amh.

AemM A›` yo_Vr` aMZm H$Í$Z oÃH$mU ZJarV 75% odembH$mZ oÃH$mU AgmdV Agm AßXmO JoUVVkmßZr Ï`∑VH$bm Amh.

da oXbbr _mohVr hr m odYmZmMr JoUVr` og’Vm Zmhr h _mÃ bjmV R>db [mohO. H$mUr gmßJmd ZdrZ o[T>rVrb EImXmJoUVk darb odYmZ Ia qH$dm ImQ> Amh V ^odÓ`H$mimV Z_H$ R>adrb. ÂhUyZM hm ‡˝ W \$∑V C[pÒWV H$bm Amh.

darb MMm© Ver oZÓ\$i Amh Ag g_Oy Z . yo_Vr¿`m ZdrZ ZdrZ aMZm_YyZ AZH$ ‡H$maMr XS>bbr _mohVr H$erCbJS>V OmV V `W g_Ob. h hr Zg WmS>H$ !

da oXbÎ`m oddMZmgmR>r Imbr Zm|Xbbm gßX © emYoZ]ßY C[`moOV H$bm Amh. À`mV AmUIr H$mhr yo_Vr` aMZm oXÎ`mAmhV.

Guy Richard K. There are three times obtuse angled triangles as there are acute angled tranigles,

Mathametics Magazine, Volume 66, No.3, June 1993 (USA)

26

6. oÌH$mU ZJarVrb EH$ Hw$Qw>§]

_r emiV B`Œmm 7 drV AgVmZm yo_VrV oÃH$mUmM jÃ\$i H$mT> `mg oeH$bm. À`mdir oejH$mßZr ∆ ABC \$˘`mdaH$mT>bm. À`m¿`m wOm 3, 4, 5 Aem hmÀ`m. gyÃm‡_mU `m oÃH$mUmM jÃ\$i 6 h CŒma H$mT>b. oÃH$mUmMr [nao_Vr oeH$dbrhmVrM. Vr 12 Amh h [U bjmV Amb. m oÃH$mUmMr [nao_Vr jÃ\$im¿`m Xwfl[Q> Amh Aghr V ÂhUmb. m JwUdŒm_wi hm gm[moÃH$mU _m¬`m bjmV amohbm hmVm. `m oÃH$mUmM JwUY_© Ag.

(1) `m oÃH$mUm¿`m wOm 3, 4 d 5 bmß]r¿`m AmhV d `m gd© Z°goJ©H$ gbJ gßª`m AmhV;

(2) hm H$mQ>H$mZ oÃH$mU Amh.

(3) `m oÃH$mUmV µ = 2 Amh; µ ÂhUO [nao_Vr AmoU jÃ\$i `mßM JwUmŒma.

(4) `m oÃH$mUmM jÃ\$i 6 Amh d hr gßª`m 3, 4, 5 `m gßª`mßZm gbJ Amh.

_hmod⁄mb`mV [ohÎ`m dfu AmÂhmbm "oÃH$mUo_Vr' ‡H$aUmV jÃ\$i H$mT> `mgmR>r ham∞ZM gyÃ C[ w∑V AgÎ`mM bjmVAmUyZ oXb. (3, 4, 5) `m oÃHy$Q>mbm bm^bbr JwUdŒmm AgUmam AgmM Xwgam oÃH$mU Amh H$m `mMm emY K `mMm _r H$mhrH$mi ‡`ÀZ H$Î`mM AmR>dV. `mgmR>r MmMUr [’V dm[aV hmVm. [U Agm oÃH$mU _bm o_imbm Zmhr.

1955 gmbr Ama.[r.[Î[ `mZ [na[yU© oÃH$mUmMr Ï`mª`m H$br Vr od^mJ-2 _‹ oXbr Amh. À`mZ Agm Xwgam [na[yU©oÃH$mU emYyZ H$mT> `mM JoUVkmßZm AmdmhZ H$b. 1956 gmbr EZ≤.O.\$mBZ `m JoUVkmZ (3, 4, 5) hm oÃH$mU ZJarVEH$_d [na[yU© oÃH$mU AgÎ`mM og’ H$b. MmMUr [’VrMr _`m©Xm, À`mßMm emY oZ]ßY dmMÎ`mda _m¬`m bjmV Ambr.

EZ≤. O. \$mBZ `mß¿`m og’V_wi AmVm `m oXeZ emY KU Wmß]b Ag dmQ>b hmV. [U 1973 gmbr EM≤.S>„by Jm°ÎS> `mZ `memY_moh_bm H$bmQ>Ur oXbr d oÃH$mUm¿`m A‰`mgmbm dJirM JVr o_imbr. À`mßZr da Zm|XbÎ`m JwUY_© mXrVrb "Vm oÃH$mUH$mQ>H$mZ oÃH$mU Agbm [mohO' h ]ßYZ H$mTy>Z Q>mH$ `mM gwMdb. AmVm \$∑V Imbrb XmZM JwUY_© Agbb oÃH$mU emY `mMH$m_ gwÍ$ Pmb. V Ag :

(1) oÃH$mUm¿`m VrZ wOm gbJ Z°goJ©H$ gßª`m AgmÏ`mV;

(2) oÃH$mUmM jÃ\$i K hr Z°goJ©H$ gßª`m Agmdr.

(1, 2, 3) hm oÃH$mU bjmV KVm `B©b [U Vm jwÑH$ oÃH$mU Amh. ÂhUyZ da Zm|XbÎ`m Xm›hr A[jm [yU© H$aUmam(3, 4, 5) hm [ohbm oÃH$mU Amh. À`mM jÃ\$i 6 Amh. MmMUr [’VrZ (13, 14, 15) d (51, 52, 53) h XmZ oÃH$mU dAZwH´$_ À`mßMr jÃ\$i 84 d 1170 o_imb. m ‡`ÀZmVyZ bjmV Amb H$s Ag AmUIrhr oÃH$mU AgmdV. [U MmMUr [’VrZAg oÃH$mU o_iU AdKS> Pmb. Z_H$ JoUVr` gyÃ hmVmer Amb VaM AmUIr oÃH$mU o_iVrb Ag bjmV Amb. [UAIarbm V gyÃ JmÎS> mßZm o_imb. V bjmV `mgmR>r o_imbÎ`m _mohVrM H$mÔ>H$ obhˇ. VgM gm rgmR>r Zdr oM›hÏ`dÒWmKD$. Am[Î`mbm VrZ gbJ Z°goJ©H$ gßª`m hÏ`mV `mgmR>r

27

(a, b, c) = (un- 1, un, un + 1) Ag g_Oy. VgM Am[Î`mbm AJmXaM (1, 2, 3), (3, 4, 5), (13, 14, 15),(51, 52, 53) hr oÃHy$Q> _mhrV AmhV. À`mMr jÃ\$i 0, 6, 84, 1170 _mhrV AmhV. hr _mohVr Imbrb H$mÔ>H$ - 3 _‹

oXbr Amh. Kn h oM›h n `m OmJda AgUmË`m oÃH$mUmM jÃ\$i Amh Ag g_Oy.

n a = un - 1 b = un C = un + 1 Kn eam

0 1 2 3 0 jwÑH$ oÃH$mU

1 3 4 5 6

2 13 14 15 84

3 51 52 53 1170

4 ? ? ? ?

5 : : : :

H$mÔ>H$ (3)

h H$mÓQ>H$ (3) AmVm Am[Î`mbm dmT>dV ›`m`M Amh. EH$m JoUVkmZ H$mÔ>H$ (3) Mm gmVÀ`mZ [mR>[wamdm H$bm d H$mÔ>H$mVrb

gßª`mßMm ZmVgß]Y emYyZ H$mT>bm. À`m¿`m bjmV Amb H$s un `m aH$m›`mVrb gßª`m

un = (2 + √3)n + (2 - √3)n, n = o, 1, 2, 3,.... (4)

(4) `m gyÃmZ o_iVmV À`m Aem :

n = 0 u0 = (2 + √3)0 + (2 - √3)0 = 1 + 1 = 2

n = 1 u1 = (2 + √3)1 + (2 - √3)1 = 2 + 2 = 4

n = 2 u2 = (2 + √3)2 + (2 - √3)2 = 14

n = 3 u3 = (2 + √3)3 + (2 - √3)4 = 52

un `m gßª`mVyZ 1 hr gßª`m dOm H$br AmoU 1 gßª Z dmT>dbr H$s un - 1 d un + 1 o_iVmV. AmVm oÃH$mUm¿`m VrZ

gbJ wOm o_imÎ`m H$s ham∞Z gyÃ dm[Í$Z Am[Î`mbm jÃ\$i Ko, K1, K2, K3 o_iVmV. AmVm H$mÔ>H$ - 3 dmT>dVm `V hbjmV Amb. CXmhaUmW© :

n = 4, u4 = (2 +√3)4 + (2 -√3)4 = 194, ÂhUyZ u4 - 1=193 AmoU u4 + 1 = 195 hm . VgM ham∞Z gyÃmZ (193,

194, 195) `m oÃH$mUmM jÃ\$i 16296 Amh. `m arVrZ u5, u6, u7.... `m gßª`m H$mT>Vm VmV d H$mÔ>H$ (3) dmT>d `mMr,Z gß[Umar dmQ> _mH$ir hmV. Aem arVrZ odÒVmnaV H$mÔ>H$ - 3 h oÃH$mUZJarVrb EH$ Hw$Qw>ß] Am[Î`m AmiIrM Pmb Amh.

H$mÔ>H$ - (3) _‹ Kn hr gßª`m dJmZ dmT>V OmV Ag ghO bjmV V. CXmhaUmW©,

28

n = 20 gmR>r u20 = (2 + √3)20 + (2 - √3)20 = 275758382274 hr 12 AßH$s gßª`m o_iV. AmVm u20 - 1 d u20 +

1 obohVm `B©b. [U K20 hr gßª`m oH$Vr _mR>r Agb V ghO bjmV `B©b. n Oa bmIm¿`m[jm _mR>r Agb Va Kn Mr gßª`m_mR>m gßJUH$ dm[Í$Z H$mT>mdr bmJb. JoUVkmßZr Aer AmUIr H$mhr Hw$Qw>ß] AerM emYyZ H$mT>br AmhV. CXmhaUmW© un - 2, un,

un + 2 hm oÃH$mU AgmM A‰`mg H$Í$Z g_OyZ KVm B©b d AJoUV oÃH$mU obohVm Vrb.

da oXbb gyÃ (4) Vg g_O `mg gm[ Amh [U n hr gßª`m _mR>r hmV Jbr H$s (2 ± √3)n hr gßª`m ‡À`j H$mT>U _mÃ

AdKS> hmV OmV. Jm°ÎS> m JoUVkmZ H$mÔ>H$ - 3 _Yrb un aH$m›`mV UmË`m gßª`mßMm A‰`mg H$bm. À`m¿`m bjmV Amb H$s`m gßª`m Imbrb gyÃm¿`m AmYma obohVm VmV. V Ag :

un + 2 = 4 un + 1 - un, uo = 2, u1 = 4 n = 2, 3, 4, 5,... (5)

Am[U H$mhr un gßª`m H$mT>Î`m AmhV À`m AmVm (5) Mm C[`mJ H$Í$Z V[mgyZ KVm Vrb. CXmhaUmW©

n = 1, u3 = 4u2 - u1 = 4 (4) - 2 = 14,

n = 2 u4 = 4u3 - u2 = 4 (14) - 4 = 52.

(5) `m gyÃmV un + 2 hr gßª`m À`m[ydu AmbÎ`m un + 1 AmoU un da Adbß]yZ Amh. d hr `mOZm AÏ`mhV MmbUma Amh.JoUVm¿`m A‰`mgmV Aem ÒdÍ$[mMr gyÃ AZH$ oR>H$mUr VmV. `m ‡H$ma¿`m gyÃmßZm "[wZamdoV©V gyÃ' (Recurrence

Relations) Ag ÂhUVmV. (5) h Aem ‡H$maM gyÃ Amh.

jÃ\$im¿`m aH$m›`mV kn `m gßª M [wZamdoV©V gyÃ

(6) Kn + 2 = 4 Kn + 1 - Kn, K0 = 0, K1 = 6; n = 2, 3, 4, 5,... Amh. CXmhaUmW©, Am[Î`mbm Ko d K1 _mhrV Amh.AmVm gyÃ (6) Mm C[`mJ H$Í$Z K2 H$mTy>. `W n = o KD$Z

K2 = 14 K1 - K0 = 14 (6) - 0 = 84 `B©b. H$mÔ>H$mV K2 = 84 hr gßª`m oXbbr Amh. K2 d K1 _mhrV Pmb H$s K3H$mT>Vm B©b. hr gmIir AIßoS>V AmhM.

AmVm[ ™V oÃH$mU ZJarVrb m Hw$Qw>ß]m¿`m wOm AmoU jÃ\$i mßMm odMma Pmbm. oÃH$mUmMr AmUIr AmiI hm `mgmR>r

r, ra, rb, rc, R `m gßª`mhr _mhrV H$Í$Z ø`mÏ`m bmJVmV. À`m o_id `mgmR>r bmJUmar gyÃ od^mJ - 2 _‹ oXbr AmhV. Vr`mOyZ `m Hw$Qw>ß]mVrb Aer dmT>rd gßª`mÀ_H$ _mohVr H$mT>Vm B©b. À`mgmR>r Oa r, ra, rb, rc, R `mgmR>r [wZamdoV©V gyÃ

o_imbr Va gd© oÃH$mUmßMr Aer dmT>Vr AmiI gwb^[U hmV OmB©b. hr gd© gyÃ W oXbbr ZmhrV [U m gßª`mßM EH$ H$mÔ>H$

(4) _mÃ Imbr oXb Amh. Ia Va, H$mÔ>H$ - (3) ¿`m COÏ`m ]mOybm H$mÔ>H$ - (4) OmS>b Va `m Hw$Qw>ß]mVrb gd© oÃH$mUmßMr_mohVr n, un - 1, un, un + 1, kn, r, ra, rb, rc, R `m ÒdÍ$[mV EH$oÃV B©b.

29

n rn rn a rn b rn c R

0 0 0 0 0 0

1 1 œ 3 2 52

2 4 6 12 21 652 8

3 15 14 45 130 9013 30

4 56 234 168 1164 125455 7 12

5 209 679 627 6878 174254 11 418

. . . . .

. . . . .

. . . . .

H$mÔ>H$ - 4 AIßS>[U dmT>dVm `B©b. H$mÔ>H$ - 4 _‹` rn AmoU rnb `m gd© Z°goJ©H$ gßª`m Amh. oZXmZ `m aH$m›`mVrbgßª`mM [wZamdoV©V gyÃ H$mT>Vm V H$m ? ‡`ÀZ H$am.

oÃH$mUZJarVrb EH$mM Hw$Qw>ß]mMr Am[U _mohVr H$mÔ>H$ - 4 ÒdÍ$[mV gmXa H$br Amh. mgß]ßYr EH$ _hÀdmMr _mohVr WZ_yX H$amdr bmJb. `m gd© [na_ gßª`m AmhV. Ag π$oMVM KS>V.

`m oÃH$mUmßMr AmUIr _mohVr o_idVm B©b. oÃH$mUmV _‹`Jm, oeambß] d H$mZo¤^mOH$ AgVmV. m gd© afmß¿`m bmß„`mH$mT>Vm U e∑` Amh. hr _mohVr W oXbbr Zmhr.

a + b + c = 2K

`m g_rH$aUmM CŒma (3, 4, 5) hm oÃH$mU Amh `mMm CÑI [ydu H$bm Amh. AmVm

a + b + c = 3K

Ag g_rH$aU oXb Va À`mM CŒma H$mT>Vm B©b H$m ? `mMm AW© [nao_Vr jÃ\$im¿`m 3 [Q> AgUmam oÃH$mU Amh H$m ?`m ‡˝mM CŒma Amh : Agm H$mUVmhr oÃH$mU ApÒVÀdmV Zmhr.

H$mÔ>H$ - (4)

30

7. H$mR>rM VwH$S> AmoU oÌH$mU oZo_©Vr

EH$ bmß] H$mR>r AJa [≈>r ø`m. `m H$mR>rMr bmß]r Z°goJ©H$ gßª`m n Agmdr. oVM 3 ^mJ H$Í$. `mgmR>r H$mR>r XmZ oR>H$mUrH$m[mdr bmJb. H$mR>rM VwH$S>hr Z°goJ©H$ gßª`m a, b, c bmß]rM AmhV. `m_wia + b + c = n............(7) Amh. `W a, b, c `m Mb gßª`m AmhV. d n hr oXbbr gßª`m Amh.`m_wi (7) h EH$ g_rH$aU Amh. h VrZ VwH$S> OmSy>Z V`ma Pmbbm oÃH$mU AmH•$Vr-29 _‹ XmIdbm Amh :

`W XP + PQ + QY = a + b + c = n.

oÃH$mU V`ma hm `mgmR>r XY hr H$mR>r `mΩ` OmJdaM VmS>mdr bmJb. H$maU (a, b, c) oÃH$mUmV a + b > c qH$dmb + c > a AmoU c + a > b h ]ßYZ A[nahm`© Amh. Oa a + b = c, qH$dm b + c = a AJa c + a = b Agb Va ∆ ABC

jwÑH$ oÃH$mU hmB©b. Aem oÃH$mUmMm A‰`mgmgmR>r H$mhrM C[`mJ Zmhr.

g_Om XY = n = 7 Amh d `m H$mR>rM VwH$S> 5, 1, 1 Aem bmß]rM H$b Va oÃH$mU (a, b, c) = ( 5, 1, 1) hmB©b. hm oÃH$mUjwÑH$ ÒdÍ$[mMm Amh. n = 7 gmR>r g_rH$aU (7) M (5, 1, 1) h CŒma hmUma Zmhr. H$maU W a + b = 5 + 1 < 1 = c Amh.[U a = 2, b = 2, c = 3 Ag VwH$S> o_idb Va oÃH$mU V`ma hmVm. H$maU Ò[Ô> Amh. `mMm AW© H$mhr dim oÃH$mU hmVmVa H$mhr dim Vm hmV Zmhr. AmVm n = 7 `m H$mR>rM oH$Vr VwH$S> V`ma hmVmV Vr gd© oÃHy$Q> Am[U _mßSy>. hr _mßS>UrH$aVmZm (a, b, c) = (b, c, a) = (c, a, b) h bjmV R>db [mohO.

n = 7 gmR>r (7, 0, 0), (6, 1, 0), (5, 2, 0), (5, 1, 1), (4, 3, 0), (4, 2, 1), (3, 2, 2), (3, 3, 1) AerVrZ oÃHy$Q> V`ma hmVmV. W EHy$U 8 oÃHy$Q> obohbr AmhV. À`m[°H$s \$∑V (3, 2, 2) d (3, 3, 1) m oÃHy$Q>mßM oÃH$mU V`mahmVmV. Va ]mH$s¿`m ghm oÃHy$Q>mßM jwÑH$ oÃH$mU hmVmV. da oXbÎ`m EHy$U oÃHy$Q>mßMr _mßS>Ur bjmV KVbr Va n = 8, n = 9,

... `mgmR>r oÃHy$Q> _mßS>Vm Vrb. ‡À H$ _mßS>UrV XmZ ‡H$maM oÃH$mU V`ma hmVmV.

(1) O oZo¸V oÃH$mU hmVmV À`mßMr gßª`m T (n) `m gßkZ obhˇ. n = 7 gmR>r T (7) = 2 h bjmV V.

(2) ]mH$sM gd© oÃH$mU jwÑH$ AmhV À`m_wi À`mßMm odMma H$a `mMr JaO Zmhr.

W T (n) hr gßª`m o_id `mgmR>r Am[U MmMUr [’V dm[abr Amh. hr [’V dm[Í$Z T (o), T (1), ..., T (14)

CB

A

AmH•$Vr - 29

c b

a

X a b cY

qp

31

gßª`mßM H$mÔ>H$-5 V`ma H$b Amh. AmUIr H$mhr gßª`m n = 15, n = 16, n = 17 ... MmMUr [’VrZ V[mgVm Vrb [U ZßVahr [’V Hß$Q>midmUr hmV OmB©b. Imbr oXbb H$mÔ>H$-5 V[mgyZ ø`md bmJb H$maU Aem gy _ [hmUrVyZ T (n) gmR>r H$mhr gyÃo_iV H$m ? V [hmd bmJb.

Imbr oXbÎ`m H$mÔ>H$-5 _‹ [ohÎ`m aH$m›`mV H$mR>rMr bmß]r n (n = 1, 2, ...) hr Z°goJ©H$ gßª`m oXbr Amh. XwgË`maH$m›`mV n Mr od^mJUr a, b, c _‹ H$Í$Z oZU© r oÃH$mU (a, b, c) oXb AmhV. [wT>rb aH$m›`mV oÃH$mUmßMr gßª`m T (n)oXbr Amh. T (n) gßª ¿`m o_iUmË`m lUrVyZ H$mhr JoUVr gyÃ o_iVmV H$m ? h JoUVkmZr emYb d À`mV À`mßZm `e Amb.

n (a, b, c) T(n) eam

0. -- 0 (000) jwÑH$ oÃH$mU1. -- 0 (1, 0 0) hm jwÑH$ oÃH$mU2. -- 0 (2, 0, 0), (1, 1, 0) jwÑH$ oÃH$mU3. (1, 1, 1) 1 (2, 1, 0), (3, 0, 0) jwÑH$ oÃH$mU4. -- 0 (4, 0, 0), (3, 1, 0), (2, 2, 0), (2, 1, 1) jwÑH$ oÃH$mU5. (2, 2, 1) 1

6. (2, 2, 2) 1

7. (3, 2, 2), (3, 3, 1) 2

8. (3, 3, 2) 1

9. (4, 4, 1), (4, 3, 2), (3, 3, 3) 3

10. (4, 4, 2), (4, 4, 3) 2

11. (5, 5, 1), (5, 4, 2), (5, 3, 3), (4, 4, 3) 4

12. (5, 5, 2), (5, 4, 3), (4, 4, 4) 3

13. (6, 6, 1), (6, 5, 2), (6, 4, 3)

(5, 4, 4), (5, 5, 3) 5

14. (6, 6, 2), (6, 5, 3), (6, 4, 4), (5, 5, 4) 4

. . . . . .

H$mÔ>H$ (5)

H$mÔ>H$ (5) oMoH$Àgm [’V dm[Í$Z H$b Amh. hr arV _mR>Ám gßª`mßZm Hß$Q>midmUr hmV OmV. n = 1235, n = 133249 Aem^Ï` gßª`mßZm H$er C[`mJr [S>Uma ? V e∑` Zmhr.

H$mÔ>H$ - 5 _‹ ‡À H$ n gmR>r T(n) gßª`m H$mT>mdr bmJV. `mVyZ T(n) Mr lUr V`ma hmV Vr Aer.

(0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4,...).

hr lUr n ¿`m dmT>À`m qH$_Vr]am]a dmT>VM Ag oXgV Zmhr T(n) gßª`m H$_r OmÒV hmV AgVmZm oXgVmV.

_yimV hm ‡˝ yo_Vr_Yrb Amh. [U Vm (7) `m g_rH$aUmV Ambm d ]rOJoUVr` ]Zbm. Vm Agm; H$mUÀ`mhr n `mZ°goJ©H$ gßª Mr VrZ Z°goJ©H$ gßª V od^mJUr H$am Vr Aer H$s,

n = a + b + c, a + b > c, b + c > a, c + a > b.

Aer oH$Vr oÃHw$Q> T(n) AgVrb ?

n = 5 V 14 gmR>r gd© oÃHw$Q> _mßSy>Z jwÑH$ oÃH$mU gamdmgmR>r H$mT>m.

32

`m g_rH$aUmMr CŒma T(n) BVH$s AgVmV. yo_Vr AmoU ]rOJoUV odf` dJi Agb Var JoUV odf` EH$_H$mß¿`mghH$m`m©Z ‡JV hmVm À`mM h EH$ CXmhaU.

T(n), n = 1, 2, 3,.... m lUrVrb gßª`mMm A‰`mg emÒÃkmßZr H$bm d VW UmË`m gßª`m JoUVr VÀdmZ o_iVmV AggwMdb. emY KVÎ`mda o_imbbr 4 gyÃ H´$_mZ [wT> oXbr AmhV.

1) Tn = Tn-3, n hr gßª`m g_ AgVmZm;

=Tn-3 +

n+

(-1)

n + 1

, OÏhm n odf_ Agb.4 4

2

da oXbÎ`m H$mÔ>H$mVrb n d Tn `m gßª`m darb gyÃmZwgma VmV. V [S>VmiyZ [mhˇ.

n = 13 hr gßª`m odf_ Amh. darb gyÃm‡_mU

T13 = T13-3 +13

+(-1)

13 + 1

4 4

2

= T10 + 13 + (-1)7

4 4

= 2 + 31 1

4 4

= 5

h [mM oÃH$mU darb H$mÔ>H$ (5) _‹ oXb AmhV.

n = 14 gmR>r da oXbb gyÃ dm[Í$Z (14 hr g_ gßª`m Amh)

Tn = Tn - 3

T14 = T14-3 = T11 = 4

gamdmgmR>r gd© H$mÔ>H$ (5) V[mgyZ ø`m.

2) Imbr oXbb Xwga gyÃ darb‡_mU dm[aVm B©b.

Tn= [ n2 ] [ n ] [n + 2]12 4 4

`W [x] = x hˇZ bhmZ qH$dm ]am]a Agbbm _mR>ÁmV _mR>m [yUm™H$ hm .

CXmhaUmW© : [ 9.5 ] = 9, [.5 7] = 0, Oa n = 13 Agb Va darb gyÃmZwgma

T13 = [ 132

] - [ 13 ] [13+2 ] = [ 169 ] - [ 13 ] [15 ]12 4 4 12 4 4

= 14 3 (3)

= 5.

33

Oa n = 14 Agb Va

T14 = [ 142

] - [ 14 ] [14+2 ] = [ 196 ] - [ 14 ] [16 ]12 4 4 12 4 4

= 16 3 (4)

= 4.

(3) Tn hr gßª`m Imbr oXbÎ`m gyÃmZ [U H$mT>Vm V.

Tn = [ n2 ] O|Ïhm n hr gßª`m g_ AgV; 48

= [ (n+3)2

] O|Ïhm n hr gßª`m odf_ AgV.48

Oa n = 9 Agb Va odf_ gßª gmR>r

T9 = [(9 + 3)2

] = [ 122

] = [144 ] = 3.

48 48 48

n = 4, Agb Va

T4 = [ 42 ] = [16 ] = [ 1 ] = 0.48 48 3

4) da oXbÎ`m H$mÔ>H$ - (5) _‹ Am[Î`mbm

( To, T1, T2, T3, T4, T5, T6, T7, T8, T9, T10, T11, T12, T13, T14 )

= ( 0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4)

Aer lUr o_iV. EH$m JoUVkmZ emY bmdbm H$s

T(x) = x3

(1-x2) (1-x3) (1-x4)

hr amer Oa _mobH$Zwgma odÒVma obohbr Va da oXbÎ`m lUrMm emY KVm Vm. WmS>∑`mV.

T(x) = x3

(1-x2) (1-x3) (1-x4)

= x3 (1 - x2)-1 (1 - x3) -1 (1 - x4) -1

= To + T1x + T2 x2 + T3 x3 + T4x4 + .... + T14x14 + ...

= x3 + x5 + x6 + 2x7 + x8 + 3x9 + 2x10 + 4x11 + 3x12 + 5x13 + 4x14 + ...

Ag og’ H$amd bmJb. `m og’VgmR>r ]rOJoUVmVrb ]m∞ Zm∞o_Ab ‡_ (Binomial Theorem) _mhrV hd. `mbm o¤[Xog’mßV Ag ÂhUVmV. h ‡_ emiV gm_m›`V: oeH$db OmV Zmhr. h gyÃ Ia Amh Ag Am[U J•ohV YÍ$. AmVm n bmß]r¿`m

34

H$mR>rM dJdJ˘`m oR>H$mUr VwH$S> H$Í$Z T(n) oH$Vr hmVmV h g_O `mgmR>r darb amerVrb xn Mm ghJwUH$ H$mUVm Amh ? `m‡˝mM CŒma Amh.

OW H$bÎ`m bIZmVyZ JoUVmM kmZ Q>flfl`mQ>flfl`mZ H$g dmT>V OmV V bjmV V. JoUVmV Oar eH$S>m CXmhaU MmMUr[’VrZ gmS>dbr Var ‡˝ g_mYmZH$maH$[U gwQ>bm Ag _mZV ZmhrV. n m gd©gmYmaU gßª bm mΩ` Agb Ag gyÃ H$mT>[ ™VoXbÎ`m ‡˝mM CŒma [yU© o_imb Ag hmV Zmhr. À`m_wi JoUVk gyÃ o_id `m¿`m _mJ bmJVmV. MmMUr [’VrZ H$bb H$mÔ>H$ (5)

AmVm da oXbÎ`m gyÃmßZr V[mgyZ KVm Vrb. `mMm AW© MmMUr [’V oZÍ$[`mJr Amh Ag _mÃ hmV Zmhr. `m [’VrZo_imbÎ`m gßª`m gyÃ o_id `mgmR>r dm[aÎ`m AmhV. AmVm 100 hr gßª`m KD$Z Am[U T(n) hr gßª`m H$mTy>. XwgË`m gyÃm‡_mU

T100 = [1002

] - [ 100 ] [100 + 2 ]12 4 4

= [ 10,000 ] - 25 [ 25.5 ]12

= 833 - 25 x 25 = 833 - 625

= 208.

AmVm 100 \y$Q> bmß]rMr H$mR>r o_iU AdKS> Amh. ÂhUyZ H$mR>r EdOr 100 \y$Q> bmß]rMm Xmam KD$. Vm ododY oR>H$mUrVmS>bm Va 208 oÃH$mU o_iVrb.

`m CbQ> [ohb gyÃ dm[ab Va

T100 = T100 - 3 = T97 hm`.

T97 Am[Î`mbm _mhrV Zmhr. ÂhUyZ T97 gmR>r Am[U Xwga gyÃ dm[Í$ :

T97 = [ 972

] - [ 97 ] [ 97 + 2 ]12 4 4

= [ 9409 ] - [ 97 ] [ 99 ] = [ 784

1 ] - [ 24

1 ] [ 24

3 ]12 4 4 12 4 4

= 784 - 24 x 24 = 784 - 576

= 208

AmVm T100 = T97 = 208 oVga gyÃ 97 `m odf_ gßª gmR>r `mOb Va

T97 = [(97 + 3)2] = [ (100)2]48 48

= [ 10,000 ] = [ 625 ] = [ 208

1 ]48 3 3

= 208.

da oXbbr VrZ gyÃ dm[Í$Z Am[Î`mbm T(100) = 208 hr gßª`m o_imbr.

35

8. Q>°·gr - yo_Vr

1 V 7 m od^mJmV Am[U oÃH$mU AmH•$VrMm A‰`mg H$bm. m od^mJmV EH$ ZdrZ ‡H$maMr yo_Vr odMmamV KUma AmhmV.

g_Vbmda AgbÎ`m q]XyVrb AßVa \y$Q>[≈>r KD$Z _mOVm V. q]Xy A [mgyZ q]Xy B [ ™V Om `mgmR>r AZH$ _mJ© H$mT>Vm`Vrb. [U darb AmH•$Vr - 35 _‹ XmIdÎ`m‡_mU Oa gai Jbm Va h AßVa gdm©V H$_r AgV. A›` _mJm©Z Jbm Va AßVadmT>V. q]Xy A d B mßZm OmS>Uma gdm©V H$_r AßVa Am[U d (A, B) m oM›hmZ Ï`∑V H$Í$. W d h Aja "AßVa' (distance)

Z_yX H$a `mgmR>r `mOb Amh. V BßM, \y$Q>, oH$bm_rQ>a, _°b `m EH$H$mZ oZXoeV H$aVm V. wp∑bS>Z AßVamMr hr Ï`mª`m gd©^yo_VrV C[`mObr Amh. yo_VrVrb AZH$ ‡_`, gyÃ Agbb "Element' m ZmdmM [wÒVH$ À`mZ obohb VW À`mZ d (A, B)

hr AßVamMr gßH$Î[Zm dm[abr. AßVa _mO `mg hr Ï`mª`m Iy[M gm[r Amh. d Vr Am[Î`m amO¿`m Ï`dhmamV C[ w∑V PmbrÂhUyZ wp∑bS>Mr yo_Vr "Ï`mdhmnaH$ yo_Vr' R>abr.

XmZ q]XyVrb AßVa d (A, B) Z Ï`∑V hmV. emb A‰`mgH´$_mV AßVam¿`m `m Ï`mª ¿`m AmYmamda Agbbr yo_VroeH$dbr OmV. [U XmZ q]Xy_Yrb AßVa AZßV [’VrZ _mOVm V. `m od^mJmV Aem gd© [’Vr oXbÎ`m ZmhrV. \$∑V EH$MZdr [’V W odMmamV KVbr Amh. yo_Vr¿`m aMZV _J H$mUV ]Xb KS>VmV mMm oZXe [U H$bm Amh. m oddMZmV "AßVa'hr _yb^yV gßH$Î[Zm Amh. À`m_wi AßVamMr Ï`mª`m ‡W_ Ò[Ô> H$am`bm hdr. ZdrZ Ï`mª`m XVmZm AßVam¿`m _yi Ï`mª`er_i gmYbm [mohO. `wp∑bS>¿`m ^yo_VrVrb AßVamVrb _yb^yV VÀd AmVm g_OyZ KD$. darb AmH•$VrV A AmoU B h q]XyoXbb AmhV. À`m_Yrb AßVambm d (A, B) = dE (A, B) Ag ÂhUy. W E h Aja Euclid M AßVa XmId `mgmR>r `mObAmh. AmVm wp∑bS> AßVamM H$mhr JwUY_© bjmV KD$.

(A) dE (A, B) > 0; mMm AW© ‡Vbmdarb XmZ q]XyVrb AßVa YZ qH$dm ey›` AgV. WmS>∑`mV V F$U gßª`m H$YrM ZgV.

(Am) Oa A = B Agb Va dE (A, B) = dE (B, A) = o `mMm AW© XmZ q]Xy A AmoU B Oa EH$Ã Amb Va À`mß¿`m_YrbAßVa ey›` hmV. VgM AßVa _mO `mgmR>r AB ¿`m oXegß]ßYr odMma H$a `mMr JaO Zmhr. `mCbQ> Oa dE (A, B) = 0

Agb Va A = B Amh; Ï`mª`m Zh_r XmZhr oXemßZr H$m_ H$aV.

(B) AmVm `m ‡VbmdaM C hm oVgam q]Xy KD$. oÃH$mU ABC _‹ dE (A, B) + dE (B, C) > dE (C, A)

h "oÃH$mUr` Ag_mZVm' VÀd gÀ` AgV. C q]Xy Oa AB afdaM Agb VaM W g_mZVm A[ojV AgV.

B

A AmH•$Vr - 35

36

AßVamMr ZdrZ Ï`mª`m H$aVmZm (A), (Am), (B) h VrZ _wb yV JwUY_© V[mgyZ ø`md bmJVrb VaM Vr Ï`mª`m AgßoXΩY[U"AßVa' `m gßkbm [mÃ R>ab. AmVm Am[U AßVamMr Xwgar Ï`mª`m KD$ d Vr darb VrZ JwUY_© [miV H$m ? h V[mgyZ KD$.

XY AmbImda AmH•$Vr - 35 _‹ XmIdÎ`m‡_mU A, B, C q]Xy Ag ø`m H$s ∠ ACB = 900 AmoU A (x1, y1),

B (x2, y2), C (x2, y1)

AmH•$Vr - 35XmZ q]Xy_Yrb AßVa _mO `mgmR>r Am[U $& . & h oM›h dm[aVm. AßVa Zh_r YZ qH$dm ey›` gßª`m AgV. m oM›hmZ Aem

gßª`m Ï`∑V H$aVmV. `mbmM H$db_yÎ` (Absolute value) ÂhUVmV.

H$db_yÎ` (Absolute value) Zh_r YZ AJa ey›` gßª`m AgV. V H$YrM F$U gßª`m ZgV. x hr H$mUVrhr gV≤(Real) gßª`m Agb Va H$db_yÎ` (absolute value) Mr Ï`mª`m Imbrb‡_mU H$aVmV ï

& x &= x, OÏhm x > o,

= -x, OÏhm x < o,

= o, OÏhm x = o.

g_Om, x = 9 Amh Va & x & = & 9 & = 9,

x = -9 Amh Va & x & = & -9 & =9,

x = 0 Amh Va & x & = & 0 & = 0.

VgM Oa x AmoU y `m H$mR>Î`mhr XmZ gV≤ gßª`m AgVrb Va

& x + y & ≤ & x & + & y & hr Ag_mZVm gÀ` AgV.

AmH•$Vr - 35 _‹ g_Om A `m oR>H$mUr Ka Amh d B `m oR>H$mUr emim Amh.

KamVyZ oZKyZ emiV Om`M Amh. Kam[mgyZ emibm OmS>Umam aÒVm gai _mJ© Agb Va AB h AßVa H$mQ>md bmJb. (As

the crow flies) h AßVa wp∑bS>r` Amh. [U ehamV XmZ oR>H$mUr gai afZ OmVm U π$oMVM e∑` hmV. eha Oa[yd©oZ`moOV Agb d aÒV H$mQ>H$mZmV ]mßYbb AgVrb Va ‡W_ A [mgyZ C [ ™V $d ZßVa H$mQ>H$mZmV diU KD$Z C [mgyZB [ ™V Omd bmJb. mMm AW© Kam[mgyZ emi[ ™V AßVa AC + CB EdT> Agb. W AC = & x2 - x1 & d CB = & y2 - y1 & Amh.h AßVa [m`r qH$dm Q>∞∑grZ Omd bmJb. `mgmR>r Ka AmoU emim _Yrb AßVa Am[U dT (A, B) `m oM›hmZ Ï`∑V H$Í$. `W dh Aja distance gmR>r d T Aja Q>∞∑gr-‡dmg oZXoeV H$a `mgmR>r `mOb Amh. AmVm darb AmH•$VrV-

dT (A, B) = dE (A, C) + dE (C, B), ÂhUOM

AmH•$Vr - 35

X

`W,

AC = & x1 - x2 &

CB = & y1 - y2 &

A (x 1, y1)

B (x 2, y2)

C (x2, y1)

900

>

< Y

37

dT ( (x1, y1), (x2, y2) ) = & x2 - x1 & + & y2 - y1 & (8)

= & x1 - x2 & + & y1 - y2&.

hr Q>∞∑gr-AßVamMr Ï`mª`m Amh,

`mMm AW© "Q>∞∑gr AßVa' H$mT>VmZm ‡W_ & x1 - x2 & H$mT>md & y1 - y2 & H$mT>md d À`mßMr ]arO H$amdr Ag h gyÃ oZXoeVH$aV. CXmhaUmW© A(3, 4) d B (9, 7) Agb Va A AmoU B _Yrb Q>∞∑gr AßVa (8) `m Ï`mª Zwgma

dT ((3, 4), (9, 7)) = 3 - 9 + 4 - 7 = -6 + - 3

= 6 + 3

= 9,

EdT> Amh. Q>∞∑grdmbm EdT>Ám AßVamM o]b Am[Î`mbm XB©b.

AmH•$Vr - 35 _‹ Am[Î`mbm wp∑bS>r` AßVahr _mOVm V. ∆ ACB hm H$mQ>H$mZ oÃH$mU Amh. ÂhUyZ

AB2 = AC2 + CA2 = & x1-x2 & 2 + & y1 - y2 &

2

qH$dm

AB = √ &x1 - x2 &2 + &y1 - y2&2 qH$dm (9)

AB = dE ((x1, y1), (x2, y2) = √ &x1 - x2 &2 + &y, - y2 &

2 hm`.

(9) hr wp∑bS>r` - AßVamMr Ï`mª`m Amh. h AßVa Am[Î`mbm [naoMV AmhM. da KVbÎ`m CXmhaUmV

dE ((3, 4), (9, 7)) = √(3 - 9)2 + (4 - 7)2 = √ 62 + 32

= √36 + 9 = √ 45 = √ 9 x 5

= 3 √ 5.

dT d dE hr AßVa dJir AmhV. `mMm AW© emiV Om `mgmR>r _wbmbm 9 EdT> AßVa H$mT>md bmJb [U H$md˘`mbm 3 √5 hAßVa CS>V Omd bmJb.

Am[U AmVm dT m AßVamgß]Yr odMma H$bm [U dT AßVamMr Or (8) _Yrb Ï`mª`m KVbr Amh Vr "AßVa' m gßkbm [mÃAmh H$m ? V ‡W_ V[mgb [mohO. `mgmR>r Am[Î`mbm (A), (Am) AmoU (B) hr VrZ _yb yV ]ßYZ dT `m Ï`mª bm _m›`H$amdr bmJVrb. Imbr oXbbr AmH•$Vr [hm :

AmH•$Vr - 36

A (x 1, y1)

B (x 2, y2)

C (x3, y3)

X

Y

(0, 0)

38

AmH•$Vr - 36 _‹ (8) hr Ï`mª`m dm[Í$Z :dT (A, B) = dT ((x1, y1), (x2, y2)) = & x1 - x2 & + & y1 - y2 &

dT (B C) = dT ((x2, y2), (x3, y3)) = & x2 - x3 & + & y2 - y3 &, (10)

dT (C A) = dT ((x3, y3), (x1, y1)) = & x3 - x1 & + & y3 - y1 &,

hr [X o_iVmV.

(A) (10) _‹` Ambbr [X V[mgbr Va bjmV V H$s dT (A, B), dT (BC), dT (CA) m gßª`m YZ qH$dm ey›` AmhV;

(Am) Oa A = B Agb Va (x1, y1) = (x2, y2), ÂhUOM x1 = x2 d y1 = y2 Amh. À`m_wi (10) _‹` dT (A, B) = 0

og’ hmV. `m¿`m CbQ> Oa dT (A, B) = 0 Agb Va & x1 - x21+ & y1-y2 & = 0 ÂhUOM x1 = x2 d y1 = y2qH$dm A = B hm`. Am[U dT Ï`mª Vrb (A) d (Am) hr Xm›hr odYmZ gÀ` AmhV h og’ H$b Amh. AmVm (B) hodYmZ gÀ` Amh H$m ? V V[mgy,

AmVm Am[U gV≤ gßª`mßZm og’ Agbb ( a AmoU b `m gV≤ gßª`m AmhV)

a + b ≤ a + b h gyÃ dm[Í$. `W x1 - x3 = x1 - x2 + x2 - x3 ≤ x1 - x2 + x2 - x3 AmoU

y1 - y3 = y1 - y2 + y2 - y3 ≤ y1 - y2 + y2 - y3, hm`.

Xm›hr ]mOyßMr ]arO H$Í$Z, Am[Î`mbm Imbrb Ag_mZ [Xr o_iV.

x1 - x3 + y1 - y3 ≤ x1 - x2 + y1 - y2 + x2 - x3 + y2 - y3

`mMm AW© dT (C, A) ≤ dT (A, B) + dT (B, C), og’ hmV.

ÂhUyZ "oÃH$mUmVrb Ag_mZVm VÀd' (B) bmJy AgÎ`mM Am[U og’ H$b Amh. AßVa hr gßkm dT gmR>r, AmVm (A),(Am) AmoU (B) og’ H$Î`m_wi, `mΩ` R>abr. dT hr [U AßVamMrM Ï`mª`m `m_wi _m›` R>abr Amh. Og wp∑bS>Mr AßVamMrÏ`mª`m yo_VrV ‡oVo>V _mZbr OmV VgmM AoYH$ma "Q>∞∑gr AßVa dT' bm [U ‡m· Pmbm Amh. `mgmR>r Am[U AmVm H$mhrCXmhaU g_OyZ KD$.

(i) g_Om A = (5, 4) AmoU B = (1, 2) Va dE (A, B) d dT (A, B) hr Xm›hr AßVa obhm.

(9) `m gyÃmZwgma

dE (A, B) = dE ((5,4), (1, 2))

= √(5 - 1)2 + (4 - 2)2 = √ 42 + 22 = √ 20 = 2 √5.

AmoU (8) `m gyÃmZwgma

dT (A, B) = dT ((5, 4), (1, 2)) = 5 - 1 + 4 - 2 = 4 + 2 = 6

`W dE (A, B) ¹ dT (A, B) Amh.

(ii) H$mR>Î`m [napÒWVrV dE (A, B) d dT (A, B) g_mZ AgVrb ?

39

A (x1, y1) AmoU B (x2, y2) h oXbb q]Xy AmhV. AmVm

(8) d (9) `m gyÃmZwgma

dE (A, B) = dE (x1, y1), (x2, y2)) = √ (x1 - x2)2 + (y1 - y2)2,

AmoU

dT (A, B) = dT ((x1, y1), (x2, y2)) = x1 - x2 + y1 - y2 Amh.

g_Om x1 - x2 = p AmoU y1 - y2 = q Agb Va p + q ≥ √ p2 + q2

hr Ag_mZVm Am[Î`mbm _mohV Amh. p = o, q = o Agb Va _mÃ g_mZVm `V. ÂhUO p = x1 - x2 = o; dq = y1 - y2 = o hm . `mMm AW© (x1, y1) = (x2, y2) qH$dm Xm›hr q]Xy A AmoU B EH$M AmhV.

`m CXmhaUmV AmUIr EH$ e∑`Vm V[mgy. Imbr H$mT>bbr AmH•$Vr - 37 [hm.

AmH•$Vr-37 _‹ A (x1, y1) AmoU B (x1, y2) h q]Xy AmhV. W AB afm Y - Ajmbm g_mßVa Amh. AmVm AB h AßVa dEd dT `m Xm›hr [’VrZ H$mTy> :

dE (A, B ) = dE ((x1, y1), (x1, y2)) = √ (x1 - x1)2 + (y1 - y2 )2

= √(y1 - y2)2 = y1 - y2 ;

dT (A, B) = dT ((x1, y1), (x1, y2)) = x1 - x1 +y1 - y2

= y1 - y2

ÂhUyZ dE (A, B) = dT (A, B) = y1 - y2 hm`.

WmS>∑`mV X - Ajmbm qH$dm Y - Ajmbm g_mßVa AgbÎ`m H$mUÀ`mhr afda XmZ q]Xy KVb Va À`m_Yrb wp∑bS>r` d Q>∞∑grAßVa g_mZ AgV.

>

> >

>

A (x 1, y1)

X

C D

Y

B (x 1, y2)

X1

Y1

(p1,q1) (p2, q1)

AmH•$Vr - 37

40

darb AmH•$Vr-37 _‹` C (p1, q1) d D (p2, q1) Agb Va CD afm X - Ajmbm g_mßVa AgV. W darb‡_mU

dE (C, D) = dT (C, D) Amh H$m ? V V[mgyZ [hm. (CŒma "hm ' Ag Amh.)

(iii) Oa A (2, 1) q]Xy oXbm Agb Va A [mgyZ 3 dT AßVamda Agbb 3 q]Xy H$mT>m.

g_Om B (x, y) q]Xy A [mgyZ 3 EH$H$ AßVamda Amh. Va Am[Î`mbm (x, y) q]Xy Ag hdV H$s,

dT (A, B) = dT ((2, 1), (x, y)) = 3

ÂhUO dT (2, 1), (x, y) = 2 - x + 1 - y = 3,

B (x, y) = B (o, o) Agb Va dT (A, B) = 2 + 1 = 3,

B (x, y) = B (-1, 1) Agb Va dT (A, B) = 2 + 1 + 1 -1 = 3,

B (2, -2) Agb Va dT (A, B) = 2 - 2 + 1 + 2 = 3

W (0, 0), (-1, 1) AmoU (2, - 2) h q]Xy A (2, 1) m q]Xy[mgyZ 3 Q>∞∑gr AßVamda AmhV. h q]Xy W AZw dmZ H$mT>bAmhV. AmUIrhr H$mT>Vm Vrb.

(iv) A (-2, -1) d B = (3, 2) h oXbb q]Xy AmhV. Va P (x, y) dT (P, A) = 3 AmoU dT (P, B) = 5 hm AmbI H$mT>m.

`m CXmhaUmV Am[Î`mbm gd© P (x, y) q]Xy H$mT>m`M AmhV. P (x, y) h gd© q]Xy A (-2, -1) `m q]Xy[mgyZ 3 Q>∞∑grAßVamda VgM B (3, 2) `m q]Xy[mgyZ 5 Q>∞∑gr AßVamda AmhV. h AßVa "Q>∞∑gr AßVa' ø`m`M Amh. ÂhUOM

dT (P, A) = dT ((x, y), (-2, -1)) = x + 2 + y + 1 = 3 ;

AmoU dT (P, B) = dT ((x, y), (3, 2)) = x - 3 - y - 2 = 5;

gm]V oXbÎ`m AmbImda QRST AmoU CDET m XmZ AmH•$À`m H$mT>Î`m AmhV. AmVm Imbr oXbbr XmZ odYmZ V[mgy

(A) QRST `m AmbImdarb gd© q]Xy P (x, y) h A (-2, -1) `m q]Xy[mgyZ 3 Q>∞∑gr - AßVamda AmhV.

(]) CDET `m AmbImdarb gd© q]Xy P (x, y) h B (3, 2) `m q]Xy[mgyZ 5 Q>∞∑gr - AßVamda AmhV.

QRST hr AmH•$Vr H$er H$mT>br Amh V ‡W_ [mhˇ. A (-2, -1) hm q]Xy oXbm Amh. `m q]XyVyZ X - Aj d Y - Aj `mßZmg_mßVa AgbÎ`m afm H$mT>m. AT, AQ, AR, AS `m gd© afm 3 - Q>∞∑gr AßVa bmß]r¿`m AmhV. (g_mßVa afmda wp∑bS>r` dQ>∞∑gr AßVa gmaIM AgV. À`m_wi Q, R, S, T h q]Xy oZXoeV H$aVm VmV. AmVm QR, RS, ST AmoU TQ `m afm H$mT>m.AmVm QRST hr AmH•$Vr V`ma Pmbr.

Q, R, S, T q]Xy A (-2, -1) [mgyZ 3 Q>∞∑gr - AßVamda AmhV V Ò[Ô> AmhV. QRST AmbImda AmVm H$mR>bmhr q]Xyø`m. h gd© P (x, y) q]Xy A (-2, -1) [mgyZ 3 AßVamda Amh. h AmVm ghO V[mgyZ [hmVm `B©b. (0, 0), (4, 0), (1,1)(-3, -3), (7/2, 5/2)... h `m AmbImdarb q]Xy A (-2, -1) [mgyZ 3 Q>∞∑gr AßVamda AmhV. `mgmR>r (8) gyÃ oXbb AmhM.(h odYmZ V[mgyZ ø`m)

AmVm B (3, 2) hm Xwgam q]Xy oXbm Amh. `m q]Xy[mgyZ 5 - Q>∞∑gr - AßVamda AgUma gd© q]XyMm gßM o_idm`Mm Amh. daoXbbr aMZm CDET AmH•$Vr H$mT> `mgmR>r W mOy. m AmH•$Vrdarb gd© q]Xy B (3, 2) [mgyZ 5 Q>∞∑gr - AßVamda AmhV. C,

D, E, T h q]Xy W oZXoeV H$b AmhVM. AmUIrhr H$mhr q]Xy oZXoeV H$am.

41

AmH•$Vr - 38 _‹ XmIdÎ`m‡_mU TS hm afMm ^mJ QRST d CDET `mß¿`mV g_mZ Amh. À`m_wi `m TS darb gd©q]Xy A (-2, -1) [mgyZ 3 Q>∞∑gr - AßVamda Amh. Va B (3, 2) [mgyZ 5 Q>∞∑gr - AßVamda AmhV. h oXbÎ`m ‡˝mM CŒma Amh.`m CXmhaUmVrb gd© Q>∞∑gr - AßVa (8) hr Ï`mª`m dm[Í$Z AmoU ‡À`j AmbImda _mOVm Vrb. Q>∞∑gr - AßVa H$mUÀ`mhr[na_ ghoZXeH$m[mgyZ _mOVm Vrb. CXmhaUmW©, TS `m afda K (0.5, 0.5) hm q]Xy Amh. `W

dT (A, K) = dT ((-2, -1), (0.5, 0.5)) = - 2 + 0.5 + -1 - 0.5

= -1.5 + -1.5

= 1.5 + 1.5

= 3

A [mgyZ K hm q]Xy 3 AßVamda Amh. VgM

dT (B, K) = dT ((3, 2) (0.5, 0.5)) = 3 + 0.5 + 2 - 0.5

= 3.5 + 1.5 = 3.5 + 1.5 = 5

K (-0.5, 0.5) hm q]Xy A (-2, -1) [mgyZ 3 AßVamda Va B (3, 2) [mgyZ 5 AßVamda Amh.

hm oZÓH$f© AmbI [mhZhr H$mT>Vm B©b.

AmH•$Vr - 38 _‹ dT (A, K) = AM + MK = 3

d dT (B, K) = KN + NB = 5 Amh.

AmH•$Vr - 38

E (3,7)

B

Y

D

X

N

S

K

T

A

R

( 8, 2 )( 3, 2 )

C ( 3, - 3 )

(1, -1)

( 0, 0 )

(-2, -1)

(-2, -4)

Y'

(- 5 , -1)

( - 2, 2 )

MQ

X'

42

ZJa oZ`mOZ AmoU Q>∞∑gr yo_VrOJmV H$mhr eha [yd©oZ`moOV AmhV `mMm AW© _wª` aÒV AmoU C[aÒV H$mQ>H$mZmV AmIbb AgVmV. W ehamVrb

gmd©OoZH$ C[`mJm¿`m B_maVr ([mÒQ>, ]gÒWmZH$, aÎd ÒQ>eZ, emim, hm∞pÒ[Q>b, ]±H$m, ]mOma BÀ`moX) oZ`moOV OmJdaM]mßYÎ`m OmVmV. Aem ehamVyZ Q>∞∑gr AßVa gßH$Î[Zm Ï`dhmamgmR>r C[ w∑V R>aV. Imbrb XmZ CXmhaUmßZr ZJaoZ`mOZ dQ>∞∑gr yo_Vr `m_Yrb gßH$Î[Zm Ò[Ô> hmVrb.

(1) EH$m [yd©oZ`moOV ehamV P (-2,4) `m OmJda JmS>rbm A[KmV Pmbm Amh. `mdir XmZ [mobgJmS>Ám A (2, 1) dB (-1, 1) `moR>H$mUr C‰`m AmhV. `m[°H$s H$mUVr JmS>r A[KmVm¿`m OmJ[mgyZ Odi Amh ?

AmbImdarb AmH•$Vr - 41 [hm.

P hr A[KmVmMr OmJm Amh. A AmoU B `moR>H$mUr [mbrg JmS>Ám AmhV. AmVm Am[U PB AmoU PA hr Q>∞∑gr AßVa_mOy. ([mbrg JmS>Ám [yd©oZ`moOV ehamVrb aÒÀ`mdÍ$Z ‡dmg H$aUma AmhV)

dT (P, A) = dT ((-2, 4), (2, 1)) = - 2 - 2 + 4 - 1

= - 4 + 3 = 7;

dT (P, B) = dT ((-2, 4), (-1, 1) = -2 + 1 + 4 - 1 = -1 + 3 = 1 + 3 = 4.

`W dT (P, B) < dT (P, A) Amh. `mMm AW© B W Agbbr [mobgmßMr JmS>r A[KmVm¿`m OmJda bdH$a OmD$Z [mMb.

da oXbb CXmhaU gm[ Amh. `m JoUVmM CŒma AZw dmZ [U XVm B©b. [U JwßVmJwßVr¿`m oR>H$mUr JoUVr` odMma AMyH$R>ab.

(2) AZw[ AmoU od⁄m m XmKmßZm amh `mgmR>r Ka emYm`M Amh. AZw[ Am°fYm¿`m H$maIm›`mV A (-3, -1) ZmH$ar H$aVm, Vaod⁄m B (3, 3) `m oR>H$mUr AgbÎ`m ]±H$V H$m_mbm OmV. XmKhr Am[Î`m H$m_m¿`m oR>H$mUr [m`r OmVmV. À`mßZr R>adb H$sXmKmßM o_iyZ EHy$U Mmb `mM AßVa H$_rV H$_r Agmd. À`mßZr Am[b Ka H$mR> emYmd ?

Oa XmKmßZmhr gmaIM AßVa Mmbmd bmJmd Ag R>adb Va KamgmR>r H$mUVr OmJm oZdS>mdr ?

AmH•$Vr - 42 _‹ AZw[ AmoU od⁄m `mß¿`m H$m_m¿`m OmJm AZwH$_ A AmoU B q]XyZr AmbImda XmIdÎ`m AmhV.XmKmß¿`m H$m_m¿`m OmJVrb Q>∞∑gr - AßVa dT

dT (A, B) = dT ((-3, -1), (3, 3)) = -3 - 3 + -1 -3

= -6 + - 4 = 6 + 4 = 10

Amh. AmH•$Vr - 42 _‹ XmIdÎ`m‡_mU PAQB `m AmH•$VrV AmbÎ`m H$mR>Î`mhr q]Xyda Ka KVb Var XmKmßZm o_iyZ EHy$UMmbm`M Q>∞∑gr AßVa 10 EdT>M Amh. À`m_wi AZw[ AmoU od⁄m `mßZr `m PAQB `m AmH•$Vr¿`m AmVrb ]mOyg H$mR>hr KaemYmd. h AßVa gd© oR>H$mUr 10 Amh. AmH•$Vr¿`m ]mha Ka emYb Va _mÃ À`mßZm 10 [jm OmÒV AßVa Mmbmd bmJb.

g_Om C (x, y) = C ( 13, 2 ) W Ka KVb10 10

W Ka KVb Va Am[U XmKmßM EHy$U Mmb `mM Q>∞∑gr AßVa H$mTy>.

43

dT (A, C) = dT (-3, -1), -13 , 2

10 10

= -3 + 13 + -1 -

2 = -

17 + -

12

10 10 10 10

= 17

+ 12

= 29

10 10 10

dT (C, B) = dT - 13, 2 , (3, 3) = -13 -3 + 2 - 3 10 10 10 10

= - 43 + - 28 = 71

10 10 10

Xm›hr Q>∞∑gr AßVamMr ]arO 29 + 71 = 10 Amh. 10 10

`m CXmhaUm¿`m XwgË`m ^mJmV AZw[ AmoU od⁄m `mßZm gmaIM AßVa Mmb `mMr AQ> Amh. PAQB `m AmH•$VrV L(x,y)

`moR>H$mUr Ka Amh Ag g_Oy Va

dT (A,L) = dT ((-3,-1), (x,y)) = l x+3 l + l y+1 l

dT (B,L) = dT ((3,3), (x,y)) = l x-3 l + l y-1 l.

Am[Î`mbm (x,y) o_iod `mgmR>r Imbrb g_rH$aU gmS>dmd bmJb. ÂhUyZ

dT (A,L) = dT (B,L)

ÂhUO l x+3 l + l y+1 l = l x-3 l + l y-3 l.

`mMm AW© (x+3) = (3-x) d

(y+1) = (3-y) hm`.

ÂhUyZ x=0 AmoU y=1 d L(x,y) = L(0,1).

AZw[ AmoU od⁄m `mßM Ka L oR>H$mUr Agb Va À`mßZm À`mß¿`m ZmH$ar¿`m oR>H$mUr g_mZ AßVa Mmbmd bmJb.

W gmS>dbbr Xm›hr CXmhaU gm[r AmhV. ZmJar dÒVrV amhUmË`m ZmJnaH$mßZm dmaßdma Aem g_Ò`mßZm Vm|S> ⁄md bmJV.Am[U emiV oeH$V AgbÎ`m yo_VrV m ÒdÍ$[mMr CXmhaU ghgm V ZmhrV. Q>∞∑gr yo_VrV dJ˘`m ‡H$maM ‡˝ AgVmV.`m ‡H$ma¿`m yo_VrMr C[ w∑VVm bjmV `mgmR>r hr CXmhaU C[ w∑V AmhV.

44

(-7 , 5)

(-3 , 3)

2

5

(- 1, 4) ( 3, 4)

( 7, 3)

3

4

X 7 31 (0, 0)

4

3

5

(-3, -5) Y'

AmH•$Vr - 39

( - 4, 9 )

S

19

P( - 6, - 8)[mÒQ> Am∞o\$g

emim 14H ( 9, 10 )Ka

33

( 0, 0 )

P = [mÒQ> Am∞o\$g

H = Ka

S = emim

Ï`mª`m dm[am (8)

d1 (H, S) = 14

d1 (S, P) = 19

d1 (P, H ) = 33

EHy$U Q>∞∑gr AßVa = 66

AmH•$Vr - 40

Y'

X'

45

[mR>Q>∞∑gr - AßVa _mO `mMm gamd hm `mgmR>r H$mhr [mR> Imbr oXb AmhV.

(1) Imbr oXbÎ`m q]Xy_Yrb Q>∞∑gr AßVa AmbImdÈZ _mOm.

A) (0, 0), (3, 4), ................... (7)

Am) (0, 0), (7, 3), .................. (10)

B) (0, 0), (-1, 4), .................. (5)

B©) (0, 0), (-3, -5) ................. (8)

D$) (-3, 3), (-7, 5) .................. (7).

(2) Imbr oXbÎ`m q]XyVrb Q>∞∑gr-AßVa Ï`mª`m (8) `mOyZ H$mT>m.

A) (4, 5), (7, 9) .................. (7)

Am) (-7, 8), (7, 5) ................. (17)

B) (-9, -10), (-3, -4) ............. (12)

B©) (11, -9), (-4, 5) ............... (29)

\$) (4, -6), (-6, 4) .............. (20).

(3) eaXM Ka (9,10) `m q]Xyda, emim (-4, 9) q]Xyda d [mÒQ> Am∞o\$g (-6, 8) `m q]Xyda Amh. À`mbm emiV OmD$Z EH$XmIbm o_idm`Mm Amh d Vm ZßVa [mÒQ>mZ [mR>dm`Mm Amh d Kar [aV `m`M Amh. À`mbm EHy$U oH$Vr Q>∞∑gr AßVa MmbmdbmJb ? eaX [mÒQ>mVyZ Kar Ambm Va Vm oH$Vr Q>∞∑gr AßVa Mmbb ? (CŒma o_id `mgmR>r AmbI H$mJXmda AmH•$Vr - 40

H$mT>mdr. VgM `mgmR>r Q>∞∑gr-AßVamMr Ï`mª`m dm[am. (CŒma ï 66, 33)

AmbI AmH$•$Vr 39 _‹` H$mT>b AmhV .

COdrH$S>rb H$ßgmV CŒma oXbr AmhV.

X


YP (2, 4)

A (2, 1)

Y'

X1

B (-1,1)

Y

B (3,3)

Y'

X'

P

A(-3,-1) Q

(0,0)

-13 2 10

, 10

C

X

46

yo_VrV dVw©i `m AmH•$Vrbm _hÀdmM ÒWmZ Amh. wp∑bS>r` yo_VrV dVw©imMr Ï`mª`m Aer oXbr OmV ï

(a, b) `m oXbÎ`m pÒWa q]Xy[mgyZ oXbÎ`m AßVa r da AgUmË`m gd© q]XyMm gßM ÂhUO dVw©i hm . `m Ï`mª V "AßVa' hme„X _hŒdmMm Amh. Am[Î`mbm AmVm (i) wp∑bS>r` AßVa d (ii) Q>∞∑gr-AßVa h ‡H$ma _mhrV AmhV À`m_wi `m od^mJmV XmZdVw©i H$mT>Vm Vrb. Ï`mª V (a, b) hm pÒWa q]Xy oXbm Amh À`mbm dVw©imMm H|$–q]Xy ÂhUVmV. Va oXbb AßVa r `m oM›hmZdVw©imMr oÃ¡`m Ï`∑V hmV. AmVm Am[U XmZ dVw©i H$mTy>.

(3) w-dVw©i (Euclidean Circle)

h dVw©i H$mT>VmZm wp∑bS>r` AßVa Am[U bjmV KUma AmhmV. ÂhUyZ `m AmH•$Vrbm w-dVw©i ÂhQ>b Amh. `m AmH•$VrMmAmbI Am[Î`m [naM`mMm Amh. Vm AmH•$Vr-43 _‹ XmIdbm Amh.

H|$–q]Xy A(a, b)[mgyZ dVw©imdarb P (x, y) hm Mbq]Xy r (oÃ¡`m) AßVamda Amh. AP = r h AßVa wp∑bS>r` Amh. ÂhUyZ

r = √( x - a )2 + ( y - b )2 hm`.

`W Am[U (9) _Yrb AßVamMr Ï`mª`m C[`mJmV AmUbr. gßMm¿`m ^mfV AmH•$Vr-43 _Yrb dVw©i Imbr oXÎ`m‡_mU obohVm`B©b.

`w- = P(x, y) : dE (P (x, y), (a, b)) = r (11)

= (P (x, y) : √( x - a )2 + ( y - b )2 = r)

(4) Q>r-dVw©i (Taxi-Circle)

`m Q>r-dVw©imV Am[U H|$–q]Xy A(a, b) d Mbq]Xy P(x, y) `m_Yrb r oÃ¡ Mr _mOUr Q>∞∑gr-AßVa `mOyZ H$aUmaAmhmV. `mMm AW© r = dT ((x, y), (a, b)) = & x - a & + & y - b & hm . h AßVa (8) _‹ oXbÎ`m Ï`mª Z W oZXoeV H$bAmh. AmVm Q>r-dVw©i gßMm¿`m ÒdÈ[mV obhˇ : V Ag :

Q>r - = (P (x, y) : dT ((x, y), (a, b)) = r) (12)

ÂhUOM

Q>r- = (P (x, y) : & x - a & + & y - b & = r ) hm`.

da (11) _‹ w-dVw©i d (12) _‹ Q>r-dVw©i `mß¿`m Ï`mª`m gßM ÒdÈ[mV oXÎ`m AmhV. `m Ï`mª`mßMr VwbZm H$br Va Xm›hr

A (a, b )

P (x , y )

AmH•$Vr - 43

r

.

.

.

47

gßM EH$mM ÒdÍ$[mM AmhV \$∑V E EdT>r T h oM›h obohb Amh. `m ]Xbm_wi Xm›hr dVw©i dJir AmhV h ghO bjmV V.

Q>r-dVw©i AmbImda H$g XmIdVm B©b V Am[U XmZ CXmhaU KD$Z [mhˇ.

g_Om, dVw©imMm H|$–q]Xy (a, b) = (-2, -1) Amh d oÃ¡`m 3 Amh. (12) _‹ oXbÎ`m dVw©im¿`m Ï`mª Zwgma

Q>r - = (P (x, y) : dT ((x, y), (-2, -1))) = 3

= (P (x,y) : & x + 2 & + & y + 1 & = 3)

`m dVw©imda AgUma H$mhr q]Xy Am[U V[mgy h q]Xy W AZw dm¿`m AmYmamZ oXb AmhV. V H$g o_idm`M V EH$Xmg_Ob H$s AmUIrhr q]Xy H$mT>Vm Vrb.

`W P (-2, -4), P (1, -1), P 1 , -3 P - 3 , -7 P (0, 0) P (-2, 2) 2 2 2 2

h q]Xy Q>∞∑gr - AßVa dVw©imda AmhV H$maU

dT ((-2, -4), (-2, -1)) = & -2 + 2 & + & -4 + 1 & = 3,

dT (( 1, -1), (-2, -1)) = & 1 + 2 & + & -1 + 1 & = 3,

dT 1 , -3 (-2, -1) = & 1 + 2 & + & -3 + 1 & = 3, 2 2 2 2

dT -3 ,-7 (-2, -1) = & -3 + 2 & + & -7 + 1 & = 3, 2 2 2 2

dT (( 0, 0) , (-2, -1)) = & 0 + 2 & + & 0 + 1 & = 3,

dT ((-2, 2) , (-2, -1)) = &-2 + 2 & + & 2 + 1 & = 3,

da oXbb gd© q]Xy Q>∞∑gr AßVa dVw©imda AmhV. `m dVw©imMm _‹`q]Xy (-2, -1) `m oR>H$mUr Amh AmoU À`mMr oÃ¡`m 3Amh. AmVm VwÂhr AmUIr H$mhr q]Xy AgM emYyZ H$mT>m. h gd© q]Xy AmbI - H$mJXda Zm|XyZ [hm. AmVm Am[Î`m ghO bjmV

B©b H$s `m Q>∞∑gr - AßVa dVw©imMr AmH•$Vr - 44 _‹ H$mT>bÎ`m AmbImgmaIr Agb.

AmH•$Vr - _Yrb QRST hm AmbI Q>r - dVw©i Amh. mMm H|$–q]Xy A (-2, -1) Amh d oÃ¡`m 3 Amh. mMm AW© m AmbImdarbH$mUVmhr q]Xy P (x, y), A (-2, -1) [mgyZ 3 T- AßVamda Amh. Am[Î`mbm [naoMV AgbÎ`m dVw©im[jm AmH•$Vr - 44 dJirdmQ>V. Ag hmUmaM H$maU W Am[U AßVamMr Ï`mª`m ]Xbbr Amh W dT (P, A) h AßVa 3 Amh Ag J•hrV Yab Amh.AmUIr EH$ Q>r - dVw©i H$mTy>.

AmH•$Vr - 45 _‹` A (3, 2) hm H|$–q]Xy d oÃ¡`m 4 Agbb Q>r - dVw©i H$mT>bb Amh. AmbImda H$mßhr q]Xy oZXoeV H$b AmhV.h gd© q]Xy A (3, 2) [mgyZ 4 `m Q>∞∑gr - AßVamda AmhV H$m V V[mgyZ [hm. hm AmbI H$mJX Amh. À`m_wi AmbI CDET daAmVm H$mUVmhr q]Xy ø`m (¡`mM ghoZXeH$ [yUm™H$ qH$dm A[yUm™H$ AmhV) d Ag gd© q]Xy A (3, 2) [mgyZ 4 - Q>∞∑gr AßVamdaAmhV H$m V V[mgyZ [hm.

.

48

AmbI CDET JoUVr` ^mfV AmVm

T- = P (x, y) : dT ((x, y), (3, 2) = 4

= p (x, y) : x - 3 + y - 2 = 4

Agm obohVm Vm.

Ia Va AmH•$Vr-44 d AmH•$Vr-45 hr Xm›hr Q>r - dVw©i Am[U AJmXaM H$mT>br AmhV. ([hm : AmH•$Vr - 38) [U Am[UÀ`mdir m AmH•$À`m dVw©i AmhV Ag ÂhQ>b ZÏhV. m AmH•$À`m H$em H$mT>mÏ`mV V XIrb VW gmßoJVb AmhM. À`m_wi AmH•$À`m- 44, 45 H$mT>U AmVm gm[ dmQ>b.

darb XmZ CXmhaUmdÍ$Z yo_VrVrb AmH$ma H$g ]XbV OmVmV V bjmV V. eha oZ`mOZmV Q>∞∑gr- yo_VrMm C[`mJhmVm. AmUIr H$mhr C[`moOVm AmhV. `m gd© W godÒVa[U oXbÎ`m ZmhrV. Imbr oXbÎ`m Q>∞∑gr - AßVa dVw©imM AmbIH$mT>m.

i) P (x, y) / dT ( (x, y), (0, 0) ) = 2

ii) P (x, y) / dT ( (x, y), (1, 1) ) = 5

hr dVw©i H$mT> `mgmR>r AmbI - H$mJX dm[am. ‡À H$ dVw©imdarb 10 q]XyM ghoZXeH$ H$mT>m.

Q>∞∑gr yo_VrV AmVm bß]dVw©i (Ellipse), A›dÒV (Parabole), A[mÒV (Hyperbola) m AmH•$À`mhr [maß[naH$ yo_Vr[jmdJ˘`m dmQ>Vrb. À`m W oXÎ`m ZmhrV. oÃo_Vr (3-dimension) _‹ T - dVw©i T - Jmb AmH•$VrV H$g [amdoV©V hmB©b`mMmhr odMma H$aVm B©b.

Am[U emiV wp∑bS>r`Z yo_Vr oeH$Vm À`m_wi dVw©imMr [aß[amJV AmH•$Vr Am[Î`m _Zmda R>gbbr Amh. [U "AßVa' hrH$Î[Zm ]Xbbr H$s hr AmH•$Vrhr ]XbV. AßVa `m gßH$Î[Zda AmYmabÎ`m gd©M AmH•$À`m AmVm dJ˘`mM oXgVrb À`m_wi[aß[aZ bjmV amohbÎ`m AmH•$À`m ÂhUO AIa¿`m, Aer _ZmMr K≈> R>dU H$Í$Z K `mM H$maU Zmhr. m gd© MMV gmßoJVbbJoUVr` VÀd EH$M Amh Zm ? h _mÃ V[mgyZ [hmd bmJb.

.

P (x,y

)

r(-2,2)

r = 3 o(0,0)

s(1,-1)A(-2,-1)

Q(-5,-1)

R (-2,-4)

(-1,2)T

0(O,O) (2,-1) (5,0)

(1,4)(5,4)

AmH•$Vr - 44

T-dVw©i, H|$–q]Xy (-2,-1)oÃ¡`m 3

E (3,6)

A (3,2)r = 4

(7,2)D

P (x,y)

C (3,-2)AmH•$Vr - (2,45)

T- dVw©i, H|$–q]Xy (3,2)oÃ¡`m 4

49

`m [wpÒVH$V XmZ AßVamß¿`m Ï`mª`m oXbÎ`m AmhV. [U XmZ q]XyV AßVa _mO `m¿`m Ï`mª`m Agßª` AmhV. À`m_wi AZßVyo_Vr ‡H$ma obohVm Vrb. Oer Oer JaO dmT>V OmB©b Vg h yo_Vr ‡H$ma CbJS>V OmVmV h bjmV KVb [mohO.

JoUVmVrb ododY emIm odÒVmaV H$em OmVmV h g_O `mgmR>r da oXbb A w∑brS>r` AßVamM CXmhaU oXbb Amh.

gw_ma XmZ hOma df wp∑bS>Mr yo_Vr JoUV oejUmV H|$–ÒWmZr amohbr `mbm H$maU amO¿`m Ï`dhmamgmR>r bmJUmayo_VrM kmZ À`mVyZ o_iV hmV. Ka]mßYUr, ZJaoZ`mOZ, Iim¿`m _°XmZmßMr aMZm, jÃ\$i, KZ\$i, dÒVyßM AmH$ma Aem

oH$VrVar JaOm `m yo_VrZ ^mJdÎ`m hmÀ`m. À`m_wi AmOhr wp∑bS>Mm _mZ JoUV jÃmV A]moYV amohbm Amh. `m bhmZem[wpÒVH$V hm gd© BoVhmg godÒVa obohU e∑` Zmhr d Ver JaOhr Zmhr. ÂhUyZ \$∑V W WmS>∑`mV CÑI H$bm. EH$_hÀdmMm _w‘m ÂhUO m yo_V¿`m C[ w∑VVda _`m©Xm AmhV. oÃH$mUZJarVrb hr yo_Vr amOaÒVm Amh Ag _mÃ _m›` H$bM[mohO.

yo_Vr` aMZmwp∑bS>Mr yo_Vr oeH$VmZm oejH$ AmoU od⁄mœ`m™Zm H$mhr AS>MUr OmUdVmV. _J AZH$ ‡_ mßMr og’Vm oboh `mgmR>r

H$mhr yo_Vr` aMZm - gmfl`m qH$dm AdKS> - bjmV ø`mÏ`m bmJVmV. CXmhaUmW© :

"oÃH$mUm¿`m VrZ AßVJ©V H$mZmßMr ]arO 2 H$mQ>H$mZ EdT>r AgV' `m ‡_ mMr og’Vm XVmZm Imbr XmIdbbr yo_Vr`aMZm H$amdr bmJV. _J og’Vm ghO obohVm V.

`m AmH•$VrV ∠A + ∠B + ∠C = 180o h Ò[Ô> bjmV V. W PQ afm BC afbm g_mßVa afm H$mT>br Amh.

B

∠ ∠ ∠ ∠ ∠ B

AP Q

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∠ ∠ ∠ ∠ ∠ C

∠∠∠∠∠A∠ ∠ ∠ ∠ ∠ B ∠ ∠ ∠ ∠ ∠ C

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9. CËH«$m§V yo_VrÀ`m H$mhr emIm

Am[U emb yo_Vrbm H$mhrgm digm KmbyZ oÃH$mUmgß]ßYr H$mhr ‡˝mßMr MMm© H$br AmoU AßVam¿`m [naoMV Ï`mª bmhrAßVamda R>db. yo_Vr hm odf` JoUVmV odef _hÀdmMm _mZbm OmVm. `m odf`mVrb ‡_ , gyÃ kmZm¿`m A›` emImVC[ w∑V R>abr AmhV. h odYmZ [wT> H$bÎ`m bIZmVyZ H´$_mZ CbJS>V OmB©b. `m H$maUmZ yo_Vr odf` gmVÀ`mZ ‡dmhramohbm Amh. `mMmhr AßXmO `m od^mJmVrb bIZm_wi B©b.

"The Elements' hm wp∑bS>Mm JßW EdT>m JmObm H$s ZßVaMr 2000 df Vm emimßVyZ H´$o_H$ JßW ÂhUyZ oZ`moOV H$bmhmVm. OJmVrb H$mUVhr H$o_H$ [wÒVH$ BVH$ df emimßVyZ bmdb Jbb Zmhr. [naUm_r wp∑bS>M Zmd JoUVr ÂhUyZ eVH$mZweVH$bhmZ Wmamß¿`m AmR>mda oQ>Hy$Z amohb Amh.

[U `m emÒÃkmZ obohbÎ`m JßWmVrb ‡_ mVyZ H$mhr J•hrV yo_Vr¿`m ‡JVrg AS>W˘`mMr R>abr. 19 Ï`m eVH$m¿`mgwadmVr[ ™V hm Kmi Mmby hmVm. À`m¿`m J•ohVm_‹ H$mhr A[yU©Vm Amh h JmD$g `m JoUVkmZ CbJSy>Z XmIdb d ZdrZ

yo_Vr obohbr. À`mbm Am[br yo_Vr oOdßV[Ur ‡H$moeV H$a `mM Y° © Pmb Zmhr. Vr À`m¿`m _•À yZßVa 1825 ¿`m gw_mamg‡og’ Pmbr. `m ‡H$meZmZßVa yo_Vr¿`m ‡JVrVrb AS>Wi Xya Pmb d yo_Vr¿`m ododY emIm emYÎ`m JÎ`m. 19 Ï`meVH$m¿`m CŒmamYm©V ar_mZ `mZ `m yo_VrV _mR>r ^a KmVbr d AdH$memgmR>r C[ w∑V yo_VrMm emY bmdbm. `m yo_Vr¿`mAmYmaZ Ab]Q>© AmBZÒQ>mBZ m emÒÃkmZ Theory of Relativity _Yrb gm[jVm og’mßV og’ H$bm. BÏhmZodM bm]mMÏhÒH$sd `mhmZ ]mÎ`mB© `m yo_Vr VkmßZr yo_Vr¿`m A‰`mgmV _hÀd[yU© ^a KmVbr.

XH$m∞V© (De Carte) `m JoUVkm¿`m bjmV EH$ _yb yV gßH$Î[Zm Ambr À`mZ JoUVmVrb gd© gV≤ (Real) gßª`mßMm gßMbjmV KVbm d À`m]am]a gai afdarb q]XyMm gßM [U bjmV KVbm. À`m¿`m ‹`mZmV Amb H$s h Xm›hr gßM VwÎ`]b AmhV.`mMm AW© afda 0 `m gßª gmR>r EH$ q]Xy oZo¸V H$bm Va ‡À H$ gßª gmR>r EH$M q]Xy afda AgVm d ‡À H$ q]XygmR>r EH$MgV≤ gßª`m oZo¸V H$aVm V. `mbm EH$mg EH$ gß]ßY (One-to-one Correspondence) Ag ÂhUVmV. `m emYm_wi ‡À H$q]Xybm EH$ gßª`m oZo¸V Pmbr d À`m_wi yo_Vr d gßª`m `mßM ZmV Ò[Ô> Pmb.

XH$m∞V© `mZ _J H$mQ>H$mZmV N>XUmË`m XmZ afm H$mT>Î`m. À`m AmH•$Vr_‹ XmIdbÎ`m AmhV.

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AmVm X AmoU Y `m Ajmß¿`m AmYmamZ A hm g_Vbmda Agbbm q]Xy A (x, y) `m ghoZXeH$ XmZ gV≤ gßª`mßZr oZo¸VH$aVm Ambm. XH$m∞V©¿`m m emYmVyZ d°õofH$ yo_Vr (Analytical Geometry) m odf`mbm gwÍ$dmV Pmbr. m yo_VrV AmVm]rOJUrVr` gßH$Î[Zm JmdÎ`m JÎ`m d `mVyZ wp∑bS>¿`m yo_Vrbm ‡JVrgmR>r ZdrZ gmYZ o_imb.

XH$m∞V© `mZ XmZ AjmßVrb H$mZ 900 Mm KVbm hmVm. H$mQ>H$mZm_wi `m yo_VrV gm[[Um Z∏$sM Ambm. [U W H$mQ>H$mZMø`m`bm hdm Aer JaO _mÃ Zmhr. hm H$mZ H$mR>bmhr Mmby eH$b. Imbr oXbbr AmH•$Vr [hm :

darb AmH•$VrV XmZ AjmßV xoy hm H$mZ J EdT>m KVbm Amh. Aer AjmßMr aMZm [U KU e∑` AgÎ`mM H$mhr JoUVkmßZrXmIdb. [U `m aMZbm bmH$_m›`Vm _mÃ o_imbr Zmhr. H$maU `m yo_VrV H$mhr oÃH$mUo_Vr gyÃ gVV dm[amdr bmJVmV.

da oXbbr yo_Vr gyÃ, ‡_ Am[U ‡Vbmda A‰`mgV AgVm. À`m_wi hr yo_Vr o¤o_VrV (Two Dimensions) C[`mJmVV. [U Ï`dhmamV AZH$ AmH$ma oÃo_Vr` (Three Dimensions) AgVmV À`m_wi o¤o_Vr` wp∑bS>r` yo_Vr A‰`mgmbm

A[war [S>V. `m JaO¿`m [yV©VgmR>r _J oÃo_Vr yo_VrMm odH$mg Pmbm. `m yo_VrV _ybV: wp∑bS>r` yo_VrMr _ybVÀdMgm_mdbr AmhV. \$∑V XmZ Ajmß¿`m OmJr VrZ Aj X, Y, Z ø`md bmJVmV. Imbr oXbbr AmH•$Vr - 48 Am[Î`m [naM`mMrAmhM.

darb AmH•$VrV VrZ Aj H$mQ>H$mZmV H$mT>bb AmhV. A q]Xy oÃo_VrV oZo¸V H$a `mgmR>r (x, y, z) h ghoZXeH$ _mhrVAgmd bmJVmV. m oÃo_Vr AdH$mem_wi AmH$memVrb J´h-Vma d À`mßM ´_UmM _mJ© oZo¸V H$aU gm[ Pmb. am∞H$Q>Mm ‡dmg-_mJ© AmIVm Ambm. oÃo_VrV AgUma gd© AmH$ma JoUVr` [’VrZ A‰`mgVm Amb.

d°kmoZH$mßZr o¤o_VrVyZ - oÃo_Vr` yo_VrMm A‰`mg ghO[U ‡JV H$bm. [U H$mhr d°kmoZH$mßZm MVwW©o_Vr, [ßMo_Vr `mAdH$memVrb yo_VrMr JaO ^mgy bmJbr. hm A‰`mg _J n o_Vr[ ™V (n hr Z°goJ©H$ gßª`m Amh) goXe (Vector) `mgßH$Î[ZVyZ ‡JV H$bm. ]ZmH$ AmoU ohb]Q>© `m emÒÃkmßZr A_yV© J•ohVH$mß¿`m ghmÊ`mZ hm A‰`mg AZßVo_Vr[ ™V dmT>dbm.

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Vm[ ™V yo_VrV wp∑bS>r` AdH$me (Euclidean space) bmH$mßZm _mhrV hmVr. ]ZmH$ d ohb]Q>© `mß¿`m emYm_wi ]ZmH$AdH$me (Banach space), ohb]Q>© AdH$me (Hilbert space) Aem A_yV© AdH$mem_‹ yo_VrMm A‰`mg ‡JV Pmbm.A_yV© AdH$me JoUVkmßZr oZ_m©U H$b. hr H$di ]m°o’H$ H$gaV Agmdr Ag dmQ> `mMm gß d Amh [U Vg Pmb Zmhr. JoUVmVrbH$mhr AoZU© rV ‡˝, eßH$m `m_wi _mJu bmJÎ`m.

Am[U amhVm Vr [•œdr OdiOdi JmbmH•$Vr Amh. À`m_wi OhmO, od_mZ, aÒV `mßM _mJ© gai afV ZgVmV. `m gd©_mJm™Zm dH´$Vm AgV. À`m_wi Jmbmdarb yo_VrMm emY KU JaOM R>ab. `m yo_Vrbm Jmbm‹`m` (spherical Geometry)

Ag ÂhUVmV. `m yo_VrV AgUmË`m H$mhr Jß_Vr, dJi[Um XmId `mgmR>r Imbr XmZ CXmhaU oXbr AmhV.

XmZ eha aÒÀ`mZ OmS>br H$s À`mß¿`m_Yrb AßVa oH$bm_rQ>a `m EH$H$mZ _mOb OmV. CXmhaUmW© : _wß]B© V [wU h AßVa190 oH$bm_rQ>a Amh. h AßVa [•œdrda H$mT>bÎ`m dH´$ _mJm©Z XmIdVm V. h AßVa XmZ ehamßZm OmS>UmË`m gai afVrb Zmhr.[U XmZ eha g_w– _mJm©Z OmS>br Jbr Va _mÃ XmZ q]Xyß_Yrb AßVa _mO `mgmR>r oH$bm_rQ>a h AßVa C[`mJr hmV Zmhr. hAßVa _mO `mgmR>r Zm∞oQ>H$b oH$bm_rQ>a (knotical kilometers) h EH$H$ dm[aVmV.

gmB©gmR>r H$mhr ‡_mU Imbr oXbr AmhV.

1 oH$bm_rQ>a = 3280 \y$Q>,

1 Zm∞Q>rH$b oH$bm_rQ>a = 3800 \y$Q>,

1 _°b = 1.61 oH$bm_rQ>a = 5280 \y$Q>,

1 Zm∞Q>rH$b _°b = 6080 \y$Q>,

hr ‡_mU gw_ma AmhV.

Am[Î`mbm [•œdrMr oÃ¡`m _mhrV Amh.

`m gßª Mm C[`mJ H$Í$Z [•œdrJmbmdarb dH´$s` AßVa _mOVm V.

[wT>rb JmbmH•$VrV [•œdrdarb dH´$_mJ© XmIdbm Amh.

`m AmH•$VrV A AmoU B hr XmZ eha Xw]B© AmoU _wß]B© XmIdbr AmhV. mßZm OmS>Umam g_w– _mJ© AB hm [•œdr¿`m [mR>rdaAgÎ`mZ Vm dH´$ Amh. wp∑bS>¿`m yo_Vr‡_mU Vm AB m gai afdÍ$Z _mOVm Uma Zmhr. hm _mJ© [•œdr¿`m [mQ>mVyZ OmUmamhmB©b d hr gßH$Î[Zm e∑` Zmhr.

AmVm OJmVrb VrZ eha - › y m∞H©$, _m∞ÒH$m AmoU _wß]B© - odMmamV KD$. hr VrZ eha OmS>Uma od_mZmM _mJ© XmIdb H$sEH$ oÃH$mU V`ma hmB©b. hm oÃH$mU AßVamimV bQ>H$bbm Agb d À`m¿`m wOm dH´$ AmH$mam¿`m AgVrb. Aem oÃH$mUmVrbH$mZmßMr ]arO 1800 [jm _mR>r Va 5400 [jm bhmZ AgV Ag `m yo_VrV og’ H$aVm V. WmS>∑`mV wp∑bS>Mr yo_VrJmbm‹`m`mV (Spherical Geomtry) H$mhr dim oZÍ$[`mJr R>aV. H$mJX dm[Í$Z ‡Vbmda oÃH$mU H$mT>bm Va XmZ q]XyZmOmS>Umam gai _mJ© d [•œdr¿`m [•>^mJmda VdT>ÁmM AßVamMm H$mT>bbm _mJ© OdiOdi gmaIm AgVm. ÂhUyZ wp∑bS>Mr

yo_Vr H$mJXmda ÂhUO EH$mM ‡Vbmda H$mT>bÎ`m AmH•$À`mßgmR>rM C[ w∑V R>aV. H$mJXmMm ‡Vb d [•œdrMm VdT>ÁmM AmH$mamMm^mJ `mV AoVgy _ \$aH$ [S>Vm. hm \$aH$ Xwb©ojV H$a `mEdT>m AgVm.

AmVm da oXbbr VrZ eha dH´$ _mJm©Z [•œdr¿`m [•>^mJmda EH$_H$mßZm OmS>br Va [•œdrdarb hm oÃH$mU oÃo_VrVXmIdVm Vm. [•œdrdarb gd© Xe q^Vrda bmdbÎ`m ZH$memVhr XmIdb OmVmV `mgmR>r hm q^Vrda bQ>H$dbbm ZH$mem

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oÃo_VrVyZ o¤o_VrV AmUmdm bmJVm. AmVm o¤o_VrV da Zm|Xbbr eha H$er XmIdmdrV hm ‡˝ Ambm. `mgmR>r yJmbmVrbJoUVr [’V oZo¸V H$amdr bmJbr. mgmR>r ‡jo[V yo_Vr (Projective Geometry) hm odf` A‰`mgmdm bmJVm. yo_VrgmR>rh H$m © emÒÃkmßZr `mΩ` [’VrZ [ma [mS>b d [•œdrMm Jmb q^Vrdarb ZH$memV XmIdVm Ambm.

[•œdrMm Jmb À`m_mZmZ AmH$mam¿`m gßX^m©V [wÓH$iM gwg_ Amh. AZH$ JoUVr` gyÃ Aem A‰`mgmgmR>r emYyZ H$mT>brAmhV. Am[U AmVm EH$ Am]S>Ym]S> XJS> odMmamV KD$ d `m XJS>mda ISy> dm[Í$Z EImXr dH´$ Agbbr AmH•$Vr H$mTy>. AmVm hoMÃ H$mJXmda aImQ>m`M Agb Va [nadV©ZmgmR>r bmJUmar ‡jo[V yo_Vr oH$Vr ‡JV Agmdr bmJb `mMr H$Î[ZmM H$amdrbmJb. `mgß]ßYr bmJUmar JoUVr` gyÃ emYU OdiOdi Ae∑` Amh. [U H$mhr ‡_mUmV AmVm gm r dmT>Î`m AmhV. AmVmoZaoZam˘`m ‡H$maM H∞$_a [U C[b„Y hmD$ bmJb AmhV. WmS>∑`mV H∞$_am `m ßÃmMm emY ‡jo[V yo_Vrer oZH$Q>Mm gß]ßYR>D$Z Amh. gy ©‡H$memV MmbV AgVmZm Am[Î`m_mJM MmbV Agbbr gmdbr [hmdr. ÂhUO Aem yo_VrMr AmiI Am[Î`mbmbdH$a [Q>b. da oXbbr Xm›hr CXmhaU ‡JV wp∑bS>r` yo_Vr ‡H$mamßVrb AmhV.

yo_Vr¿`m da H$bÎ`m qMVZmVyZ _J H$mhr JoUVkmßZr yo_Vrodf`r AZH$ gßH$Î[Zm AmUIr ‡JV H$Î`m. EImXm IS>]S>rV,S>m|Jami, MT>CVmamßMm ‡Xe oZ`mOZmgmR>r AmH•$Vr]’ H$am`Mr Agb Va H$mUVr yo_Vr C[`mJr hmB©b hm odMma [wT> Ambm.Imbrb AmH•$À`m-51 [hm.

oÃo_VrV H$mT>bÎ`m `m VrZ AmH•$VrVyZ oZaoZamir dH´$Vm XmIdbr Amh. [ohÎ`m oMÃmV A [mgyZ B [ ™V OmUmam gaig[mQ> aÒVm ‡Vbmda XmIdbm Amh. XwgË`m oMÃmV EH$ CßMdQ>m Ambbm dH´$ ÒdÍ$[mMm AmH$ma XmIdbm Amh Va oVgË`moMÃmV S>m|JamßMr amßJ, CßM d Imb XË`mImË`mßMm ‡Xe XmIdbm Amh. `m oV›hr oMÃmßVrb ‡Xe EH$mM g_Vb H$mJXmdaaImQ>m`Mm Amh.

`m aImQ>ZmgmR>r Am[Î`mbm oZaoZam˘`m dH´$ AmbImßMr, diU, Imbr, CßMr `mßMm oÃo_VrV gßª`mÀ_H$ A‰`mg H$amdmbmJb. W H$mhr diU odoMÃ AdÒWVhr AgVrb. [U AmVm `m[°H$s ]arM aMZm JoUVr` [’VrZ gmßJVm V. › yQ>Z db]ZrO `m 17 Ï`m eVH$mVrb XmZ emÒÃkmßZr H$bZemÒÃ (Calculus) `m JoUVm¿`m emIMm emY bmdbm. `m emYm_wiJoUVmV H´$mßVr Pmbr Ag ÂhUVm B©b. EIm⁄m IS>]S>rV AmH$mamVrb dÒVy_‹ Agbb MT>CVma, CßMr, Imbr `mßMm AmVmemY KVm D$ bmJbm. À`m_wi ‡Vbmda Aem dÒVyßM (S>m|JamVrb XË`m-Ima, gwiH$) AmVm oMÃmßH$Z H$aU e∑` Pmb. AerododYVm JoUVr` [’VrZ A‰`mgUmar emIm ÂhUO H$bZ- yo_Vr (Differential Geometry) hm`.

yo_Vr oZaoZam˘`m AmH$mamßMm emY KV. odúmV AZH$ dÒVy AmhV d À`mßMm AmH$ma yo_Vr¿`m _m‹`_mVyZ g_OyZ KVmVm. `m od^mJmV H$mhr WmS>Ám dÒVyßM AmH$ma bjmV KVb AmhV.

AmH•$Vr - 51

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emiV oejUmbm gwÍ$dmV H$Î`m[mgyZ Am[U [wÒVH$ dm[am`bm gwÍ$dmV H$aVm. `m[wÒVH$m¿`m AmH$mamH$S> Am[U H$Yr oZaIyZ [mohb Amh H$m ? Am[U Or [wÒVH$ hmVmiVmÀ`m[°H$s ]hVH$mßMm gmYmaU AmH$ma Imbrb AmH•$À`m-52 _‹ XmIdbm Amh :

AWm©V gd©M [wÒVH$ `mM AmH$mamMr AgVmV Ag R>mgyZ gmßJm`M Zmhr. hm AmH$mabß]-Mm°H$mZmMm Amh. EH$ _mÃ Ia H$s AJXr Mm°ag AmH$mamV N>m[bbr [wÒVH$ AJXrπ$oMVM AmT>iVrb. da H$mT>bbm Mm°H$mZ AßXmO 6 : 10 m AmH$mamMm Amh. Am[U dm[aVmÀ`m dhrMm AmH$mahr bß]-Mm°H$mZmgmaIm AgVm. (Mm°ag ZgVm) KamV q^Vrda bmdbb\$mQ>m, q^VrdaM bQ>H$dbbr H$mbXoe©H$m AÎ]_ _‹ oMH$Q>dbb \$mQ>m, Am[Î`m X·amMmAmH$ma, Kam¿`m oIS>∑`mßM AmH$ma, OdUm¿`m, A‰`mgm¿`m Q>]bmßMm AmH$ma bß]-Mm°H$mZm¿`m AmH•$VrgmaI AgVmV. (ghgmMm°ag ZgVmV.) Ï`dhmamV Am[U ¡`m Mm°H$mZr dÒVy dm[aVm À`m[°H$s ]h˛VH$ dÒVy bß]-Mm°H$mZmH•$Vr AgVmV.

emÒÃkmßZr AmVm Aem AZH$ bmI dÒVyßMr Zm|X H$br Amh W \$∑V H$mhr Zmd oXbr AmhV.

_mUgm¿`m earamVrb AZH$ Ad`dmV 6 : 10 ‡_mU AmT>iyZ Amb Amh. CXmhaUmW© _mUgm¿`m Hß$]a[mgyZ Imß⁄m[ ™Veara aMZm bjmV KVbr Va EH$ Mm°H$mZ bjmV Vm. hm Mm°H$mZ 6 x 10 `m ÒdÍ$[mMm Agb Va Vm gwS>m°b AgÎ`mM Am[Î`mbjmV V. ]wQ>∑`m _mUgm¿`m ]m]VrV hm Mm°H$mZ Mm°agmH$S> PwH$Vm Va AoV CßM AgUmam _mUgmV h 6 : 10 ‡_mU o\$gH$Q>bbAgV. BQ>mob`Z JoUVk o\$]mZmH$s `mZ ‡W_M 6 : 10 h ‡_mU _hÀdmM AgÎ`mM emYyZ H$mT>b.

bm pÏh›gr `m H$bmH$mamZ Am[Î`m [|qQ>J_‹ `m ‡_mUmMm C[`mJ H$Í$Z KVbm d Vm OJ‡og’ Pmbm.

AmVm Am[U yo_VrVrb `m ‡_mUmM JoUV g_OyZ KD$. AB `m AmH$mamMr afm ø`m.

AB `m afda C q]Xy Agm KD$ H$s AC =

CB

AB AC

h ‡_mU og’ hmV. AmVm g_Om AB = 1 AmoU AC = x Amh Va CB = 1 - x hm . da KVbb ‡_mU x

= 1 - x

1 x

Ag obohVm B©b. `mdÍ$Z Am[U x Mr qH$_V H$mTy>. W x2 + x - 1 = 0 h g_rH$aU gmS>dyZ Am[Î`mbm x = -1 + √5 hr 2

qH$_V H$mT>Vm V. JUH$ ßÃ dm[Í$Z x = -1 + √5

= 0.6180339 = 0.6 qH$dm 6/10 Amh. 2

`m 6 : 10 ‡_mUm]‘b AJXr ‡mMrZ H$mim[mgyZ AZH$ VÀdk, H$bmH$ma, JoUVk Hw$Vwhb Ï`∑V H$arV AgV. h ‡_mU"B©úar' Amh Ag AZH$ JßWmVyZ À`mßZr Z_yX H$b Amh. `m ‡_mUmbm "gmZar ‡_mU' (Golden Ratio) AWdm aÂ` JwUmŒmaAghr ÂhQ>b Amh. `mbm "aÂ` JwUmŒma' Aghr ÂhUVmV. `m ‡_mUmgß]Yr AmVm H$mhr JßW obohb AmhV Va A_naH$VyZ `m‡_mUmMm Imbda emY KUma Fibonocci Quarterly `m ZmdmM oZ`VH$mobH$ ‡og’ hmV.

Am[Î`m ‡À`H$m¿`m KamV Amagm (Mirror) AgVmM. À`mMm C[`mJ Hw$Qw>ß]mVrb gd©M _mUg oXdgmVyZ H$mhr dim H$aVmV.H$mhtZm Va AmaemV S>mH$md `mMm N>ßXM AgVm. Amagm ‡Vbmg_mZ Agb Va Am[Î`mbm MmßJb ‡oVq]] oXgV. JoUVm¿`m

10

6

[wÒVH$mMm AmH$ma

AmH•$Vr - 52

A BCx 1-x

55

^mfV AmaemV [hmU ÂhUO ÒdV: ( I ) d AmaemVrb ‡oVq]] g_Om A Amh. Amagm Oa CŒm_ ‡Vb Agb Va ‡oVq]]mM‡oVq]] ÂhUOM A2 = I hm . Ia Va ‡oVq]]mMr hrM JoUVr` Ï`mª`m H$aVm B©b. W ÒdV: d ‡oVq]] `mV EH$dm∑`VmqH$dm g_mZVm VÀd bjmV V. hr EH$Í$[Vm h EH$ yo_Vr` VÀd Amh.

ododY H$maUmßZr dVw©imH•$VrMm C[`mJ H$bmÀ_H$ Ï`dhmamV OmÒV H$bm OmVm. H$maU gm¢X © emÒÃmV EH$dm∑`Vm - g_mZVm- ‡_mU]’Vm - ]mßYgyX[Um `m JoUVr` - yo_Vr` VÀdmZm _hÀdmM ÒWmZ Amh. Imbr oXbbr dVw©imH•$Vr [hm :

darb AmH•$VrV "g_mZVm VÀd' (Symmetry) VßVmVßV ^abb bjmV V. AY©dVw©i ACB d ADB hr X AjmV ‡oVq]]

AmhV. VgM AY©dVw©i CBD AmoU CAD hr Y - AjmV ‡oVq]] AmhV. ‡À H$ Ï`mg afbm AgmM gß]ßY Ï`∑V H$aVm B©b.

W AZH$ ‡H$maMr ‡oVq]o]V - g_mZVm (Mirror Symmetry) XmIdVm V. A D, D B, B C AmoU C A hrdVw©ir` [nadV©Z g_mZVmhr (rotational Symmelry) ApÒVÀdmV Amh. h VÀdhr dVw©imdarb ‡À H$ q]Xybm bm^b Amh.

À`m_wiM gm›`m-_mÀ`mßM XmoJZ H$aVmZm dVw©i hm yo_Vr` AmH$ma _mR>Ám ‡_mUmV dm[abm OmVm. gm›`mV _T>dbbm _mÀ`mMm

hma pÒÃ`m J˘`mV KmbVmV VÏhm ‡oVq]o]V g_mZVm, [nadV©Z g_mZVm Aer VÀd `m XmoJ›`mV [hm`bm o_iVmV. _mÀ`mß¿`mAmH$mamV Agbbr g_mZVm, EH$dm∑`Vm, ‡_mU]’Vm, ]mßYgyXVm hr gmZmamZ C[`moObbr hr VÀd À`m_wi hmamM gm¢X © d

À`m ]am]aM ÒÃrM gm¢X ©hr dmT>dVmV. AbßH$mamVyZ XS>bbr hr H$bm Ornamental Symmetry ÂhUyZ AmiIbr OmV.

Am[b earahr `m ‡oVq]o]V g_mZVM EH$ CXmhaU Amh. `mV earamMm S>mdm AmoU COdm ^mJ g_modÔ> hmVm. earamVMhË`mda ZmH$ hm Ad`d gdm©V Dß$M Ad`d AgVm. g_mZVm VÀd AZw d `mgmR>r Am[Î`mbm _‹` AjmMr JaO AgV. hr afm

Òdm^modH$M (‡À`j Z H$mT>Vm) ZmH$mdÍ$Z H$mT>br OmV. m AjmdÍ$ZM earamVrb g_mZVm VÀd OmIVm V. Am[Î`m gßÒH•$VrV

_wbrM gm¢X © dU©Z H$aVmZm "oVM ZmH$ Mm\$H$irgmaI gai Amh' Ag ÂhQ>b OmV, V `m_wiM. ZmH$m¿`m afdÍ$Z H$mZm-Jmbm[ ™V AmKiUmar dH´$Vm _J AoYH$ aIrd AmoU Ò[Ô> ]ZV. ]T>] ZmH$mMr _mUg `m_wiM g_mZVm VÀdmbm _wH$VmV.

Aem yo_Vr` AmH$mamßM kmZ H$Í$Z K `mgmR>r AmVm JßW C[b„Y AmhV.

yo_VrMr CÀH$mßV ‡JVr AmVm Z°goJ©H$ yo_Vr (Natural Geometry), Fractal yo_Vr, Neural Networks, DNA

maps, Aem odf`mVyZ ‡JV hmV Amh.

`m od^mJm¿`m edQ>r yo_VrVrb ¤°V og’mßV (Duality Principal) `mM EH$ CXmhaU XD$Z hm odf` gß[dy.

`m [wpÒVH$Vrb [ohÎ`m od^mJmV q]Xy AmoU afm mßM gß]ßY ¤°V-ÒdÍ$[mM AmhV Ag ÂhQ>b hmV. ([hm : AmH•$Vr - 4 d 5)

(1) XmZ q]Xy_wi EH$M afm oZp˚MV hmV.

AmH•$Vr - 53

XX&

A B

Y

D

Y&

C

g_mZVm VŒd

56

(2) [aÒ[amßZm N>XUmË`m XmZ afda EH$M q]Xy AgVm

Ag h gß]ßY AmhV. wp∑bS>r` yo_Vr `m gß]ßYmda AmYmnaV Amh. W Am[U q]Xy AmoU afm `mß¿`m Ï`mª`m `mOÎ`m hmÀ`m.

CÀH´$mßV yo_VrV AmVm _yb yV gßkmß¿`m Ï`mª`mM H$aV ZmhrV. À`mß¿`m_Yrb gß]ßY oXb OmVmV À`mßZm J´ohV (axioms,

propositions) ÂhUVmV. `m AmYmamda _J odf` [wT> \w$bV OmVm. `m odf`mbm _J A_yV© ÒdÍ$[ ‡m· hmV. [U H$mhrdim

odúmVrb EImX H$mS>, ‡˝ gmS>d `mgmR>r Ag A_yV© odf` C[ w∑V R>aVmV. mgmR>r Imbr EH$ CXmhaU oXb Amh. g_Om q]Xyd afm `m gßkmß¿`m Ï`mª`m oXbÎ`m ZmhrV. AmoU da Zm|Xbbr J•ohV (1 d 2) À`mßMm gß]ßY Ò[Ô> H$aVmV. AmVm Imbr oXbbr

AmH•$Vr [hm :

da H$mT>bÎ`m AmH•$Vr-54 _‹ EH$ Jmb H$mT>bm Amh. AB d CD h `m JmbmM XmZ Ï`mg AmhV. ABCD h `m JmbmM EH$AoVdVw©i (Great Circle) Amh. (`m dVw©imMm H|$–q]Xy d JmbmMm _‹`q]Xy EH$M AgVm ) ABCD m Mma q]XyVyZ OmUma \$∑V

EH$M dVw©i AgV. XwgË`m AmH•$VrV XmZ AerM AoVdVw©i oXbr AmhV `m XmZ dVw©imßMm Ï`mg AB Amh. XmZ AoVdVw©im_wi

\$∑V EH$M Ï`mg oZo¸V hmVm. AmVm Am[U

Ï`mg = q]Xy

AoVdVw©i = afm

Aer H$Î[Zm H$Í$. W q]Xy AmoU afm mßMr oMÃ Am[U Xwb©ojV H$Í$. da oXbÎ`m g_mZVVyZ Imbr oXbbr ¤°V J•ohVH$ oZo¸V

H$aVm VmV.

(1) XmZ q]Xy_wi (Ï`mgm_wi) EH$M afm (AoVdVw©i) oZo¸V hmV.

(2) XmZ afmß_wi (AoVdVw©im_wi) EH$M q]Xy (Ï`mg) oZo¸V hmVm.

h ]Xb H$Î`mda EH$ ZdrZM yo_Vr V`ma hmV h ghO bjmV V. `m CXmhaUmdÍ$Z bjmV V H$s darb J•ohV wp∑bS>r`yo_Vr gmaIrM AmhV. `m ¤°VmVyZ yo_Vr¿`m OmUrdm _mÃ odÒV•V PmÎ`m AmhV. Aem A_yV© ÒdÍ$[mVyZ yo_VrMr ]mJ ododY

‡H$ma \w$bV OmV.

AmVm edQ>r _bm EdT>M gwMdm`M Amh. gd© odúmV JoUV VÀd VwSw>ß] ab Amh. S>mig[U m odúmH$S>, À`mVrb g•Ô>rH$S>,

dÒVyßH$S> [mhV amhm. [naUm_r VwÂhmbm Wrb JoUVÀdmMm ]mY hmB©b. ZdoZo_©VrMm AmZßX A_`m©X AgVm.

C

A

D

B

O

A

B

AmH•$Vr - 54

AjaOwiUrQ>mB©[ BZmÏhQ>g©, gmVmam. \$mZ ï- (02162) 234372

© dmB© VmbwH$m JoUV A‹`m[H$ _ßS>i, dmB©

gß[mXH$ZmJe eßH$a _mZ

gß[mXZ gm¯lr. AÍ$U gmdßVlr. ^JdmZ wO]igm°. AZwamYm Omer

‡H$meH$lr.oXZH$a od. \$amßXA‹`j, dmB© VmbwH$m JoUV A‹`m[H$ _ßS>idmB©

‡H$meZ df©18 gflQ>|]a 2008

bIH$‡m.S>m∞.gXmoed Xd12, ‡gr o]ÎS>tJ,_im [UOr ï 403001, Jmdm.Xya‹dZr - (0832)-2225816

_yÎ` Í$[ - 50/-

‡ÒVwV [wpÒVH$m V`ma H$a `m¿`m H$m_mV S>m∞.lr.adt– Hw$bH$Uu, [wU `mßZr d À`mß¿`mQ≠>ÒQ>Z H$bÎ`m AmoW©H$ _XVr_wiM, AÎ[ oH$_VrV hr [wpÒVH$m Am[Î`m[ ™V V Amh.

dmB© VmbwH$m JoUV A‹`m[H$ _ßS>i, dmB©1. o_l gßª`m - ‡m._.am.amB©bH$a 15.002. od^mJUr d oVMr ^mdßS> - S>m∞.d.J.oQ>H$H$a 15.003. JoUVr wp∑VdmX - ‡m.`.Zm.dmbmdbH$a 15.004. JoUV _m°O - lr.Zm.eß._mZ 15.005. H$mZmMß oÃ^mOZ - ‡m._.am.amB©bH$a 15.006. gßª`mZJarV ^Q>Hß$Vr - lr.[r.H$.lroZdmgZ≤ 20.00

AZwdmX : S>m∞._YwH$a Xe[mßS>7. JoUVmVrb H$`mg, Ia d MwH$bb - S>m∞.d.J.oQ>H$H$a 20.008. jÃ\$i AmoU KZ\$i, H$mhr VmpŒdH$ [°by - S>m∞.adt– ]m[Q> 20.009. F$U gßª`m - ‡m._.am.amB©bH$a 20.00

lr.Zm.eß._mZ10. yo_Vr` aMZm - lr.Zm.eß._mZ 20.0011. g_o_Vr AmoU BVa - ‡m._.am.amB©bH$a 20.0012. oXZXoe©H$_Ybr OmXy - lr.[r.H$.lroZdmgZ 20.00

AZwdmX : S>m∞._YwH$a Xe[mßS>13. EH$mM _miM _Ur - lr.Zm.eß._mZ 20.0016. XmZ _wbmIVr - gßH$bZ : lr.Zm.eß._mZ 20.0017. JoUVtM oH$Òg - S>m∞.d.J.oQ>H$H$a 20.0018. oZX©eH$ yo_Vr - ‡m._.am.amB©bH$a 20.0019. oÃH$mU ZJargh yo_VrMr ododYVm - ‡m.S>m∞.gXmoed Xd 50.0020. gßª`m_mobH$m - lr.oXbr[ JmQ>qIS>rH$a 40.00

21. A[yUm™H$ : AmOrH$Sy>Z oeH$m (gr.S>r.) - ‡m._.am.amB©bH$a 40.0022. H$m[m AmoU OmS>m (gr.S>r.) - ‡m._.am.amB©bH$a 50.00

JoUV A‹``Z - A‹`m[Z - odH$gZ gßÒWm, ZmoeH$

1. M.T.S. / N.T.S. d A›` Ò[Ym™gmR>r - lr.b._. wO]i,JoUV ‡ÌZgßJh Xdimbr H±$[ 100.00

2. Question Bank (Mathematics) For M.T.S. / - L.M.Bhujbal, 100.00N.T.S. and other competitive examinations. Bhas Bhamre,Nasik

3. JoUV _mVr - odMmamßZm MmbZm XUma - S>m∞.d.J.oQ>H$H$a, 45.00JoUV odf`H$ bI

4. ObX AmH$S>_mS>r¿`m 13 w∑À`m - _yi bIH$-lr.XŒmmÃ` H$m[aH$a, 25.00AZwdmX : lr.oXbr[ JmQ>qIS>rH$a

5. JoUV : N>ßX-AmZßX Ã°_mogH$ (dmof©H$ dJ©Ur) 75.00

1. JoUV JwUJmZ - lr.Zm.eß._mZ 50.00

2. CŒmamVyZ ‡ÌZoZo_©Vr (8 dr, 9 dr, 10 dr gmR>r) - lr.Zm.eß._mZ 125.00yo_Vrodf`H$ ‡ÌZmßMr odef MMm©.

3. odkmZ ÂhUO H$m` ? - lr.Zm.eß._mZ 5.00

4. e°joUH$ gßX © : odkmZ odf`H$ ¤°_mogH$ (dmof©H$ dJ©Ur) 125.00

gd© [wÒVH$mßgmR>r lr.Zm.eß._mZ, 1123, ^mΩ`mX`, ]m˜Uemhr, dmB©.Xya‹dZr : (02167) 220766. _m]mB©b : 9226283203. `mß¿`mer gß[H©$ gmYmdm.

bIH$modf`r :

‡m.S>m∞.gXmoed Xd

od⁄mdmMÒ[Vr [Xdr

[XÏ wŒma JoUV odf`mM ‡m‹`m[H$, gßemYH$, _mJ©Xe©H$.

^maV, H∞$ZS>m AmoU A_naH$m W ZmH$ar.

[XÏ wŒma JoUV JßWmßM bIH$.

oZaoZam˘`m gßÒWmßM AmOrd gXÒ`.

_hmamÔ≠> JoUV oejH$ [nafXM A‹`j (1990)

_amR>r ^mfm ï gmohpÀ`H$, bIH$ AmoU gßemYH$.

[wpÒVH$m H$. 19

dmB© VmbwH$m JoUV AÜ`m[H$ _§S>i, dmB©¤mam : lr.Zm.eß._mZ, 1123, ^mΩ`mX`, ]m˜Uemhr, dmB© - 412 803.

Xya‹dZr : (02167) 220766, Email : [email protected]

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