The moving power of mathematical invention is not reasoning but imagination. – A.DEMORGAN 

11.1 Hkwfedk (Introduction)

d{kk XI esa] oS'ysf"kd T;kfefr dk vè;;u djrs le; f}&foeh; vkSj f=k&foeh; fo"k;ksa osQ ifjp; esa geus Lo;a dks osQoy dkrhZ; fof/ rd lhfer j[kk gSA bl iqLrd osQ fiNys vè;k; esa geus lfn'kksa dh ewy ladYiukvksa dk vè;;u fd;k gSA vc ge lfn'kksa osQ chtxf.kr dk f=k&foeh; T;kfefr esa mi;ksx djsaxsA f=k&foeh; T;kfefr esa bl mikxe dk mn~ns'; gS fd ;g blosQ vè;;u dks vR;ar ljy ,oa lq#fpiw.kZ (lqxzkg;) cuk nsrk gSA*

bl vè;k; esa ge nks ¯cnqvksa dks feykus okyh js[kk osQ fnd~&dksT;k o fno~Q&vuqikr dk vè;;u djsaxs vkSj fofHkUu fLFkfr;ksa esa varfj{k esa js[kkvksa vkSj ryksa osQ lehdj.kksa] nks js[kkvksa] nks ryksa o ,d js[kk vkSj ,d ry osQ chp dk dks.k] nks fo"keryh; js[kkvksa osQ chp U;wure nwjh o ,d ry dh ,d ¯cnq ls nwjh osQ fo"k; esa Hkh fopkj foe'kZ djsaxsA mijksDr ifj.kkeksa esa ls vf/dka'k ifj.kkeksa dks lfn'kksa osQ :i esa izkIr djrs gSaA rFkkfi ge budk dkrhZ; :i esa Hkh vuqokn djsaxs tks dkykarj esa fLFkfr dk Li"V T;kferh; vkSj fo'ys"k.kkRed fp=k.k izLrqr dj losQxkA

11.2 js[kk osQ fno~Q&dkslkbu vkSj fno~Q&vuqikr (Direction Cosines and Direction Ratios of a Line)

vè;k; 10 esa] Lej.k dhft,] fd ewy ¯cnq ls xqtjus okyh lfn'k js[kk L }kjk x, y vkSj z-v{kksa osQ lkFk Øe'k α] β vkSj γ cuk, x, dks.k fno~Q&dks.k dgykrs gSa rc bu dks.kksa dh dkslkbu uker% cosα, cosβ vkSj cosγ js[kk L osQ fno~Q&dkslkbu (direction cosines or dc's)dgykrh gSaA

;fn ge L dh fn'kk foijhr dj nsrs gSa rks fno~Q&dks.k] vius laiwjdksa esa vFkkZr~ π-α, π-β vkSj π-γ ls cny tkrs gSaA bl izdkj] fno~Q&dkslkbu osQ fpÉ cny tkrs gSaA

è;ku nhft,] varfj{k esa nh xbZ js[kk dks nks foijhr fn'kkvksa esa c<+k ldrs gSa vkSj blfy, blosQ fno~Q&dkslkbu osQ nks lewg gSaA blfy, varfj{k esa Kkr js[kk osQ fy, fno~Q&dkslkbu osQ vf}rh; lewg osQ fy,] gesa Kkr js[kk dks ,d lfn'k js[kk ysuk pkfg,A bu vf}rh; fno~Q&dkslkbu dks  l, m vkSj n osQ }kjk fufnZ"V fd, tkrs gSaA

fVIi.kh varfj{k esa nh xbZ js[kk ;fn ewy ¯cnq ls ugha xqtjrh gS rks bldh fno~Q&dkslkbu dks Kkr djus osQ fy,] ge ewy ¯cnq ls nh xbZ js[kk osQ lekarj ,d js[kk [khaprs gSaA vc ewy ¯cnq ls buesa ls ,d lfn'k js[kk osQ fno~Q&vuqikr Kkr djrs gSa D;ksafd nks lekarj js[kkvksa osQ fno~Q&vuqikrksa osQ lewg leku (ogh) gksrs gSaA

,d js[kk osQ fno~Q&dkslkbu osQ lekuqikrh la[;kvksa dks js[kk osQ fno~Q&vuqikr (direction ratios or dr's) dgrs gSaA ;fn ,d js[kk osQ fno~Q&dkslkbu l, m, n o fno~Q&vuqikr a, b, c gksa rc fdlh 'kwU;srj λ ∈ R osQ fy, a = λl, b=λm vkSj c = λn

eku yhft, ,d js[kk osQ fno~Q&vuqikr a, b, c vkSj js[kk dh fno~Q&dkslkbu l, m, n gSA rc

= = (eku yhft,), k ,d vpj gSA

blfy, l = ak, m = bk, n = ck ... (1)

ijarq l2 + m2 + n2 = 1

blfy, k2 (a2 + b2 + c2) = 1

;k k =

vr% (1) ls] js[kk dh fno~Q&dkslkbu (d.c.’s )

fdlh js[kk osQ fy, ;fn js[kk osQ fno~Q&vuqikr Øe'k% a, b, c gS] rks ka, kb, kc; k ≠ 0 Hkh fno~Q&vuqikrksa dk ,d lewg gSA blfy, ,d js[kk osQ fno~Q&vuqikrksa osQ nks lewg Hkh lekuqikrh gksaxsA vr% fdlh ,d js[kk osQ fno~Q&vuqikrksa osQ vla[; lewg gksrs gSaA

11.2.1 nks ¯cnqvksa dks feykus okyh js[kk dh fno~Q&dkslkbu (Direction cosines of a line passing through two points)

D;ksafd nks fn, ¯cnqvksa ls gksdj tkus okyh js[kk vf}rh; gksrh gSA blfy, nks fn, x, ¯cnqvksa
P(x1, y1, z1) vkSj Q(x2, y2, z2) ls xqtjus okyh js[kk dh fno~Q&dkslkbu dks fuEu izdkj ls Kkr dj ldrs gSa (vko`Qfr 11-3(a)A

eku yhft, fd js[kk PQ dh fno~Q&dkslkbu l, m, n gSa vkSj ;g x, y vkSj z-v{k osQ lkFk dks.k Øe'k% α, β, γ cukrh gSaA

eku yhft, P vkSj Q ls yac [khafp, tks XY-ry dks R rFkk S ij feyrs gSaA P ls ,d vU; yac [khafp, tks QS dks N ij feyrk gSA vc ledks.k f=kHkqt PNQ esa, ∠PQN = γ (vko`Qfr 11.3 (b)) blfy,

cosγ =

blh izdkj cosα =

vr% ¯cnqvksa P(x1, y1, z1) rFkk Q(x2, y2, z2) dks tksM+us okys js[kk[kaM PQ fd fno~Q&dkslkbu

, , gSaA

tgk¡ PQ =

fVIi.kh ¯cnqvksa P(x1, y1, z1) rFkk Q(x2, y2, z2) dks tksM+us okys js[kk[kaM osQ fno~Q&vuqikr fuEu izdkj ls fy, tk ldrs gSaA

x2 – x1, y2 – y1, z2 – z1, ;k x1 – x2, y1 – y2, z1 – z2

mnkgj.k 1 ;fn ,d js[kk x, y rFkk z-v{kksa dh /ukRed fn'kk osQ lkFk Øe'k% 90°, 60° rFkk 30° dk dks.k cukrhs gS rks fno~Q&dkslkbu Kkr dhft,A

gy eku yhft, js[kk dh fno~Q&dkslkbu l, m, n gSA rc l = cos 90° = 0, m = cos 60° = ,
n = cos 30° =

mnkgj.k 2 ;fn ,d js[kk osQ fno~Q&vuqikr 2] &1] &2 gSa rks bldh fno~Q&dkslkbu Kkr dhft,A

gy fno~Q&dkslkbu fuEuor~ gSa

, ,

vFkkZr~

mnkgj.k 3 nks ¯cnqvksa (– 2, 4, – 5) vkSj (1, 2, 3) dks feykus okyh js[kk dh fno~Q&dkslkbu Kkr dhft,A

gy ge tkurs gSa fd nks ¯cnqvksa P (x1, y1, z1) vkSj Q(x2, y2, z2) dks feykus okyh js[kk dh fno~Q&dkslkbu

gSa] tgk¡ PQ =

;gk¡ P vkSj Q Øe'k% (– 2, 4, – 5) vkSj (1, 2, 3) gSaA

blfy, PQ = =

blfy, nks ¯cnqvksa dks feykus okyh js[kk dh fno~Q&dkslkbu gSa%

mnkgj.k 4 x, y vkSj z-v{kksa dh fno~Q&dkslkbu Kkr dhft,A

gy x-v{k Øe'k% x, y vkSj z-v{k osQ lkFk 0°, 90° vkSj 90° osQ dks.k cukrk gSA blfy, x-v{k dh fno~Q&dkslkbu cos 0°, cos 90°, cos 90° vFkkZr~ 1,0,0 gSaA

blh izdkj y-v{k vkSj z-v{k dh fno~Q&dkslkbu Øe'k% 0, 1, 0 vkSj 0, 0, 1 gSaA

mnkgj.k 5 n'kkZb, fd ¯cnq A (2, 3, – 4), B (1, – 2, 3) vkSj C (3, 8, – 11) lajs[k gSaA

gy A vkSj B (dks feykus okyh js[kk osQ fno~Q&vuqikr

1 –2, –2 –3, 3 + 4 vFkkZr~ – 1, – 5, 7 gSaA

B (vkSj C dks feykus okyh js[kk osQ fno~Q&vuqikr 3 –1, 8 + 2, – 11 – 3, vFkkZr~, 2, 10, – 14 gSaA

Li"V gS fdABvkSj BC osQ fno~Q&vuqikr lekuqikrh gSaA vr% AB vkSj BC lekarj gSaAq ijarq ABvkSj BC nksuksa esa B (mHk;fu"B gSA vr% A, B, vkSj C lajs[k ¯cnq gSaA

iz'ukoyh 11-1

1. ;fn ,d js[kk x, y vkSj z-v{k osQ lkFk Øe'k% 90°, 135°, 45° osQ dks.k cukrh gS rks bldh fno~Q&dkslkbu Kkr dhft,A

2. ,d js[kk dh fno~Q&dkslkbu Kkr dhft, tks funsZ'kka{kksa osQ lkFk leku dks.k cukrh gSA

3. ;fn ,d js[kk osQ fno~Q&vuqikr –18, 12, – 4, gSa rks bldh fno~Q&dkslkbu D;k gSa\

4. n'kkZb, fd ¯cnq (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) lajs[k gSaA

5. ,d f=kHkqt dh Hkqtkvksa dh fno~Q&dkslkbu Kkr dhft, ;fn f=kHkqt osQ 'kh"kZ ¯cnq
(3, 5, – 4), (– 1, 1, 2) vkSj (– 5, – 5, – 2) gSaA

11.3 varfj{k esa js[kk dk lehdj.k (Equation of a Line in Space)

d{kk XI esa f}&foeh; ry esa js[kkvksa dk vè;;u djus osQ i'pkr~ vc ge varfj{k esa ,d js[kk osQ lfn'k rFkk dkrhZ; lehdj.kksa dks Kkr djsaxsA

,d js[kk vf}rh;r% fu/kZfjr gksrh gS] ;fn

(i);g fn, ¯cnq ls nh xbZ fn'kk ls gksdj tkrh gS] ;k

(ii);g nks fn, x, ¯cnqvksa ls gksdj tkrh gSA

11.3.1 fn, x, ¯cnq A ls tkus okyh rFkk fn, x, lfn'k osQ lekarj js[kk dk lehdj.k (Equation of a line through a given point A and parallel to a given vector )

ledksf.kd funsZ'kka{k fudk; osQ ewy ¯cnq O osQ lkis{k eku yhft, fd ¯cnq A dk lfn'k   gSA eku yhft, fd ¯cnq A ls tkus okyh rFkk fn, x, lfn'k  osQ lekarj js[kk l gSA eku yhft, fd l ij fLFkr fdlh LosPN ¯cnq P dk fLFkfr lfn'k  gS (vko`Qfr 11-3)A

rc  lfn'k   osQ lekarj gS vFkkZr~ =λ, tgk¡ λ ,d okLrfod la[;k gSA

ijarq

vFkkZr~ λ = 

foykser% izkpy λ osQ izR;sd eku osQ fy, ;g lehdj.k js[kk osQ fdlh ¯cnq P dh fLFkfr iznku djrk gSA vr% js[kk dk lfn'k lehdj.k gS%

= ... (1)

fVIi.kh ;fn  gS rks js[kk osQ fno~Q&vuqikr a, b, c gS vkSj foykser% ;fn ,d js[kk osQ fno~Q&vuqikr a, b, c gksa rksjs[kk osQ lekarj gksxkA ;gk¡ b dks u le>k tk,A

lfn'k :i ls dkrhZ; :i O;qRiUu djuk (Derivation of Cartesian Form from Vector Form)

a, b, c gSa eku yhft, fdlh ¯cnq P osQ funsZ'kkad (x, y, z) gSaA rc

vkSj 

bu ekuksa dks (1) esa izfrLFkkfir djosQ vkSj , osQ xq.kkadksa dh rqyuk djus ij ge ikrs gSa fd

x = x1 + λa;y = y1 + λ b;z = z1+ λc ... (2)

;s js[kk osQ izkpy lehdj.k gSaA (2) ls izkpy λ dk foyksiu djus ij] ge ikrs gSa%

= ... (3)

;g js[kk dk dkrhZ; lehdj.k gSA

fVIi.kh ;fn js[kk dh fno~Q&dkslkbu l, m, n gSa] rks js[kk dk lehdj.k

= gSaA

mnkgj.k 6 ¯cnq (5, 2, – 4) ls tkus okyh rFkk lfn'k osQ lekarj js[kk dk lfn'k rFkk dkrhZ; lehdj.kksa dks Kkr dhft,A

gy gesa Kkr gS] fd

 =

blfy,] js[kk dk lfn'k lehdj.k gS%

= [(1) ls]

pw¡fd js[kk ij fLFkr fdlh ¯cnq P x, y, z) dh fLFkfr lfn'k  gS] blfy,

=

λ dk foyksiu djus ij ge ikrs gSa fd

=

tks js[kk osQ lehdj.k dk dkrhZ; :i gSA

  

11.4 nks js[kkvksa osQ eè; dks.k (Angle between two lines)

eku yhft, fdL1 vkSjL2 ewy ¯cnq ls xqtjus okyh nks js[kk,¡ gSa ftuosQ fno~Q&vuqikr Øe'k% a1, b1, c1 vkSj a2, b2, c2, gSA iqu% eku yhft,fd L1 ij ,d ¯cnq P rFkk L2 ij ,d ¯cnq QgSA vko`Qfr 11-6 esa fn, x, lfn'k OP vkSj OQ ij fopkj dhft,A eku yhft, fd OP vkSj OQ osQ chp U;wu dks.k θ gSA vc Lej.k dhft, fd lfn'kksa OP vkSj OQ osQ ?kVd Øe'k% a1, b1, c1 vkSj a2, b2, c2 gSaA blfy, muosQ chp dk dks.k θ

 cosθ = }kjk iznÙk gSA

iqu% sin θosQ :i esa] js[kkvksa osQ chp dk dks.k

sin θ=

.... (2)

cosθ = |l1 l2 + m1m2 + n1n2| (D;ksafd)... (3)

vkSj sin θ =... (4)

fno~Q&vuqikr a1, b1, c1 vkSj a2, b2, c2 okyh js[kk,¡

(i)yacor~~ gS] ;fn θ = 90°, vFkkZr~ (1) ls a1a2 + b1b2 + c1c2 = 0

(ii)lekarj gS] ;fn θ = 0, vFkkZr~ (2) ls 

vc ge nks js[kkvksa osQ chp dk dks.k Kkr djsaxs ftuosQ lehdj.k fn, x, gSaA ;fn mu js[kkvksa  =vkSj = osQ chp U;wu dks.k θ है 

rc 

dkrhZ; :i esa ;fn js[kkvksa%=... (1)

vkSj=... (2)

cos θ ==

osQ chp dk dks.k θ gS tgk¡ js[kk,¡ (1) o (2) osQ fno~Q&vuqikr Øe'k% a1, b1, c1 rFkk a2, b2, c2 gS rc

mnkgj.k 7  fn, x, js[kk&;qXe

osQ eè; dks.k Kkr dhft,

gy eku yhft, 

nksuksa js[kkvksa osQ eè; dks.k θ gS] blfy,

cosθ =

=

vr% θ = cos–1 

mnkgj.k 8 js[kk&;qXe%

=

vkSj ==

osQ eè; dks.k Kkr dhft,A

gy igyh js[kk osQ fno~Q&vuqikr 3] 5] 4 vkSj nwljh js[kk osQ fno~Q&vuqikr 1] 1] 2 gSaA ;fn muosQ chp dk dks.k θ gks rc

cosθ =  =

vr% vHkh"V dks.k cos–1 gSA

11.5 nks js[kkvksa osQ eè; U;wure nwjh (Shortest Distance between two lines)

varfj{k esa ;fn nks js[kk,¡ ijLij izfrPNsn djrh gS rks muosQ chp dh U;wure nwjh 'kwU; gSA vkSj varfj{k esa ;fn nks js[kk,¡ lekarj gS rks muosQ chp dh U;wure nwjh] muosQ chp yacor~~ nwjh gksxh vFkkZr~ ,d js[kk osQ ,d ¯cnq ls nwljh js[kk ij [khapk x;k yacA

blosQ vfrfjDr varfj{k esa] ,slh Hkh js[kk,¡ gksrh gS tks u rks izfrPNsnh vkSj u gh lekarj gksrh gSA okLro esa ,slh js[kkvksa osQ ;qXe vleryh; gksrs gSa vkSj bUgsa fo"keryh; js[kk,¡ (skew lines) dgrs gSaA mnkgj.kr;k ge vko`Qfr 11-7 esa x, y vkSj z-v{k osQ vuqfn'k Øe'k% 1] 3] 2 bdkbZ osQ vkdkj okys dejs ij fopkj djrs gSaA

js[kk GE Nr osQ fod.kZ osQ vuqfn'k gS vkSj js[kk DB, A osQ Bhd Åij Nr osQ dksus ls xqtjrh gqbZ nhokj osQ fod.kZ osQ vuqfn'k gSA ;s js[kk,¡ fo"keryh; gSa D;ksafd os lekarj ugha gS vkSj dHkh feyrh Hkh ugha gSaA

nks js[kkvksa osQ chp U;wure nwjh ls gekjk vfHkizk; ,d ,sls js[kk[kaM ls gS tks ,d js[kk ij fLFkr ,d ¯cnq dks nwljh js[kk ij fLFkr vU; ¯cnq dks feykus ls izkIr gksa rkfd bldh yackbZ U;wure gksA

U;wure nwjh js[kk[kaM nksuksa fo"keryh; js[kkvksa ij yac gksxkA

11.5.1 nks fo"keryh; js[kkvksa osQ chp dh nwjh (Distance between two skew lines)

vc ge js[kkvksa osQ chp dh U;wure nwjh fuEufyf[kr fof/ ls Kkr djrs gSaA eku yhft, l1 vkSj l2 nks fo"keryh; js[kk,¡ gS ftuosQ lehdj.k (vko`Qfr 11.8) fuEufyf[kr gSa%

js[kk l1 ij dksbZ ¯cnq S ftldh fLFkfr lfn'k vkSj l2 ij dksbZ ¯cnq T ftldh fLFkfr lfn'k . gS] yhft,A rc U;wure nwjh lfn'k dk ifjek.k] ST dk U;wure nwjh dh fn'kk esa iz{ksi dh eki osQ leku gksxk (vuqPNsn 10.6.2)A

;fn l1 vkSj l2 osQ chp dh U;wure nwjh lfn'k gS rks ;g nksuksa b1 vkSj b2 ij yac gksxhA  dh fn'kk esa bdkbZ lfn'k  bl izdkj gksxh fd

 =

blfy, vHkh"V U;wure nwjh 

dkrhZ; :i (Cartesian Form)

js[kkvksa%

l2 :=

osQ chp dh U;wure nwjh gS%

11.5.2 lekarj js[kkvksa osQ chp dh nwjh (Distance between parallel lines)

;fn nks js[kk,¡ l1 ;fn l2 lekarj gSa rks os leryh; gksrh gSaA ekuk nh xbZ js[kk,¡ Øe'k%

... (1)

vkSj

gSa] tgk¡ l1 ij ¯cnq S dk fLFkfr lfn'kvkSj l2 ij ¯cnq T dk fLFkfr lfn'k gS (vko`Qfr 11.9)

D;ksafd l1, vkSj l2 leryh; gSA ;fn ¯cnq T ls l1 ij Mkys x, yac dk ikn P gS rc js[kkvksa l1 vkSj l2 osQ chp dh nwjh = |TP|

eku yhft, fd lfn'kksa vkSj osQ chp dk dks.k θ gSA rc]

stgk¡ js[kkvksa l1 vkSj l2 osQ ry ij yac bdkbZ lfn'k gSa

ijarq 

blfy, (3) ls ge ikrs gSa fd

blfy, Kkr js[kkvksa osQ chp U;wure nwjh

mnkgj.k 9 js[kkvksa l1 vkSj l2 osQ chp dh U;wure nwjh Kkr dhft, ftuosQ lfn'k lehdj.k gS%

vkSj 

gy lehdj.k (1) o (2) dh  vkSj , ls rqyuk djus ij ge ikrs gSa fd

blfy,

vkSj

=

bl izdkj  =

blfy, nh xbZ js[kkvksa osQ chp dh U;wure nwjh

mnkgj.k 10 fuEufyf[kr nh xbZ js[kkvksa l1 vkSj l2 :

vkSj ==osQ chp U;wure nwjh Kkr dhft,A

gy  nksuksa js[kk,¡ lekraj gSaA (D;ksa\) gesa izkIr gS fd

blfy, js[kkvksa osQ chp dh nwjh

d=  =  

=  gSA

iz'ukoyh 11-2

1. n'kkZb, fd fno~Q&dkslkbu  okyh rhu js[kk,¡ ijLij yacor~ gSaA

2. n'kkZb, fd ¯cnqvksa (1, – 1, 2), (3, 4, – 2) ls gksdj tkus okyh js[kk ¯cnqvksa (0, 3, 2) vkSj (3, 5, 6) ls tkus okyh js[kk ij yac gSA

3. n'kkZb, fd ¯cnqvksa (4, 7, 8), (2, 3, 4) ls gksdj tkus okyh js[kk] ¯cnqvksa (– 1, – 2, 1), (1, 2, 5) ls tkus okyh js[kk osQ lekarj gSA

4. बिदु (1,2,3) से गुजरने वाली रेखा का समीकरण ज्ञात कीजिए जो सदिश शरप के समांतर है।

5. ¯cnq ftldh fLFkfr lfn'kls xq”kjus o lfn'k  dh fn'kk esa tkus okyh js[kk dk lfn'k vkSj dkrhZ; :iksa esa lehdj.k Kkr dhft,A

6. ml js[kk dk dkrhZ; lehdj.k Kkr dhft, tks ¯cnq (– 2, 4, – 5) ls tkrh gS vkSjosQ lekarj gSA

7. ,d js[kk dk dkrhZ; lehdj.k  gSA bldk lfn'k lehdj.k Kkr dhft,A

8. fuEufyf[kr js[kk&;qXeksa osQ chp dk dks.k Kkr dhft,%

9.  fuEufyf[kr js[kk&;qXeksa osQ chp dk dks.k Kkr dhft,%

(i)

10. p dk eku Kkr dhft, rkfd js[kk,¡  vkSj  ijLij yac gksaA

11. fn[kkb, fd js[kk,¡   ijLij yac gSaA

12. js[kkvksa  osQ chp dh U;wure nwjh Kkr dhft,%

13. js[kkvksa  osQ chp dh U;wure nwjh Kkr dhft,A

14. js[kk,¡] ftuosQ lfn'k lehdj.k fuEufyf[kr gS] osQ chp dh U;wure nwjh Kkr dhft,%

15. js[kk,¡] ftudh lfn'k lehdj.k fuEufyf[kr gSa] osQ chp dh U;wure Kkr dhft,%

vè;k; 11 ij fofo/ iz'ukoyh

1. mu js[kkvksa osQ eè; dks.k Kkr dhft,] ftuosQ fno~Q&vuqikr a, b, c vkSj b – c, c – a, a-b gSaA

2. x-v{k osQ lekarj rFkk ewy&¯cnq ls tkus okyh js[kk dk lehdj.k Kkr dhft,A

3. ;fn js[kk,¡  ijLij yac gksa rks k dk eku Kkr dhft,A

4.

osQ chp dh U;wure nwjh Kkr dhft,A

5. ¯cnq (1, 2, – 4) ls tkus okyh vkSj nksuksa js[kkvksa  vkSj 

 ij yac js[kk dk lfn'k lehdj.k Kkr dhft,A