NCERT पाठ्यपुस्तक पाठ 9 - त्रिकोणमिति के कुछ अनुप्रयोग, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 9 - त्रिकोणमिति के कुछ अनुप्रयोग, कक्षा 10, गणित Class 10 Notes | EduRev

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216 xf.kr
9
9.1 Hkwfedk
fiNys vè;k; esa vkius f=kdks.kferh; vuqikrksa osQ ckjs esa vè;;u fd;k gSA bl vè;k;
esa vki oqQN mu fof/;ksa osQ ckjs esa vè;;u djsaxs ftuesa f=kdks.kfefr dk iz;ksx vkiosQ
vkl&ikl osQ thou ls tqM+k gksrk gSA f=kdks.kfefr ,d izkphure fo"k; gS ftldk vè;;u
iwjs txr osQ fo}ku djrs vk, gSaA tSlk fd ge vè;k; 8 esa crk pqosQ gSa fd f=kdks.kfefr
dk vkfo"dkj bl ckr dks è;ku esa j[kdj fd;k x;k Fkk fd bldh [kxksydh esa
vko';drk iM+rh FkhA rc ls vkt rd [kxksyfon~ bldk iz;ksx i`Foh ls xzgksa vkSj rkjksa
dh nwfj;k¡ ifjdfyr djus esa djrs vk, gSaA f=kdks.kfefr dk iz;ksx Hkwxksy vkSj ukSpkyu
esa Hkh fd;k tkrk gSA f=kdks.kfefr osQ Kku dk iz;ksx ekufp=k cukus vkSj ns'kkarj (longitude)
vkSj v{kka'k (latitude) osQ lkis{k ,d }hi dh fLFkfr Kkr djus esa dh tkrh gSA
losZ{kd 'krkfCn;ksa ls f=kdks.kfefr dk iz;ksx djrs
vk jgs gSaA mUuhloha 'krkCnh dh ^c`gr~
f=kdks.kferh; losZ{k.k* fczrkuh Hkkjr dh ,d
,slh fo'kky losZ{k.k ifj;kstuk Fkh ftlosQ fy,
nks c`gÙke fFk;ksMksykbV dk fuekZ.k fd;k x;k
FkkA 1852 esa losZ{k.k djus osQ nkSjku fo'o osQ
lcls Å¡ps ioZr dh [kkst dh x;h FkhA 160 km
ls Hkh vf/d nwjh ij fLFkr vyx&vyx N%
osaQnzksa ls bl ioZr osQ f'k[kj dk izs{k.k fd;k
x;kA 1856 esa bl f'k[kj dk ukedj.k lj tkWtZ
,ojsLV osQ uke ij fd;k x;k ftlus losZizFke
fo'kky fFk;ksMksykbV dks vf/Ñr fd;k vkSj
budk iz;ksx fd;kA (lkeus cuh vko`Qfr
nsf[k,)A vc ;s fFk;ksMksykbV nsgjknwu esa fLFkr
Hkkjr losZ{k.k osQ laxzgky; esa izn'kZu osQ fy,
j[ks x, gSaA
f=kdks.kfefr osQ oqQN vuqiz;ksx
fFk;ksMksykbV
,d losZ{k.k ;a=k] tks f=kdks.kfefr osQ
fu;eksa ij vk/kfjr gS] dk iz;ksx
,d ?kw.khZ VsyhLdksi ls dks.kksa dk
ekiu djus esa fd;k tkrk gSA
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216 xf.kr
9
9.1 Hkwfedk
fiNys vè;k; esa vkius f=kdks.kferh; vuqikrksa osQ ckjs esa vè;;u fd;k gSA bl vè;k;
esa vki oqQN mu fof/;ksa osQ ckjs esa vè;;u djsaxs ftuesa f=kdks.kfefr dk iz;ksx vkiosQ
vkl&ikl osQ thou ls tqM+k gksrk gSA f=kdks.kfefr ,d izkphure fo"k; gS ftldk vè;;u
iwjs txr osQ fo}ku djrs vk, gSaA tSlk fd ge vè;k; 8 esa crk pqosQ gSa fd f=kdks.kfefr
dk vkfo"dkj bl ckr dks è;ku esa j[kdj fd;k x;k Fkk fd bldh [kxksydh esa
vko';drk iM+rh FkhA rc ls vkt rd [kxksyfon~ bldk iz;ksx i`Foh ls xzgksa vkSj rkjksa
dh nwfj;k¡ ifjdfyr djus esa djrs vk, gSaA f=kdks.kfefr dk iz;ksx Hkwxksy vkSj ukSpkyu
esa Hkh fd;k tkrk gSA f=kdks.kfefr osQ Kku dk iz;ksx ekufp=k cukus vkSj ns'kkarj (longitude)
vkSj v{kka'k (latitude) osQ lkis{k ,d }hi dh fLFkfr Kkr djus esa dh tkrh gSA
losZ{kd 'krkfCn;ksa ls f=kdks.kfefr dk iz;ksx djrs
vk jgs gSaA mUuhloha 'krkCnh dh ^c`gr~
f=kdks.kferh; losZ{k.k* fczrkuh Hkkjr dh ,d
,slh fo'kky losZ{k.k ifj;kstuk Fkh ftlosQ fy,
nks c`gÙke fFk;ksMksykbV dk fuekZ.k fd;k x;k
FkkA 1852 esa losZ{k.k djus osQ nkSjku fo'o osQ
lcls Å¡ps ioZr dh [kkst dh x;h FkhA 160 km
ls Hkh vf/d nwjh ij fLFkr vyx&vyx N%
osaQnzksa ls bl ioZr osQ f'k[kj dk izs{k.k fd;k
x;kA 1856 esa bl f'k[kj dk ukedj.k lj tkWtZ
,ojsLV osQ uke ij fd;k x;k ftlus losZizFke
fo'kky fFk;ksMksykbV dks vf/Ñr fd;k vkSj
budk iz;ksx fd;kA (lkeus cuh vko`Qfr
nsf[k,)A vc ;s fFk;ksMksykbV nsgjknwu esa fLFkr
Hkkjr losZ{k.k osQ laxzgky; esa izn'kZu osQ fy,
j[ks x, gSaA
f=kdks.kfefr osQ oqQN vuqiz;ksx
fFk;ksMksykbV
,d losZ{k.k ;a=k] tks f=kdks.kfefr osQ
fu;eksa ij vk/kfjr gS] dk iz;ksx
,d ?kw.khZ VsyhLdksi ls dks.kksa dk
ekiu djus esa fd;k tkrk gSA
f=kdks.kfefr osQ oqQN vuqiz;ksx 217
mUu;u dks.k
n`f"V js[kk
n`f"V js[kk
mUu;u dks.k
oLrq
n`f"V js[kk
mUu;u dks.k
{kSfrt Lrj
bl vè;k; esa ge ;g ns[ksaxs fd fdl izdkj okLro esa ekiu fd, fcuk gh
f=kdks.kfefr dk iz;ksx fofHkUu oLrqvksa dh Å¡pkb;k¡ vkSj nwfj;k¡ Kkr djus esa fd;k tkrk gSA
9.2 Å¡pkb;k¡ vkSj nwfj;k¡
vkb, ge vè;k; 8 esa nh xbZ vko`Qfr 8-1 ij fopkj djas] ftls uhps vko`Qfr 9-1 esa
iqu% [khapk x;k gSA
vko`Qfr 9.1
bl vko`Qfr esa] Nk=k dh vk¡[k ls ehukj osQ f'k[kj rd [khaph xbZ js[kk  AC  dks
n`f"V&js[kk (line of sight) dgk tkrk gSA Nk=k ehukj osQ f'k[kj dh vksj ns[k jgk gSA
n`f"V&js[kk vkSj {kSfrt js[kk ls cus dks.k BAC dks Nk=k dh vk¡[k ls ehukj osQ f'k[kj dk
mUu;u dks.k (angle of elevation) dgk tkrk gSA
bl izdkj] n`f"V&js[kk izs{kd dh vk¡[k osQ ml oLrq osQ ¯cnq dks feykus okyh js[kk
gksrh gS ftls izs{kd ns[krk gSA ns[ks x, ¯cnq dk mUu;u dks.k ml fLFkfr esa] n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS] tcfd ns[kk tk jgk ¯cnq {kSfrt Lrj ls Åij gksrk
gS vFkkZr~ og fLFkfr tcfd oLrq dks ns[kus osQ fy, gesa viuk flj mBkuk gksrk gSA
(nsf[k, vko`Qfr 9-2)A
vko`Qfr 9.2
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216 xf.kr
9
9.1 Hkwfedk
fiNys vè;k; esa vkius f=kdks.kferh; vuqikrksa osQ ckjs esa vè;;u fd;k gSA bl vè;k;
esa vki oqQN mu fof/;ksa osQ ckjs esa vè;;u djsaxs ftuesa f=kdks.kfefr dk iz;ksx vkiosQ
vkl&ikl osQ thou ls tqM+k gksrk gSA f=kdks.kfefr ,d izkphure fo"k; gS ftldk vè;;u
iwjs txr osQ fo}ku djrs vk, gSaA tSlk fd ge vè;k; 8 esa crk pqosQ gSa fd f=kdks.kfefr
dk vkfo"dkj bl ckr dks è;ku esa j[kdj fd;k x;k Fkk fd bldh [kxksydh esa
vko';drk iM+rh FkhA rc ls vkt rd [kxksyfon~ bldk iz;ksx i`Foh ls xzgksa vkSj rkjksa
dh nwfj;k¡ ifjdfyr djus esa djrs vk, gSaA f=kdks.kfefr dk iz;ksx Hkwxksy vkSj ukSpkyu
esa Hkh fd;k tkrk gSA f=kdks.kfefr osQ Kku dk iz;ksx ekufp=k cukus vkSj ns'kkarj (longitude)
vkSj v{kka'k (latitude) osQ lkis{k ,d }hi dh fLFkfr Kkr djus esa dh tkrh gSA
losZ{kd 'krkfCn;ksa ls f=kdks.kfefr dk iz;ksx djrs
vk jgs gSaA mUuhloha 'krkCnh dh ^c`gr~
f=kdks.kferh; losZ{k.k* fczrkuh Hkkjr dh ,d
,slh fo'kky losZ{k.k ifj;kstuk Fkh ftlosQ fy,
nks c`gÙke fFk;ksMksykbV dk fuekZ.k fd;k x;k
FkkA 1852 esa losZ{k.k djus osQ nkSjku fo'o osQ
lcls Å¡ps ioZr dh [kkst dh x;h FkhA 160 km
ls Hkh vf/d nwjh ij fLFkr vyx&vyx N%
osaQnzksa ls bl ioZr osQ f'k[kj dk izs{k.k fd;k
x;kA 1856 esa bl f'k[kj dk ukedj.k lj tkWtZ
,ojsLV osQ uke ij fd;k x;k ftlus losZizFke
fo'kky fFk;ksMksykbV dks vf/Ñr fd;k vkSj
budk iz;ksx fd;kA (lkeus cuh vko`Qfr
nsf[k,)A vc ;s fFk;ksMksykbV nsgjknwu esa fLFkr
Hkkjr losZ{k.k osQ laxzgky; esa izn'kZu osQ fy,
j[ks x, gSaA
f=kdks.kfefr osQ oqQN vuqiz;ksx
fFk;ksMksykbV
,d losZ{k.k ;a=k] tks f=kdks.kfefr osQ
fu;eksa ij vk/kfjr gS] dk iz;ksx
,d ?kw.khZ VsyhLdksi ls dks.kksa dk
ekiu djus esa fd;k tkrk gSA
f=kdks.kfefr osQ oqQN vuqiz;ksx 217
mUu;u dks.k
n`f"V js[kk
n`f"V js[kk
mUu;u dks.k
oLrq
n`f"V js[kk
mUu;u dks.k
{kSfrt Lrj
bl vè;k; esa ge ;g ns[ksaxs fd fdl izdkj okLro esa ekiu fd, fcuk gh
f=kdks.kfefr dk iz;ksx fofHkUu oLrqvksa dh Å¡pkb;k¡ vkSj nwfj;k¡ Kkr djus esa fd;k tkrk gSA
9.2 Å¡pkb;k¡ vkSj nwfj;k¡
vkb, ge vè;k; 8 esa nh xbZ vko`Qfr 8-1 ij fopkj djas] ftls uhps vko`Qfr 9-1 esa
iqu% [khapk x;k gSA
vko`Qfr 9.1
bl vko`Qfr esa] Nk=k dh vk¡[k ls ehukj osQ f'k[kj rd [khaph xbZ js[kk  AC  dks
n`f"V&js[kk (line of sight) dgk tkrk gSA Nk=k ehukj osQ f'k[kj dh vksj ns[k jgk gSA
n`f"V&js[kk vkSj {kSfrt js[kk ls cus dks.k BAC dks Nk=k dh vk¡[k ls ehukj osQ f'k[kj dk
mUu;u dks.k (angle of elevation) dgk tkrk gSA
bl izdkj] n`f"V&js[kk izs{kd dh vk¡[k osQ ml oLrq osQ ¯cnq dks feykus okyh js[kk
gksrh gS ftls izs{kd ns[krk gSA ns[ks x, ¯cnq dk mUu;u dks.k ml fLFkfr esa] n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS] tcfd ns[kk tk jgk ¯cnq {kSfrt Lrj ls Åij gksrk
gS vFkkZr~ og fLFkfr tcfd oLrq dks ns[kus osQ fy, gesa viuk flj mBkuk gksrk gSA
(nsf[k, vko`Qfr 9-2)A
vko`Qfr 9.2
218 xf.kr
{kSfrt Lrj
voueu dks.k
n`f"V js[kk
oLrq
vkb, vc ge vko`Qfr 8-2 esa nh xbZ fLFkfr ij fopkj djsaA ckyduh esa cSBh
yM+dh eafnj dh lh<+h ij j[ks xeys dks uhps dh vksj ns[k jgh gSA bl fLFkfr esa]
n`f"V&js[kk {kSfrt Lrj ls uhps gSA n`f"V&js[kk vkSj {kSfrt js[kk ls bl izdkj cus dks.k dks
voueu dks.k (angle of depression) dgk tkrk gSA
vr% ns[kh tk jgh oLrq ij fLFkr ¯cnq dk voueu dks.k ml fLFkfr esa n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS tcfd ¯cnq {kSfrt js[kk ls uhps gksrk gS vFkkZr~ og
fLFkfr tcfd ns[ks tkus okys ¯cnq dks ns[kus osQ fy, gesa viuk flj uhps >qdkuk gksrk gS
(nsf[k, vko`Qfr 9-3)A
vko`Qfr 9.3
vc vki vko`Qfr 8-3 esa cuh n`f"V&js[kk,¡ vkSj bl rjg cus dks.kksa dks igpku ldrs
gSaA ;s dks.k mUu;u dks.k gSa ;k voueu dks.k\
vkb, ge vko`Qfr 9-1 dks iqu% ns[ksaA ;fn vki lgh ek;us esa fcuk ekis gh ehukj
dh Å¡pkbZ  CD Kkr djuk pkgrs gSa rks blosQ fy, vkidks fdl tkudkjh dh vko';drk
gksrh gS\ blosQ fy, fuEufyf[kr rF;ksa dk Kku gksuk vko';d gksrk gS%
(i) nwjh DE tgk¡ Nk=k ehukj osQ ikn&¯cnq ls bl nwjh ij [kM+k gSA
(ii) ehukj osQ f'k[kj dk mUu;u dks.k ? BAC
(iii) Nk=k dh Å¡pkbZ  AE
;g ekudj fd Åij crk;h x;ha rhuksa tkudkfj;k¡ gesa Kkr gSa rks ge fdl izdkj
ehukj dh Å¡pkbZ Kkr dj ldrs gSa\
vko`Qfr esa CD = CB + BD ;gk¡ BD = AE gS tks fd Nk=k dh Å¡pkbZ gSA
BC Kkr djus osQ fy, ge ? BAC ;k ? A  osQ f=kdks.kfefr vuqikrksa dk iz;ksx djsaxsA
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216 xf.kr
9
9.1 Hkwfedk
fiNys vè;k; esa vkius f=kdks.kferh; vuqikrksa osQ ckjs esa vè;;u fd;k gSA bl vè;k;
esa vki oqQN mu fof/;ksa osQ ckjs esa vè;;u djsaxs ftuesa f=kdks.kfefr dk iz;ksx vkiosQ
vkl&ikl osQ thou ls tqM+k gksrk gSA f=kdks.kfefr ,d izkphure fo"k; gS ftldk vè;;u
iwjs txr osQ fo}ku djrs vk, gSaA tSlk fd ge vè;k; 8 esa crk pqosQ gSa fd f=kdks.kfefr
dk vkfo"dkj bl ckr dks è;ku esa j[kdj fd;k x;k Fkk fd bldh [kxksydh esa
vko';drk iM+rh FkhA rc ls vkt rd [kxksyfon~ bldk iz;ksx i`Foh ls xzgksa vkSj rkjksa
dh nwfj;k¡ ifjdfyr djus esa djrs vk, gSaA f=kdks.kfefr dk iz;ksx Hkwxksy vkSj ukSpkyu
esa Hkh fd;k tkrk gSA f=kdks.kfefr osQ Kku dk iz;ksx ekufp=k cukus vkSj ns'kkarj (longitude)
vkSj v{kka'k (latitude) osQ lkis{k ,d }hi dh fLFkfr Kkr djus esa dh tkrh gSA
losZ{kd 'krkfCn;ksa ls f=kdks.kfefr dk iz;ksx djrs
vk jgs gSaA mUuhloha 'krkCnh dh ^c`gr~
f=kdks.kferh; losZ{k.k* fczrkuh Hkkjr dh ,d
,slh fo'kky losZ{k.k ifj;kstuk Fkh ftlosQ fy,
nks c`gÙke fFk;ksMksykbV dk fuekZ.k fd;k x;k
FkkA 1852 esa losZ{k.k djus osQ nkSjku fo'o osQ
lcls Å¡ps ioZr dh [kkst dh x;h FkhA 160 km
ls Hkh vf/d nwjh ij fLFkr vyx&vyx N%
osaQnzksa ls bl ioZr osQ f'k[kj dk izs{k.k fd;k
x;kA 1856 esa bl f'k[kj dk ukedj.k lj tkWtZ
,ojsLV osQ uke ij fd;k x;k ftlus losZizFke
fo'kky fFk;ksMksykbV dks vf/Ñr fd;k vkSj
budk iz;ksx fd;kA (lkeus cuh vko`Qfr
nsf[k,)A vc ;s fFk;ksMksykbV nsgjknwu esa fLFkr
Hkkjr losZ{k.k osQ laxzgky; esa izn'kZu osQ fy,
j[ks x, gSaA
f=kdks.kfefr osQ oqQN vuqiz;ksx
fFk;ksMksykbV
,d losZ{k.k ;a=k] tks f=kdks.kfefr osQ
fu;eksa ij vk/kfjr gS] dk iz;ksx
,d ?kw.khZ VsyhLdksi ls dks.kksa dk
ekiu djus esa fd;k tkrk gSA
f=kdks.kfefr osQ oqQN vuqiz;ksx 217
mUu;u dks.k
n`f"V js[kk
n`f"V js[kk
mUu;u dks.k
oLrq
n`f"V js[kk
mUu;u dks.k
{kSfrt Lrj
bl vè;k; esa ge ;g ns[ksaxs fd fdl izdkj okLro esa ekiu fd, fcuk gh
f=kdks.kfefr dk iz;ksx fofHkUu oLrqvksa dh Å¡pkb;k¡ vkSj nwfj;k¡ Kkr djus esa fd;k tkrk gSA
9.2 Å¡pkb;k¡ vkSj nwfj;k¡
vkb, ge vè;k; 8 esa nh xbZ vko`Qfr 8-1 ij fopkj djas] ftls uhps vko`Qfr 9-1 esa
iqu% [khapk x;k gSA
vko`Qfr 9.1
bl vko`Qfr esa] Nk=k dh vk¡[k ls ehukj osQ f'k[kj rd [khaph xbZ js[kk  AC  dks
n`f"V&js[kk (line of sight) dgk tkrk gSA Nk=k ehukj osQ f'k[kj dh vksj ns[k jgk gSA
n`f"V&js[kk vkSj {kSfrt js[kk ls cus dks.k BAC dks Nk=k dh vk¡[k ls ehukj osQ f'k[kj dk
mUu;u dks.k (angle of elevation) dgk tkrk gSA
bl izdkj] n`f"V&js[kk izs{kd dh vk¡[k osQ ml oLrq osQ ¯cnq dks feykus okyh js[kk
gksrh gS ftls izs{kd ns[krk gSA ns[ks x, ¯cnq dk mUu;u dks.k ml fLFkfr esa] n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS] tcfd ns[kk tk jgk ¯cnq {kSfrt Lrj ls Åij gksrk
gS vFkkZr~ og fLFkfr tcfd oLrq dks ns[kus osQ fy, gesa viuk flj mBkuk gksrk gSA
(nsf[k, vko`Qfr 9-2)A
vko`Qfr 9.2
218 xf.kr
{kSfrt Lrj
voueu dks.k
n`f"V js[kk
oLrq
vkb, vc ge vko`Qfr 8-2 esa nh xbZ fLFkfr ij fopkj djsaA ckyduh esa cSBh
yM+dh eafnj dh lh<+h ij j[ks xeys dks uhps dh vksj ns[k jgh gSA bl fLFkfr esa]
n`f"V&js[kk {kSfrt Lrj ls uhps gSA n`f"V&js[kk vkSj {kSfrt js[kk ls bl izdkj cus dks.k dks
voueu dks.k (angle of depression) dgk tkrk gSA
vr% ns[kh tk jgh oLrq ij fLFkr ¯cnq dk voueu dks.k ml fLFkfr esa n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS tcfd ¯cnq {kSfrt js[kk ls uhps gksrk gS vFkkZr~ og
fLFkfr tcfd ns[ks tkus okys ¯cnq dks ns[kus osQ fy, gesa viuk flj uhps >qdkuk gksrk gS
(nsf[k, vko`Qfr 9-3)A
vko`Qfr 9.3
vc vki vko`Qfr 8-3 esa cuh n`f"V&js[kk,¡ vkSj bl rjg cus dks.kksa dks igpku ldrs
gSaA ;s dks.k mUu;u dks.k gSa ;k voueu dks.k\
vkb, ge vko`Qfr 9-1 dks iqu% ns[ksaA ;fn vki lgh ek;us esa fcuk ekis gh ehukj
dh Å¡pkbZ  CD Kkr djuk pkgrs gSa rks blosQ fy, vkidks fdl tkudkjh dh vko';drk
gksrh gS\ blosQ fy, fuEufyf[kr rF;ksa dk Kku gksuk vko';d gksrk gS%
(i) nwjh DE tgk¡ Nk=k ehukj osQ ikn&¯cnq ls bl nwjh ij [kM+k gSA
(ii) ehukj osQ f'k[kj dk mUu;u dks.k ? BAC
(iii) Nk=k dh Å¡pkbZ  AE
;g ekudj fd Åij crk;h x;ha rhuksa tkudkfj;k¡ gesa Kkr gSa rks ge fdl izdkj
ehukj dh Å¡pkbZ Kkr dj ldrs gSa\
vko`Qfr esa CD = CB + BD ;gk¡ BD = AE gS tks fd Nk=k dh Å¡pkbZ gSA
BC Kkr djus osQ fy, ge ? BAC ;k ? A  osQ f=kdks.kfefr vuqikrksa dk iz;ksx djsaxsA
f=kdks.kfefr osQ oqQN vuqiz;ksx 219
? ABC esa] Hkqtk BC Kkr dks.k ? A osQ laca/ esa lEeq[k Hkqtk gSA ;gk¡ ge fdu&fdu
f=kdks.kfefr vuqikrksa dk iz;ksx dj ldrs gSa\ buesa ls fdlosQ nks eku gesa Kkr gS vkSj gesa
fdldk eku Kkr djus dh vko';drk gksrh gS\ tan A  ;k  cot A dk iz;ksx djus ls gekjh
[kkst dk {ks=k de gks tkrk gS] D;ksafd bu vuqikrksa esa  AB vkSj BC dk iz;ksx gksrk gSA
vr% tan A = 
BC
AB
;k cot A = 
AB
,
BC
 ftls gy djus ij gesa BC  izkIr gks tk,xkA
BC vkSj AE tksM+us ij ehukj dh Å¡pkbZ izkIr gks tk,xhA
vkb, vc ge oqQN mnkgj.k gy djosQ vHkh&vHkh p£pr fd, x, izØe dh
O;k[;k djsaA
mnkgj.k 1 : /jrh ij ,d ehukj ÅèokZ/j [kM+h gSA /jrh osQ ,d ¯cnq ls] tks ehukj osQ
ikn&¯cnq ls 15 m nwj gS] ehukj osQ f'k[kj dk mUu;u dks.k 60°  gSA ehukj dh Å¡pkbZ Kkr
dhft,A
gy : vkb, igys ge iz'u dks fu:fir djus osQ fy,
,d ljy vkjs[k cuk,¡ (nsf[k, vko`Qfr 9-4)A ;gk¡
AB ehukj dks fu:fir djrk gS] CB ehukj ls ¯cnq dh
nwjh gS vkSj ? ACB mUu;u dks.k gSA ge ehukj dh
Å¡pkbZ vFkkZr~  AB Kkr djuk pkgrs gSa vkSj] ;gk¡  ACB
,d f=kHkqt gS tks B ij ledks.k gSA
iz'u dks gy djus osQ fy, ge f=kdks.kferh; vuqikr
tan 60° (;k cot 60°) ysrs gSa] D;ksafd bl vuqikr esa AB
vkSj BC nksuksa gksrs gSa
vc tan 60° =
AB
BC
vFkkZr~ 3 =
AB
15
vFkkZr~ AB = 15 3
vr% ehukj dh Å¡pkbZ 15 3 m gSA
vko`Qfr 9.4
Page 5


216 xf.kr
9
9.1 Hkwfedk
fiNys vè;k; esa vkius f=kdks.kferh; vuqikrksa osQ ckjs esa vè;;u fd;k gSA bl vè;k;
esa vki oqQN mu fof/;ksa osQ ckjs esa vè;;u djsaxs ftuesa f=kdks.kfefr dk iz;ksx vkiosQ
vkl&ikl osQ thou ls tqM+k gksrk gSA f=kdks.kfefr ,d izkphure fo"k; gS ftldk vè;;u
iwjs txr osQ fo}ku djrs vk, gSaA tSlk fd ge vè;k; 8 esa crk pqosQ gSa fd f=kdks.kfefr
dk vkfo"dkj bl ckr dks è;ku esa j[kdj fd;k x;k Fkk fd bldh [kxksydh esa
vko';drk iM+rh FkhA rc ls vkt rd [kxksyfon~ bldk iz;ksx i`Foh ls xzgksa vkSj rkjksa
dh nwfj;k¡ ifjdfyr djus esa djrs vk, gSaA f=kdks.kfefr dk iz;ksx Hkwxksy vkSj ukSpkyu
esa Hkh fd;k tkrk gSA f=kdks.kfefr osQ Kku dk iz;ksx ekufp=k cukus vkSj ns'kkarj (longitude)
vkSj v{kka'k (latitude) osQ lkis{k ,d }hi dh fLFkfr Kkr djus esa dh tkrh gSA
losZ{kd 'krkfCn;ksa ls f=kdks.kfefr dk iz;ksx djrs
vk jgs gSaA mUuhloha 'krkCnh dh ^c`gr~
f=kdks.kferh; losZ{k.k* fczrkuh Hkkjr dh ,d
,slh fo'kky losZ{k.k ifj;kstuk Fkh ftlosQ fy,
nks c`gÙke fFk;ksMksykbV dk fuekZ.k fd;k x;k
FkkA 1852 esa losZ{k.k djus osQ nkSjku fo'o osQ
lcls Å¡ps ioZr dh [kkst dh x;h FkhA 160 km
ls Hkh vf/d nwjh ij fLFkr vyx&vyx N%
osaQnzksa ls bl ioZr osQ f'k[kj dk izs{k.k fd;k
x;kA 1856 esa bl f'k[kj dk ukedj.k lj tkWtZ
,ojsLV osQ uke ij fd;k x;k ftlus losZizFke
fo'kky fFk;ksMksykbV dks vf/Ñr fd;k vkSj
budk iz;ksx fd;kA (lkeus cuh vko`Qfr
nsf[k,)A vc ;s fFk;ksMksykbV nsgjknwu esa fLFkr
Hkkjr losZ{k.k osQ laxzgky; esa izn'kZu osQ fy,
j[ks x, gSaA
f=kdks.kfefr osQ oqQN vuqiz;ksx
fFk;ksMksykbV
,d losZ{k.k ;a=k] tks f=kdks.kfefr osQ
fu;eksa ij vk/kfjr gS] dk iz;ksx
,d ?kw.khZ VsyhLdksi ls dks.kksa dk
ekiu djus esa fd;k tkrk gSA
f=kdks.kfefr osQ oqQN vuqiz;ksx 217
mUu;u dks.k
n`f"V js[kk
n`f"V js[kk
mUu;u dks.k
oLrq
n`f"V js[kk
mUu;u dks.k
{kSfrt Lrj
bl vè;k; esa ge ;g ns[ksaxs fd fdl izdkj okLro esa ekiu fd, fcuk gh
f=kdks.kfefr dk iz;ksx fofHkUu oLrqvksa dh Å¡pkb;k¡ vkSj nwfj;k¡ Kkr djus esa fd;k tkrk gSA
9.2 Å¡pkb;k¡ vkSj nwfj;k¡
vkb, ge vè;k; 8 esa nh xbZ vko`Qfr 8-1 ij fopkj djas] ftls uhps vko`Qfr 9-1 esa
iqu% [khapk x;k gSA
vko`Qfr 9.1
bl vko`Qfr esa] Nk=k dh vk¡[k ls ehukj osQ f'k[kj rd [khaph xbZ js[kk  AC  dks
n`f"V&js[kk (line of sight) dgk tkrk gSA Nk=k ehukj osQ f'k[kj dh vksj ns[k jgk gSA
n`f"V&js[kk vkSj {kSfrt js[kk ls cus dks.k BAC dks Nk=k dh vk¡[k ls ehukj osQ f'k[kj dk
mUu;u dks.k (angle of elevation) dgk tkrk gSA
bl izdkj] n`f"V&js[kk izs{kd dh vk¡[k osQ ml oLrq osQ ¯cnq dks feykus okyh js[kk
gksrh gS ftls izs{kd ns[krk gSA ns[ks x, ¯cnq dk mUu;u dks.k ml fLFkfr esa] n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS] tcfd ns[kk tk jgk ¯cnq {kSfrt Lrj ls Åij gksrk
gS vFkkZr~ og fLFkfr tcfd oLrq dks ns[kus osQ fy, gesa viuk flj mBkuk gksrk gSA
(nsf[k, vko`Qfr 9-2)A
vko`Qfr 9.2
218 xf.kr
{kSfrt Lrj
voueu dks.k
n`f"V js[kk
oLrq
vkb, vc ge vko`Qfr 8-2 esa nh xbZ fLFkfr ij fopkj djsaA ckyduh esa cSBh
yM+dh eafnj dh lh<+h ij j[ks xeys dks uhps dh vksj ns[k jgh gSA bl fLFkfr esa]
n`f"V&js[kk {kSfrt Lrj ls uhps gSA n`f"V&js[kk vkSj {kSfrt js[kk ls bl izdkj cus dks.k dks
voueu dks.k (angle of depression) dgk tkrk gSA
vr% ns[kh tk jgh oLrq ij fLFkr ¯cnq dk voueu dks.k ml fLFkfr esa n`f"V&js[kk
vkSj {kSfrt js[kk ls cuk dks.k gksrk gS tcfd ¯cnq {kSfrt js[kk ls uhps gksrk gS vFkkZr~ og
fLFkfr tcfd ns[ks tkus okys ¯cnq dks ns[kus osQ fy, gesa viuk flj uhps >qdkuk gksrk gS
(nsf[k, vko`Qfr 9-3)A
vko`Qfr 9.3
vc vki vko`Qfr 8-3 esa cuh n`f"V&js[kk,¡ vkSj bl rjg cus dks.kksa dks igpku ldrs
gSaA ;s dks.k mUu;u dks.k gSa ;k voueu dks.k\
vkb, ge vko`Qfr 9-1 dks iqu% ns[ksaA ;fn vki lgh ek;us esa fcuk ekis gh ehukj
dh Å¡pkbZ  CD Kkr djuk pkgrs gSa rks blosQ fy, vkidks fdl tkudkjh dh vko';drk
gksrh gS\ blosQ fy, fuEufyf[kr rF;ksa dk Kku gksuk vko';d gksrk gS%
(i) nwjh DE tgk¡ Nk=k ehukj osQ ikn&¯cnq ls bl nwjh ij [kM+k gSA
(ii) ehukj osQ f'k[kj dk mUu;u dks.k ? BAC
(iii) Nk=k dh Å¡pkbZ  AE
;g ekudj fd Åij crk;h x;ha rhuksa tkudkfj;k¡ gesa Kkr gSa rks ge fdl izdkj
ehukj dh Å¡pkbZ Kkr dj ldrs gSa\
vko`Qfr esa CD = CB + BD ;gk¡ BD = AE gS tks fd Nk=k dh Å¡pkbZ gSA
BC Kkr djus osQ fy, ge ? BAC ;k ? A  osQ f=kdks.kfefr vuqikrksa dk iz;ksx djsaxsA
f=kdks.kfefr osQ oqQN vuqiz;ksx 219
? ABC esa] Hkqtk BC Kkr dks.k ? A osQ laca/ esa lEeq[k Hkqtk gSA ;gk¡ ge fdu&fdu
f=kdks.kfefr vuqikrksa dk iz;ksx dj ldrs gSa\ buesa ls fdlosQ nks eku gesa Kkr gS vkSj gesa
fdldk eku Kkr djus dh vko';drk gksrh gS\ tan A  ;k  cot A dk iz;ksx djus ls gekjh
[kkst dk {ks=k de gks tkrk gS] D;ksafd bu vuqikrksa esa  AB vkSj BC dk iz;ksx gksrk gSA
vr% tan A = 
BC
AB
;k cot A = 
AB
,
BC
 ftls gy djus ij gesa BC  izkIr gks tk,xkA
BC vkSj AE tksM+us ij ehukj dh Å¡pkbZ izkIr gks tk,xhA
vkb, vc ge oqQN mnkgj.k gy djosQ vHkh&vHkh p£pr fd, x, izØe dh
O;k[;k djsaA
mnkgj.k 1 : /jrh ij ,d ehukj ÅèokZ/j [kM+h gSA /jrh osQ ,d ¯cnq ls] tks ehukj osQ
ikn&¯cnq ls 15 m nwj gS] ehukj osQ f'k[kj dk mUu;u dks.k 60°  gSA ehukj dh Å¡pkbZ Kkr
dhft,A
gy : vkb, igys ge iz'u dks fu:fir djus osQ fy,
,d ljy vkjs[k cuk,¡ (nsf[k, vko`Qfr 9-4)A ;gk¡
AB ehukj dks fu:fir djrk gS] CB ehukj ls ¯cnq dh
nwjh gS vkSj ? ACB mUu;u dks.k gSA ge ehukj dh
Å¡pkbZ vFkkZr~  AB Kkr djuk pkgrs gSa vkSj] ;gk¡  ACB
,d f=kHkqt gS tks B ij ledks.k gSA
iz'u dks gy djus osQ fy, ge f=kdks.kferh; vuqikr
tan 60° (;k cot 60°) ysrs gSa] D;ksafd bl vuqikr esa AB
vkSj BC nksuksa gksrs gSa
vc tan 60° =
AB
BC
vFkkZr~ 3 =
AB
15
vFkkZr~ AB = 15 3
vr% ehukj dh Å¡pkbZ 15 3 m gSA
vko`Qfr 9.4
220 xf.kr
vko`Qfr 9.5
mnkgj.k 2 : ,d fctyh feL=kh dks ,d 5m Å¡ps [kaHks
ij vk xbZ [kjkch dh ejEer djuh gSA ejEer dk
dke djus osQ fy, mls [kaHks osQ f'k[kj ls 1.3m
uhps ,d ¯cnq rd og igq¡puk pkgrh gS (nsf[k,
vko`Qfr 9-5)A ;gk¡ rd igq¡pus osQ fy, iz;qDr lh<+h
dh yackbZ fdruh gksuh pkfg, ftlls fd {kSfrt ls 60º
osQ dks.k ls >qdkus ij og visf{kr fLFkfr rd igqq¡p
tk,\ vkSj ;g Hkh crkb, fd [kaHks dk ikn&¯cnq fdruh
nwjh ij lh<+h osQ ikn&¯cnq ls gksuk pkfg,\ (;gk¡ vki
3
 = 1.73 ys ldrs gSaA)
gy : vko`Qfr 9-5 esa] fctyh feL=kh dks [kaHks AD ij ¯cnq B rd igq¡puk gSA
vr% BD = AD – AB = (5 – 1.3)m = 3.7 m
;gk¡ BC lh<+h dks izdV djrk gSA gesa bldh yackbZ vFkkZr~ ledks.k f=kHkqt BDC dk d.kZ
Kkr djuk gSA
vc] D;k vki ;g crk ldrs gSa fd gesa fdl f=kdks.kfefr vuqikr dk iz;ksx djuk
pkfg,\
;g f=kdks.kfefr vuqikr sin 60°  gksuk pkfg,A
vr%
BD
BC
 = sin 60° ;k
3.7
BC
 = 
3
2
blfy, BC =
3.7 2
3
×
 = 4.28 m (yxHkx)
vFkkZr~ lh<+h dh yackbZ 4.28 m  gksuh pkfg,A
vc
DC
BD
 = cot 60° = 
1
3
vFkkZr~ DC =
3.7
3
 = 2.14 m (yxHkx)
vr% mls lh<+h osQ ikn dks [kaHks ls 2.14 m dh nwjh ij j[kuk pkfg,A
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