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smStI ArQSwsqr dy isDWq
È؇ jyVw Awpxy Awp aqpwdn dI pRkirAw dy nwl pYdw huMdw hY[” jd aqpwdk aqpwdn pRkirAw iv`c pRXog hox
vwlI ivivD AwgqW (Buim, Srm Aqy puMjI) ƒ aplbD krdy hn[ qW auh loxmMd Awmdn dw srjn krdy hn jo
lgwn, mjdurI Aqy ibAwj dy rup iv`c swDn svwimAW ƒ pRwpq huMdI hY[ iehI Agy aqpwidq vsquAW dy leI
mMg pYdw krdI hY[ ies qrW 5UrqI Awp AwpxI mMg pYdw krdI hY[ ieh qrk ies Dwrxw qyy AwDirq hY ik
swDn – mwlkW rwhI Arjq swrI Awmdn aun vsquAw dy krX dy iv`c Krc ho jWdI hY ijnW dy aqpwdn dy iv`c
auh mddgwr sn[ Awmdn dw jyVw ihSw Krc nhI huMdw, auh bc jwdw hY Aqy ausdw invyS ho jwdW hY[ ies
qrW bcq inScX qo invyS dy brwbr hovygI[ jykr dovy iv`co koeI AMqr rwihdW hy qW ibAwj dI dr dy mwDXm
nwl smwnqw sDwipq ho jWdI hY[ klwiskI ArQ-Swp`DrI ibAwj ƒ bcq dw ienWm mwndy hn[ ibAwj dI
dr ijnI aucI hovygI, bcq vI aunI hI v`D hovygI Aqy ivlomS vI[ iedy ivprIq ivAwj dI dr ijnI nIvI
hovygI , invyS dy leI isDiqAW dI mMg vI aunI hI v`D hovygI Aqy ivlomS vI[ jykr iksy idqy hoey smy iv`c
invyS iv`c bcq v`D jWdI hY, qW ibAwj dI dr igr jwvygI[ invyS v`Dygw Aqy bcq qd G`t jwvygI jd qk
ik purx rojgwr dy p`Dr qy puMj kr dovy smwn nhI ho jWdy[ ieho ijhW ies leI ik bcq ibAwj dr dw v`Ddw
Pln mwnI jWdI hY Aqy invyS ibAwj dr dw G`tdw Pln (function) mwnI jWdI hY[
bcq Aqy invyS iv`c smwnqw dw qMqr icqr 5.1 iv`c ivKwieAw igAw
hY ijQy ss bcq vkr hY Aqy II invyS vkr hY[ dovy vkr E ibMdu qy
iek dujy ƒ kwtdy hn ijQy Or ibAwj dr hY Aqy bcq Aor invyS dovy
OA dy brwbr hn[ jykr invyS iv`c v`DI huMdI hY qW invyS vkr sjy
pwsy ƒ srk kr I’I’ ho jwdW hY Aqy Or ibAwj qy OC invyS bcq
OA qo v`D hY[ klwiskI ArQ-Swp`DrI dy muqbwk, bcq vkr SS
AwpxI pwhl vwlI isDqI iv`c hI rwihdW hY qd invyS iv`c v`DI huMdI
hY[ bcq Aqy invyS iv`c smwnqw rKx dy leI ibAwj dr v`DygI[ies
icqr iv`c Or qo v`Dkr Or ivKweI geI rY[ies ivAwj dr qy bcq
1
vkr invyS SS invyS vkr I’I’ ƒ E′ qy k`tdw hY[iesleI bcq Aqy
invyS dovy OB dy brwbr hMudy hn[
icqr 5.1
mudrw ArQ-ivvsQw iv`c sy dw inXm dI vYDqw mudrw dy klwiskI
pirmwx isDWq qy vI inrBr krdI hY jyVw ieh dsdw hY ik kImq-p`Dr mudrw dI purqI dw Pln hMudw hY[
bIjgixqIX rup iv`c, MV = PT ijQy M, V, P Aqy T krmwnUsr mudrw dI purqI, mudrw dw sMcln vyg, kImq
p`Dr Aqy mudrw rwhI kIqw igAw lyx-dyx (jW kul aqpwdn) hY[ ieh smIkrx d`sdw hY ik ArQ-ivvsQw iv`c
kul mudrw sPIiq MV brwbr hn aqpwdn dw kul mu`l PT ikh mMn kr kI V Aqy T siQr hn mudrW dI
1
purqI (M) iv`c bdlwv nwl kImq p`Dr (P) iv`c smwnupwiqk bdlwv hUuMdw hY[ieh ies mwinAqw qy AwDwirq
hY ik mudrw ivinXm dw mwiDXm hY[
mudrw dI mwqrw , kul aqpwdn Aqy kImq p`Dr ƒ icqr 5.2 iv`c
ivKwieAw igAw hY ijQy kImq p`Dr ƒ SIiqj-AS qy ilAw igAw hY
Aqy kul aqpwdn ƒ Anulmb AS qy ilAw igAw hY[ MV mudrw purqI
vkr hY jyVw ik ryktyngulr hwieprbolw (rectangular hyperbola)
huMdw hY[ ieho ijhw iesleI ik smIkrx MV=PT vkr dy swry ibMduAw
qy ivDmwn hY[ auqpwdn p`Dr OQ idqW hox qy, mudrw dI mqrw dy
nwl myl KwdW hoieAw kyvl iek kImq p`Dr qy OP hovygw ijvyjw MV
vkr qy m ibMdu hY[ jykr mudrw dI mwqrw v`DdI hY qW MV vkr sjI
Aor srk kr M V vkr ho jwvygw[ pirxwmsrup, kImq p`Dr OP icqr 5.2
1
v`Dkr OP ho jwvygw, aqpwdn p`Dr OQ idqw hox qy[ kImq p`Dr
1
iv`c ieh virDI mudrw dI mwqrw iv`c virDI dy iblkul smwnupwiqk
hY, Bwv PP = MM .
1 1
klwiskI ArQ-SwsqrI dy muqbwk, jykr mudrw dI mwqrw ƒ duguxw kIqw jwvy qW kImq p`Dr vI doguxw ho
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