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# NCERT पाठ्यपुस्तक पाठ 7 - निर्देशांक ज्यामिति, कक्षा 10, गणित Class 10 Notes | EduRev

## Class 10 : NCERT पाठ्यपुस्तक पाठ 7 - निर्देशांक ज्यामिति, कक्षा 10, गणित Class 10 Notes | EduRev

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7
7.1 Hkwfedk
d{kk IX esa] vki i<+ pqosQ gSa fd ,d ry ij fdlh ¯cnq dh fLFkfr fu/kZfjr djus osQ
fy,] gesa funsZ'kkad v{kksa osQ ,d ;qXe dh vko';drk gksrh gSA fdlh ¯cnq dh y-v{k ls
nwjh ml ¯cnq dk x-funsZ'kkad ;k Hkqt (abscissa) dgykrh gSA fdlh ¯cnq dh  x-v{k ls
nwjh] ml ¯cnq dk y-funsZ'kkad ;k dksfV (ordinate) dgykrh gSA x-v{k ij fLFkr fdlh
¯cnq osQ funsZ'kkad (x, 0) osQ :i osQ gksrs gSa rFkk y-v{k ij fLFkr fdlh ¯cnq osQ funsZ'kkad
(0, y) osQ :i osQ gksrs gSaA
;gk¡ vkiosQ fy, ,d [ksy fn;k tk jgk gSA ,d vkys[k dkxt+ ij ykafcd v{kksa
(perpendicular axes) dk ,d ;qXe [khafp,A vc fuEufyf[kr ¯cnqvksa dks vkysf[kr
dhft, vkSj fn, x, funsZ'kksa osQ vuqlkj mUgsa feykb,A ¯cnq  A(4, 8) dks  B(3, 9) ls] B dks
C(3, 8) ls] C dks D(1, 6) ls] D dks E(1, 5) ls] E dks F(3, 3) ls] F dks G(6, 3) ls] G dks
H(8, 5) ls]  H dks I(8, 6) ls] I dks J(6, 8) ls] J dks K(6, 9) ls] K dks L(5, 8) ls vkSj L dks
A ls feykb,A blosQ ckn] ¯cnqvksa P(3.5, 7), Q (3, 6) vkSj R(4, 6) dks tksM+ dj ,d f=kHkqt
cukb,A lkFk gh] ,d f=kHkqt cukus osQ fy, ¯cnqvksa  X(5.5, 7), Y(5, 6) vkSj Z(6, 6) dks
feykb,A vc ,d vkSj f=kHkqt cukus osQ fy,] ¯cnqvksa S(4, 5), T(4.5, 4) vkSj U(5, 5) dks
feykb,A var esa] ¯cnq S dks ¯cnqvksa (0, 5) vkSj (0, 6) ls feykb, rFkk ¯cnq U dks ¯cnqvksa
(9, 5) vkSj (9, 6) ls feykb,A vkidks dkSu&lk fp=k izkIr gksrk gS\
lkFk gh] vki ;g Hkh ns[k pqosQ gSa fd ax + by + c = 0 (tgk¡ a vkSj b ,d lkFk 'kwU;
u gksa) osQ :i dh nks pjksa okyh ,d lehdj.k dks tc vkys[kh; :i ls fu:fir
djrs gSa] rks ,d ljy js[kk izkIr gksrh gSA lkFk gh] vè;k; 2 esa vki ns[k pqosQ gSa fd
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7.1 Hkwfedk
d{kk IX esa] vki i<+ pqosQ gSa fd ,d ry ij fdlh ¯cnq dh fLFkfr fu/kZfjr djus osQ
fy,] gesa funsZ'kkad v{kksa osQ ,d ;qXe dh vko';drk gksrh gSA fdlh ¯cnq dh y-v{k ls
nwjh ml ¯cnq dk x-funsZ'kkad ;k Hkqt (abscissa) dgykrh gSA fdlh ¯cnq dh  x-v{k ls
nwjh] ml ¯cnq dk y-funsZ'kkad ;k dksfV (ordinate) dgykrh gSA x-v{k ij fLFkr fdlh
¯cnq osQ funsZ'kkad (x, 0) osQ :i osQ gksrs gSa rFkk y-v{k ij fLFkr fdlh ¯cnq osQ funsZ'kkad
(0, y) osQ :i osQ gksrs gSaA
;gk¡ vkiosQ fy, ,d [ksy fn;k tk jgk gSA ,d vkys[k dkxt+ ij ykafcd v{kksa
(perpendicular axes) dk ,d ;qXe [khafp,A vc fuEufyf[kr ¯cnqvksa dks vkysf[kr
dhft, vkSj fn, x, funsZ'kksa osQ vuqlkj mUgsa feykb,A ¯cnq  A(4, 8) dks  B(3, 9) ls] B dks
C(3, 8) ls] C dks D(1, 6) ls] D dks E(1, 5) ls] E dks F(3, 3) ls] F dks G(6, 3) ls] G dks
H(8, 5) ls]  H dks I(8, 6) ls] I dks J(6, 8) ls] J dks K(6, 9) ls] K dks L(5, 8) ls vkSj L dks
A ls feykb,A blosQ ckn] ¯cnqvksa P(3.5, 7), Q (3, 6) vkSj R(4, 6) dks tksM+ dj ,d f=kHkqt
cukb,A lkFk gh] ,d f=kHkqt cukus osQ fy, ¯cnqvksa  X(5.5, 7), Y(5, 6) vkSj Z(6, 6) dks
feykb,A vc ,d vkSj f=kHkqt cukus osQ fy,] ¯cnqvksa S(4, 5), T(4.5, 4) vkSj U(5, 5) dks
feykb,A var esa] ¯cnq S dks ¯cnqvksa (0, 5) vkSj (0, 6) ls feykb, rFkk ¯cnq U dks ¯cnqvksa
(9, 5) vkSj (9, 6) ls feykb,A vkidks dkSu&lk fp=k izkIr gksrk gS\
lkFk gh] vki ;g Hkh ns[k pqosQ gSa fd ax + by + c = 0 (tgk¡ a vkSj b ,d lkFk 'kwU;
u gksa) osQ :i dh nks pjksa okyh ,d lehdj.k dks tc vkys[kh; :i ls fu:fir
djrs gSa] rks ,d ljy js[kk izkIr gksrh gSA lkFk gh] vè;k; 2 esa vki ns[k pqosQ gSa fd
2018-19
vko`Qfr 7.1
vko`Qfr 7.2
y = ax
2
+ bx + c (a ? 0) dk vkys[k ,d ijoy; (parabola) gksrk gSA oLrqr%] vko`Qfr;ksa
dh T;kfefr dk vè;;u djus osQ fy,] funsZ'kkad T;kfefr (coordinate geometry) ,d
chth; lk/u (algebraic tool) osQ :i esa fodflr dh xbZ gSA ;g chtxf.kr dk iz;ksx
djosQ T;kfefr dk vè;;u djus esa lgk;rk djrh gS rFkk chtxf.kr dks T;kfefr }kjk
le>us esa Hkh lgk;d gksrh gSA blh dkj.k] funsZ'kkad T;kfefr osQ fofHkUu {ks=kksa esa O;kid
vuqiz;ksx gSa] tSls HkkSfrdh] bathfu;¯jx] leqnzh&ifjogu (;k ukS&xeu) (navigation), HkwoaQi
'kkL=k laca/h (seismology) vkSj dykA
bl vè;k; esa] vki ;g lh[ksaxs fd nks ¯cnqvksa] ftuosQ funsZ'kkad fn, gq, gksa] osQ chp
dh nwjh fdl izdkj Kkr dh tkrh gS rFkk rhu fn, gq, ¯cnqvksa ls cus f=kHkqt dk {ks=kiQy
fdl izdkj Kkr fd;k tkrk gSA vki bldk Hkh vè;;u djsaxs fd fn, gq, nks ¯cnqvksa dks
feykus ls cus js[kk[kaM dks ,d fn, x, vuqikr esa foHkkftr djus okys ¯cnq osQ funsZ'kkad
fdl izdkj Kkr fd, tkrs gSaA
7.2 nwjh lw=k
vkb, fuEufyf[kr fLFkfr ij fopkj djsa%
,d 'kgj B ,d vU; 'kgj A ls 36 km iwoZ
(east) vkSj 15 km mÙkj (north) dh vksj gSA vki
'kgj B dh 'kgj A ls nwjh fcuk okLrfod ekiu osQ
fdl izdkj Kkr dj ldrs gSa\ vkb, ns[ksaA bl
fLFkfr dks] vkys[kh; :i ls] vko`Qfr 7.1 dh rjg
n'kkZ;k tk ldrk gSA vc] vki okafNr nwjh Kkr djus
osQ fy,] ikbFkkxksjl izes; dk iz;ksx dj ldrs gSaA
vc] eku yhft, nks ¯cnq  x–v{k ij fLFkr gSaA
D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\
mnkgj.kkFkZ] vko`Qfr 7.2 osQ nks ¯cnqvksa A(4, 0) vkSj
B(6, 0) ij fopkj dhft,A ¯cnq A vkSj B, x-v{k ij
fLFkr gSA
vko`Qfr ls vki ns[k ldrs gSa fd OA =
4 ek=kd (bdkbZ) vkSj OB = 6 ek=kd gSaA
vr%] A ls B dh nwjh AB = OB – OA =
(6 – 4) ek=kd = 2 ek=kd gSA
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7.1 Hkwfedk
d{kk IX esa] vki i<+ pqosQ gSa fd ,d ry ij fdlh ¯cnq dh fLFkfr fu/kZfjr djus osQ
fy,] gesa funsZ'kkad v{kksa osQ ,d ;qXe dh vko';drk gksrh gSA fdlh ¯cnq dh y-v{k ls
nwjh ml ¯cnq dk x-funsZ'kkad ;k Hkqt (abscissa) dgykrh gSA fdlh ¯cnq dh  x-v{k ls
nwjh] ml ¯cnq dk y-funsZ'kkad ;k dksfV (ordinate) dgykrh gSA x-v{k ij fLFkr fdlh
¯cnq osQ funsZ'kkad (x, 0) osQ :i osQ gksrs gSa rFkk y-v{k ij fLFkr fdlh ¯cnq osQ funsZ'kkad
(0, y) osQ :i osQ gksrs gSaA
;gk¡ vkiosQ fy, ,d [ksy fn;k tk jgk gSA ,d vkys[k dkxt+ ij ykafcd v{kksa
(perpendicular axes) dk ,d ;qXe [khafp,A vc fuEufyf[kr ¯cnqvksa dks vkysf[kr
dhft, vkSj fn, x, funsZ'kksa osQ vuqlkj mUgsa feykb,A ¯cnq  A(4, 8) dks  B(3, 9) ls] B dks
C(3, 8) ls] C dks D(1, 6) ls] D dks E(1, 5) ls] E dks F(3, 3) ls] F dks G(6, 3) ls] G dks
H(8, 5) ls]  H dks I(8, 6) ls] I dks J(6, 8) ls] J dks K(6, 9) ls] K dks L(5, 8) ls vkSj L dks
A ls feykb,A blosQ ckn] ¯cnqvksa P(3.5, 7), Q (3, 6) vkSj R(4, 6) dks tksM+ dj ,d f=kHkqt
cukb,A lkFk gh] ,d f=kHkqt cukus osQ fy, ¯cnqvksa  X(5.5, 7), Y(5, 6) vkSj Z(6, 6) dks
feykb,A vc ,d vkSj f=kHkqt cukus osQ fy,] ¯cnqvksa S(4, 5), T(4.5, 4) vkSj U(5, 5) dks
feykb,A var esa] ¯cnq S dks ¯cnqvksa (0, 5) vkSj (0, 6) ls feykb, rFkk ¯cnq U dks ¯cnqvksa
(9, 5) vkSj (9, 6) ls feykb,A vkidks dkSu&lk fp=k izkIr gksrk gS\
lkFk gh] vki ;g Hkh ns[k pqosQ gSa fd ax + by + c = 0 (tgk¡ a vkSj b ,d lkFk 'kwU;
u gksa) osQ :i dh nks pjksa okyh ,d lehdj.k dks tc vkys[kh; :i ls fu:fir
djrs gSa] rks ,d ljy js[kk izkIr gksrh gSA lkFk gh] vè;k; 2 esa vki ns[k pqosQ gSa fd
2018-19
vko`Qfr 7.1
vko`Qfr 7.2
y = ax
2
+ bx + c (a ? 0) dk vkys[k ,d ijoy; (parabola) gksrk gSA oLrqr%] vko`Qfr;ksa
dh T;kfefr dk vè;;u djus osQ fy,] funsZ'kkad T;kfefr (coordinate geometry) ,d
chth; lk/u (algebraic tool) osQ :i esa fodflr dh xbZ gSA ;g chtxf.kr dk iz;ksx
djosQ T;kfefr dk vè;;u djus esa lgk;rk djrh gS rFkk chtxf.kr dks T;kfefr }kjk
le>us esa Hkh lgk;d gksrh gSA blh dkj.k] funsZ'kkad T;kfefr osQ fofHkUu {ks=kksa esa O;kid
vuqiz;ksx gSa] tSls HkkSfrdh] bathfu;¯jx] leqnzh&ifjogu (;k ukS&xeu) (navigation), HkwoaQi
'kkL=k laca/h (seismology) vkSj dykA
bl vè;k; esa] vki ;g lh[ksaxs fd nks ¯cnqvksa] ftuosQ funsZ'kkad fn, gq, gksa] osQ chp
dh nwjh fdl izdkj Kkr dh tkrh gS rFkk rhu fn, gq, ¯cnqvksa ls cus f=kHkqt dk {ks=kiQy
fdl izdkj Kkr fd;k tkrk gSA vki bldk Hkh vè;;u djsaxs fd fn, gq, nks ¯cnqvksa dks
feykus ls cus js[kk[kaM dks ,d fn, x, vuqikr esa foHkkftr djus okys ¯cnq osQ funsZ'kkad
fdl izdkj Kkr fd, tkrs gSaA
7.2 nwjh lw=k
vkb, fuEufyf[kr fLFkfr ij fopkj djsa%
,d 'kgj B ,d vU; 'kgj A ls 36 km iwoZ
(east) vkSj 15 km mÙkj (north) dh vksj gSA vki
'kgj B dh 'kgj A ls nwjh fcuk okLrfod ekiu osQ
fdl izdkj Kkr dj ldrs gSa\ vkb, ns[ksaA bl
fLFkfr dks] vkys[kh; :i ls] vko`Qfr 7.1 dh rjg
n'kkZ;k tk ldrk gSA vc] vki okafNr nwjh Kkr djus
osQ fy,] ikbFkkxksjl izes; dk iz;ksx dj ldrs gSaA
vc] eku yhft, nks ¯cnq  x–v{k ij fLFkr gSaA
D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\
mnkgj.kkFkZ] vko`Qfr 7.2 osQ nks ¯cnqvksa A(4, 0) vkSj
B(6, 0) ij fopkj dhft,A ¯cnq A vkSj B, x-v{k ij
fLFkr gSA
vko`Qfr ls vki ns[k ldrs gSa fd OA =
4 ek=kd (bdkbZ) vkSj OB = 6 ek=kd gSaA
vr%] A ls B dh nwjh AB = OB – OA =
(6 – 4) ek=kd = 2 ek=kd gSA
2018-19
172 xf.kr
bl izdkj] ;fn nks ¯cnq x–v{k ij fLFkr gksa] rks ge muosQ chp dh nwjh ljyrk ls
Kkr dj ldrs gSaA
vc] eku yhft,] ge y–v{k ij fLFkr dksbZ nks ¯cnq ysrs gSaA D;k ge buosQ chp
dh nwjh Kkr dj ldrs gSa\ ;fn ¯cnq C(0, 3) vkSj D(0, 8), y–v{k ij fLFkr gksa] rks ge
nwjh Åij dh Hkk¡fr Kkr dj ldrs gSa vFkkZr~ nwjh CD = (8 – 3) ek=kd = 5 ek=kd gS (nsf[k,
vko`Qfr 7.2)A
iqu%] D;k vki vko`Qfr 7.2 esa] ¯cnq  C ls ¯cnq A dh nwjh Kkr dj ldrs gSa\ pw¡fd
OA = 4 ek=kd vkSj OC = 3 ek=kd gSa] blfy, C ls A dh nwjh AC =
2 2
3 4 + = 5 ek=kd
gSA blh izdkj] vki D ls B dh nwjh BD = 10 ek=kd Kkr dj ldrs gSaA
vc] ;fn ge ,sls nks ¯cnqvksa ij fopkj djsa] tks funsZ'kkad v{kksa ij fLFkr ugha gSa]
rks D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\ gk¡! ,slk djus osQ fy,] ge ikbFkkxksjl
izes; dk iz;ksx djsaxsA vkb, ,d mnkgj.k ysdj ns[ksaA
vko`Qfr 7.3 esa] ¯cnq P(4, 6) vkSj Q(6, 8) izFke prqFkk±'k (first quadrant) esa fLFkr gSaA
buosQ chp dh nwjh Kkr djus osQ fy,] ge ikbFkkxksjl izes; dk iz;ksx oSQls djrs gSa\
vkb, P vkSj Q ls x-v{k ij Øe'k% yac PR vkSj QS [khpsaA lkFk gh] P ls QS ij ,d yac
Mkfy, tks QS dks T ij izfrPNsn djsA rc R vkSj S osQ funsZ'kkad Øe'k% (4, 0) vkSj (6, 0) gSaA
vr%] RS = 2 ek=kd gSA lkFk gh] QS = 8 ek=kd vkSj TS = PR = 6 ek=kd gSA
Li"V gS fd QT = 2 ek=kd vkSj PT = RS = 2 ek=kdA
vc] ikbFkkxksjl izes; osQ iz;ksx ls] gesa izkIr
gksrk gS%
PQ
2
= PT
2
+ QT
2
= 2
2
+ 2
2
= 8
vr% PQ =
22
ek=kd gqvkA
vki nks fHkUu&fHkUu prqFkk±'kksa esa fLFkr ¯cnqvksa
osQ chp dh nwjh oSQls Kkr djsaxs\
¯cnqvksa P(6, 4) vkSj Q(–5, –3) ij fopkj dhft,
(nsf[k, vko`Qfr 7-4)A x-v{k ij yac QS [khafp,A
lkFk gh] ¯cnq P ls c<+kbZ gqbZ QS ij PT yac [khafp,
tks y-v{k dks ¯cnq R ij izfrPNsn djsA vko`Qfr 7.3
2018-19
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7.1 Hkwfedk
d{kk IX esa] vki i<+ pqosQ gSa fd ,d ry ij fdlh ¯cnq dh fLFkfr fu/kZfjr djus osQ
fy,] gesa funsZ'kkad v{kksa osQ ,d ;qXe dh vko';drk gksrh gSA fdlh ¯cnq dh y-v{k ls
nwjh ml ¯cnq dk x-funsZ'kkad ;k Hkqt (abscissa) dgykrh gSA fdlh ¯cnq dh  x-v{k ls
nwjh] ml ¯cnq dk y-funsZ'kkad ;k dksfV (ordinate) dgykrh gSA x-v{k ij fLFkr fdlh
¯cnq osQ funsZ'kkad (x, 0) osQ :i osQ gksrs gSa rFkk y-v{k ij fLFkr fdlh ¯cnq osQ funsZ'kkad
(0, y) osQ :i osQ gksrs gSaA
;gk¡ vkiosQ fy, ,d [ksy fn;k tk jgk gSA ,d vkys[k dkxt+ ij ykafcd v{kksa
(perpendicular axes) dk ,d ;qXe [khafp,A vc fuEufyf[kr ¯cnqvksa dks vkysf[kr
dhft, vkSj fn, x, funsZ'kksa osQ vuqlkj mUgsa feykb,A ¯cnq  A(4, 8) dks  B(3, 9) ls] B dks
C(3, 8) ls] C dks D(1, 6) ls] D dks E(1, 5) ls] E dks F(3, 3) ls] F dks G(6, 3) ls] G dks
H(8, 5) ls]  H dks I(8, 6) ls] I dks J(6, 8) ls] J dks K(6, 9) ls] K dks L(5, 8) ls vkSj L dks
A ls feykb,A blosQ ckn] ¯cnqvksa P(3.5, 7), Q (3, 6) vkSj R(4, 6) dks tksM+ dj ,d f=kHkqt
cukb,A lkFk gh] ,d f=kHkqt cukus osQ fy, ¯cnqvksa  X(5.5, 7), Y(5, 6) vkSj Z(6, 6) dks
feykb,A vc ,d vkSj f=kHkqt cukus osQ fy,] ¯cnqvksa S(4, 5), T(4.5, 4) vkSj U(5, 5) dks
feykb,A var esa] ¯cnq S dks ¯cnqvksa (0, 5) vkSj (0, 6) ls feykb, rFkk ¯cnq U dks ¯cnqvksa
(9, 5) vkSj (9, 6) ls feykb,A vkidks dkSu&lk fp=k izkIr gksrk gS\
lkFk gh] vki ;g Hkh ns[k pqosQ gSa fd ax + by + c = 0 (tgk¡ a vkSj b ,d lkFk 'kwU;
u gksa) osQ :i dh nks pjksa okyh ,d lehdj.k dks tc vkys[kh; :i ls fu:fir
djrs gSa] rks ,d ljy js[kk izkIr gksrh gSA lkFk gh] vè;k; 2 esa vki ns[k pqosQ gSa fd
2018-19
vko`Qfr 7.1
vko`Qfr 7.2
y = ax
2
+ bx + c (a ? 0) dk vkys[k ,d ijoy; (parabola) gksrk gSA oLrqr%] vko`Qfr;ksa
dh T;kfefr dk vè;;u djus osQ fy,] funsZ'kkad T;kfefr (coordinate geometry) ,d
chth; lk/u (algebraic tool) osQ :i esa fodflr dh xbZ gSA ;g chtxf.kr dk iz;ksx
djosQ T;kfefr dk vè;;u djus esa lgk;rk djrh gS rFkk chtxf.kr dks T;kfefr }kjk
le>us esa Hkh lgk;d gksrh gSA blh dkj.k] funsZ'kkad T;kfefr osQ fofHkUu {ks=kksa esa O;kid
vuqiz;ksx gSa] tSls HkkSfrdh] bathfu;¯jx] leqnzh&ifjogu (;k ukS&xeu) (navigation), HkwoaQi
'kkL=k laca/h (seismology) vkSj dykA
bl vè;k; esa] vki ;g lh[ksaxs fd nks ¯cnqvksa] ftuosQ funsZ'kkad fn, gq, gksa] osQ chp
dh nwjh fdl izdkj Kkr dh tkrh gS rFkk rhu fn, gq, ¯cnqvksa ls cus f=kHkqt dk {ks=kiQy
fdl izdkj Kkr fd;k tkrk gSA vki bldk Hkh vè;;u djsaxs fd fn, gq, nks ¯cnqvksa dks
feykus ls cus js[kk[kaM dks ,d fn, x, vuqikr esa foHkkftr djus okys ¯cnq osQ funsZ'kkad
fdl izdkj Kkr fd, tkrs gSaA
7.2 nwjh lw=k
vkb, fuEufyf[kr fLFkfr ij fopkj djsa%
,d 'kgj B ,d vU; 'kgj A ls 36 km iwoZ
(east) vkSj 15 km mÙkj (north) dh vksj gSA vki
'kgj B dh 'kgj A ls nwjh fcuk okLrfod ekiu osQ
fdl izdkj Kkr dj ldrs gSa\ vkb, ns[ksaA bl
fLFkfr dks] vkys[kh; :i ls] vko`Qfr 7.1 dh rjg
n'kkZ;k tk ldrk gSA vc] vki okafNr nwjh Kkr djus
osQ fy,] ikbFkkxksjl izes; dk iz;ksx dj ldrs gSaA
vc] eku yhft, nks ¯cnq  x–v{k ij fLFkr gSaA
D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\
mnkgj.kkFkZ] vko`Qfr 7.2 osQ nks ¯cnqvksa A(4, 0) vkSj
B(6, 0) ij fopkj dhft,A ¯cnq A vkSj B, x-v{k ij
fLFkr gSA
vko`Qfr ls vki ns[k ldrs gSa fd OA =
4 ek=kd (bdkbZ) vkSj OB = 6 ek=kd gSaA
vr%] A ls B dh nwjh AB = OB – OA =
(6 – 4) ek=kd = 2 ek=kd gSA
2018-19
172 xf.kr
bl izdkj] ;fn nks ¯cnq x–v{k ij fLFkr gksa] rks ge muosQ chp dh nwjh ljyrk ls
Kkr dj ldrs gSaA
vc] eku yhft,] ge y–v{k ij fLFkr dksbZ nks ¯cnq ysrs gSaA D;k ge buosQ chp
dh nwjh Kkr dj ldrs gSa\ ;fn ¯cnq C(0, 3) vkSj D(0, 8), y–v{k ij fLFkr gksa] rks ge
nwjh Åij dh Hkk¡fr Kkr dj ldrs gSa vFkkZr~ nwjh CD = (8 – 3) ek=kd = 5 ek=kd gS (nsf[k,
vko`Qfr 7.2)A
iqu%] D;k vki vko`Qfr 7.2 esa] ¯cnq  C ls ¯cnq A dh nwjh Kkr dj ldrs gSa\ pw¡fd
OA = 4 ek=kd vkSj OC = 3 ek=kd gSa] blfy, C ls A dh nwjh AC =
2 2
3 4 + = 5 ek=kd
gSA blh izdkj] vki D ls B dh nwjh BD = 10 ek=kd Kkr dj ldrs gSaA
vc] ;fn ge ,sls nks ¯cnqvksa ij fopkj djsa] tks funsZ'kkad v{kksa ij fLFkr ugha gSa]
rks D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\ gk¡! ,slk djus osQ fy,] ge ikbFkkxksjl
izes; dk iz;ksx djsaxsA vkb, ,d mnkgj.k ysdj ns[ksaA
vko`Qfr 7.3 esa] ¯cnq P(4, 6) vkSj Q(6, 8) izFke prqFkk±'k (first quadrant) esa fLFkr gSaA
buosQ chp dh nwjh Kkr djus osQ fy,] ge ikbFkkxksjl izes; dk iz;ksx oSQls djrs gSa\
vkb, P vkSj Q ls x-v{k ij Øe'k% yac PR vkSj QS [khpsaA lkFk gh] P ls QS ij ,d yac
Mkfy, tks QS dks T ij izfrPNsn djsA rc R vkSj S osQ funsZ'kkad Øe'k% (4, 0) vkSj (6, 0) gSaA
vr%] RS = 2 ek=kd gSA lkFk gh] QS = 8 ek=kd vkSj TS = PR = 6 ek=kd gSA
Li"V gS fd QT = 2 ek=kd vkSj PT = RS = 2 ek=kdA
vc] ikbFkkxksjl izes; osQ iz;ksx ls] gesa izkIr
gksrk gS%
PQ
2
= PT
2
+ QT
2
= 2
2
+ 2
2
= 8
vr% PQ =
22
ek=kd gqvkA
vki nks fHkUu&fHkUu prqFkk±'kksa esa fLFkr ¯cnqvksa
osQ chp dh nwjh oSQls Kkr djsaxs\
¯cnqvksa P(6, 4) vkSj Q(–5, –3) ij fopkj dhft,
(nsf[k, vko`Qfr 7-4)A x-v{k ij yac QS [khafp,A
lkFk gh] ¯cnq P ls c<+kbZ gqbZ QS ij PT yac [khafp,
tks y-v{k dks ¯cnq R ij izfrPNsn djsA vko`Qfr 7.3
2018-19
vko`Qfr 7.4
rc]  PT = 11 ek=kd vkSj QT = 7 ek=kd gS (D;ksa\)
ledks.k f=kHkqt  PTQ esa] ikbFkkxksjl izes; osQ iz;ksx ls] gesa izkIr gksrk gS%
PQ =
2 2
1 1 7 + =
170
ek=kd
vkb,] vc fdUgha nks ¯cnqvksa  P(x
1
, y
1
) vkSj
Q(x
2
, y
2
) osQ chp dh nwjh Kkr djsaA x-v{k ij yac PR
vkSj QS [khafp,A P ls QS ij ,d yac [khafp,] tks mls
T ij izfrPNsn djs (nsf[k, vko`Qfr 7-5)A
rc] OR = x
1
, OS = x
2
gSA vr%] RS = x
2
– x
1
= PT gSA
lkFk gh] SQ = y
2
vkSj ST = PR = y
1
gSA  vr%] QT = y
2
– y
1
gSA
vc] ? PTQ esa] ikbFkkxksjl izes; osQ iz;ksx ls] gesa
izkIr gksrk gS%
PQ
2
= PT
2
+ QT
2
= (x
2
– x
1
)
2
+ (y
2
– y
1
)
2
vr% PQ =
( ) ( )
2 2
2 1 2 1
x x y y - + -
è;ku nsa fd pw¡fd nwjh lnSo ½.ksrj gksrh gS] ge osQoy /ukRed oxZewy ysrs gSaA
vko`Qfr 7.5
2018-19
Page 5

170 xf.kr
7
7.1 Hkwfedk
d{kk IX esa] vki i<+ pqosQ gSa fd ,d ry ij fdlh ¯cnq dh fLFkfr fu/kZfjr djus osQ
fy,] gesa funsZ'kkad v{kksa osQ ,d ;qXe dh vko';drk gksrh gSA fdlh ¯cnq dh y-v{k ls
nwjh ml ¯cnq dk x-funsZ'kkad ;k Hkqt (abscissa) dgykrh gSA fdlh ¯cnq dh  x-v{k ls
nwjh] ml ¯cnq dk y-funsZ'kkad ;k dksfV (ordinate) dgykrh gSA x-v{k ij fLFkr fdlh
¯cnq osQ funsZ'kkad (x, 0) osQ :i osQ gksrs gSa rFkk y-v{k ij fLFkr fdlh ¯cnq osQ funsZ'kkad
(0, y) osQ :i osQ gksrs gSaA
;gk¡ vkiosQ fy, ,d [ksy fn;k tk jgk gSA ,d vkys[k dkxt+ ij ykafcd v{kksa
(perpendicular axes) dk ,d ;qXe [khafp,A vc fuEufyf[kr ¯cnqvksa dks vkysf[kr
dhft, vkSj fn, x, funsZ'kksa osQ vuqlkj mUgsa feykb,A ¯cnq  A(4, 8) dks  B(3, 9) ls] B dks
C(3, 8) ls] C dks D(1, 6) ls] D dks E(1, 5) ls] E dks F(3, 3) ls] F dks G(6, 3) ls] G dks
H(8, 5) ls]  H dks I(8, 6) ls] I dks J(6, 8) ls] J dks K(6, 9) ls] K dks L(5, 8) ls vkSj L dks
A ls feykb,A blosQ ckn] ¯cnqvksa P(3.5, 7), Q (3, 6) vkSj R(4, 6) dks tksM+ dj ,d f=kHkqt
cukb,A lkFk gh] ,d f=kHkqt cukus osQ fy, ¯cnqvksa  X(5.5, 7), Y(5, 6) vkSj Z(6, 6) dks
feykb,A vc ,d vkSj f=kHkqt cukus osQ fy,] ¯cnqvksa S(4, 5), T(4.5, 4) vkSj U(5, 5) dks
feykb,A var esa] ¯cnq S dks ¯cnqvksa (0, 5) vkSj (0, 6) ls feykb, rFkk ¯cnq U dks ¯cnqvksa
(9, 5) vkSj (9, 6) ls feykb,A vkidks dkSu&lk fp=k izkIr gksrk gS\
lkFk gh] vki ;g Hkh ns[k pqosQ gSa fd ax + by + c = 0 (tgk¡ a vkSj b ,d lkFk 'kwU;
u gksa) osQ :i dh nks pjksa okyh ,d lehdj.k dks tc vkys[kh; :i ls fu:fir
djrs gSa] rks ,d ljy js[kk izkIr gksrh gSA lkFk gh] vè;k; 2 esa vki ns[k pqosQ gSa fd
2018-19
vko`Qfr 7.1
vko`Qfr 7.2
y = ax
2
+ bx + c (a ? 0) dk vkys[k ,d ijoy; (parabola) gksrk gSA oLrqr%] vko`Qfr;ksa
dh T;kfefr dk vè;;u djus osQ fy,] funsZ'kkad T;kfefr (coordinate geometry) ,d
chth; lk/u (algebraic tool) osQ :i esa fodflr dh xbZ gSA ;g chtxf.kr dk iz;ksx
djosQ T;kfefr dk vè;;u djus esa lgk;rk djrh gS rFkk chtxf.kr dks T;kfefr }kjk
le>us esa Hkh lgk;d gksrh gSA blh dkj.k] funsZ'kkad T;kfefr osQ fofHkUu {ks=kksa esa O;kid
vuqiz;ksx gSa] tSls HkkSfrdh] bathfu;¯jx] leqnzh&ifjogu (;k ukS&xeu) (navigation), HkwoaQi
'kkL=k laca/h (seismology) vkSj dykA
bl vè;k; esa] vki ;g lh[ksaxs fd nks ¯cnqvksa] ftuosQ funsZ'kkad fn, gq, gksa] osQ chp
dh nwjh fdl izdkj Kkr dh tkrh gS rFkk rhu fn, gq, ¯cnqvksa ls cus f=kHkqt dk {ks=kiQy
fdl izdkj Kkr fd;k tkrk gSA vki bldk Hkh vè;;u djsaxs fd fn, gq, nks ¯cnqvksa dks
feykus ls cus js[kk[kaM dks ,d fn, x, vuqikr esa foHkkftr djus okys ¯cnq osQ funsZ'kkad
fdl izdkj Kkr fd, tkrs gSaA
7.2 nwjh lw=k
vkb, fuEufyf[kr fLFkfr ij fopkj djsa%
,d 'kgj B ,d vU; 'kgj A ls 36 km iwoZ
(east) vkSj 15 km mÙkj (north) dh vksj gSA vki
'kgj B dh 'kgj A ls nwjh fcuk okLrfod ekiu osQ
fdl izdkj Kkr dj ldrs gSa\ vkb, ns[ksaA bl
fLFkfr dks] vkys[kh; :i ls] vko`Qfr 7.1 dh rjg
n'kkZ;k tk ldrk gSA vc] vki okafNr nwjh Kkr djus
osQ fy,] ikbFkkxksjl izes; dk iz;ksx dj ldrs gSaA
vc] eku yhft, nks ¯cnq  x–v{k ij fLFkr gSaA
D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\
mnkgj.kkFkZ] vko`Qfr 7.2 osQ nks ¯cnqvksa A(4, 0) vkSj
B(6, 0) ij fopkj dhft,A ¯cnq A vkSj B, x-v{k ij
fLFkr gSA
vko`Qfr ls vki ns[k ldrs gSa fd OA =
4 ek=kd (bdkbZ) vkSj OB = 6 ek=kd gSaA
vr%] A ls B dh nwjh AB = OB – OA =
(6 – 4) ek=kd = 2 ek=kd gSA
2018-19
172 xf.kr
bl izdkj] ;fn nks ¯cnq x–v{k ij fLFkr gksa] rks ge muosQ chp dh nwjh ljyrk ls
Kkr dj ldrs gSaA
vc] eku yhft,] ge y–v{k ij fLFkr dksbZ nks ¯cnq ysrs gSaA D;k ge buosQ chp
dh nwjh Kkr dj ldrs gSa\ ;fn ¯cnq C(0, 3) vkSj D(0, 8), y–v{k ij fLFkr gksa] rks ge
nwjh Åij dh Hkk¡fr Kkr dj ldrs gSa vFkkZr~ nwjh CD = (8 – 3) ek=kd = 5 ek=kd gS (nsf[k,
vko`Qfr 7.2)A
iqu%] D;k vki vko`Qfr 7.2 esa] ¯cnq  C ls ¯cnq A dh nwjh Kkr dj ldrs gSa\ pw¡fd
OA = 4 ek=kd vkSj OC = 3 ek=kd gSa] blfy, C ls A dh nwjh AC =
2 2
3 4 + = 5 ek=kd
gSA blh izdkj] vki D ls B dh nwjh BD = 10 ek=kd Kkr dj ldrs gSaA
vc] ;fn ge ,sls nks ¯cnqvksa ij fopkj djsa] tks funsZ'kkad v{kksa ij fLFkr ugha gSa]
rks D;k ge buosQ chp dh nwjh Kkr dj ldrs gSa\ gk¡! ,slk djus osQ fy,] ge ikbFkkxksjl
izes; dk iz;ksx djsaxsA vkb, ,d mnkgj.k ysdj ns[ksaA
vko`Qfr 7.3 esa] ¯cnq P(4, 6) vkSj Q(6, 8) izFke prqFkk±'k (first quadrant) esa fLFkr gSaA
buosQ chp dh nwjh Kkr djus osQ fy,] ge ikbFkkxksjl izes; dk iz;ksx oSQls djrs gSa\
vkb, P vkSj Q ls x-v{k ij Øe'k% yac PR vkSj QS [khpsaA lkFk gh] P ls QS ij ,d yac
Mkfy, tks QS dks T ij izfrPNsn djsA rc R vkSj S osQ funsZ'kkad Øe'k% (4, 0) vkSj (6, 0) gSaA
vr%] RS = 2 ek=kd gSA lkFk gh] QS = 8 ek=kd vkSj TS = PR = 6 ek=kd gSA
Li"V gS fd QT = 2 ek=kd vkSj PT = RS = 2 ek=kdA
vc] ikbFkkxksjl izes; osQ iz;ksx ls] gesa izkIr
gksrk gS%
PQ
2
= PT
2
+ QT
2
= 2
2
+ 2
2
= 8
vr% PQ =
22
ek=kd gqvkA
vki nks fHkUu&fHkUu prqFkk±'kksa esa fLFkr ¯cnqvksa
osQ chp dh nwjh oSQls Kkr djsaxs\
¯cnqvksa P(6, 4) vkSj Q(–5, –3) ij fopkj dhft,
(nsf[k, vko`Qfr 7-4)A x-v{k ij yac QS [khafp,A
lkFk gh] ¯cnq P ls c<+kbZ gqbZ QS ij PT yac [khafp,
tks y-v{k dks ¯cnq R ij izfrPNsn djsA vko`Qfr 7.3
2018-19
vko`Qfr 7.4
rc]  PT = 11 ek=kd vkSj QT = 7 ek=kd gS (D;ksa\)
ledks.k f=kHkqt  PTQ esa] ikbFkkxksjl izes; osQ iz;ksx ls] gesa izkIr gksrk gS%
PQ =
2 2
1 1 7 + =
170
ek=kd
vkb,] vc fdUgha nks ¯cnqvksa  P(x
1
, y
1
) vkSj
Q(x
2
, y
2
) osQ chp dh nwjh Kkr djsaA x-v{k ij yac PR
vkSj QS [khafp,A P ls QS ij ,d yac [khafp,] tks mls
T ij izfrPNsn djs (nsf[k, vko`Qfr 7-5)A
rc] OR = x
1
, OS = x
2
gSA vr%] RS = x
2
– x
1
= PT gSA
lkFk gh] SQ = y
2
vkSj ST = PR = y
1
gSA  vr%] QT = y
2
– y
1
gSA
vc] ? PTQ esa] ikbFkkxksjl izes; osQ iz;ksx ls] gesa
izkIr gksrk gS%
PQ
2
= PT
2
+ QT
2
= (x
2
– x
1
)
2
+ (y
2
– y
1
)
2
vr% PQ =
( ) ( )
2 2
2 1 2 1
x x y y - + -
è;ku nsa fd pw¡fd nwjh lnSo ½.ksrj gksrh gS] ge osQoy /ukRed oxZewy ysrs gSaA
vko`Qfr 7.5
2018-19
174 xf.kr
vr% P(x
1
, y
1
) vkSj Q(x
2
, y
2
) osQ ¯cnqvksa osQ chp dh nwjh gS
PQ =
( ) ( )
2 2
2 1 2 1
x x y y - + -
tks nwjh lw=k (distance formula) dgykrk gSA
fVIif.k;k¡ :
1. fo'ks"k :i ls] ¯cnq P(x, y) dh ewy ¯cnq O(0, 0) ls nwjh
OP =
2 2
x y + gksrh gSA
2. ge PQ =
( ) ( )
2 2
1 2 1 2
x x y y - + -
Hkh fy[k ldrs gSa (D;ksa\)
mnkgj.k 1 : D;k ¯cnq (3, 2), (–2, –3) vkSj (2, 3) ,d f=kHkqt cukrs gSa\ ;fn gk¡] rks crkb,
fd fdl izdkj dk f=kHkqt curk gSA
gy : vkb, PQ, QR vkSj PR Kkr djus osQ fy, nwjh lw=k dk iz;ksx djsa] tgk¡ P(3, 2),
Q(–2, –3) vkSj R(2, 3) fn, gq, ¯cnq gSaA gesa izkIr gksrk gS%
PQ =
2 2 2 2
(3 2) (2 3 ) 5 5 50 + + + = + = = 7.07 (yxHkx)
QR =
2 2 2 2
(–2 – 2) (–3 – 3) (– 4) (– 6) 5 2 + = + = = 7.21 (yxHkx)
PR =
2 2 2 2
(3 – 2 ) (2 – 3) 1 ( 1)2 + = + - = = 1.41 (yxHkx)
pw¡fd bu rhu nwfj;ksa esa ls fdUgha nks dk ;ksx rhljh nwjh ls vf/d gS] blfy, bu ¯cnqvksa
P, Q vkSj R ls ,d f=kHkqt curk gSA
lkFk gh] ;gk¡ PQ
2
+ PR
2
= QR
2
gSA vr%] ikbFkkxksjl izes; osQ foykse ls] gesa Kkr gksrk
gS fd ? P = 90° gSA
blfy,] PQR ,d ledks.k f=kHkqt gSA
mnkgj.k 2 : n'kkZb, fd ¯cnq (1, 7), (4, 2), (–1, –1) vkSj (– 4, 4) ,d oxZ osQ 'kh"kZ gSaA
gy : eku yhft, fn, gq, ¯cnq A(1, 7), B(4, 2), C(–1, –1) vkSj D(– 4, 4) gSaA ABCD dks
,d oxZ n'kkZus dh ,d fof/ ;g gS fd mldk xq.k/eZ tSlk fd oxZ dh lHkh Hkqtk,¡
cjkcj rFkk nksuksa fod.kZ cjkcj gksrh gSa] dk iz;ksx fd;k tk,A vc]
AB =
2 2
(1 – 4) (7 2) 9 25 34 + - = + =
2018-19
```
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## गणित कक्षा 10

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