2. Coordinate Geometry

8/12/2019 2. Coordinate Geometry

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GC-CRFMKHVA @ACIAVR]

MKVRCFQGVMCKMk tna pravmcus gehss, ycu nhva eahrkt tc ecghta tna pcsmtmck cl h pcmkt mk h gccrfmkhta pehka cl twc fmiaksmcks, mtaris cl twc gccrfmkhtas. ]cu nhva eahrkt tnht h emkahr aquhtmck mk twc vhrmhbeas, cl tna lcri hx + g 9 4 (amtnar h 4 cr b 4) ghk ba raprasaktaf `rhpnmgheey hs h strhm`nt emka mk tna gccrfmkhta pehka cl x hkf y gccrfmkhtas. Mkgnhptar 6, ycu nhva eahrkt tnht `rhpn cl h aquhtmck hx 2 + bx + g 9 4, h 4 ms hk upwhrf phrhbceh ml h ; 4 hkf hfcwkwhrf phrhbceh ml h 5 4.Mk tnms gnhptar, ycu wmee eahrk tc lmkf tna fmsthkga batwaak twc `mvak pcmkts mk taris cl tnamr gccrfmkhtas hkf hesctna gccrfmkhtas cl tna pcmkt wnmgn fmvmfas tna emka sa`iakt ocmkmk` tna twc `mvak pcmkts mktarkheey mk tna `mvak rht

NMWVCRMGHE LHGVWRaka Fasghrtas (81=< - 8


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. Vna y-gc-crfmkhta hky pcmkt ck x-hxms ms zarc.=. Vna x-gc-crfmkhta hky pcmkt ck x-hxms ms zarc.84. Hky pcmkt mk tna xy-pehka, wncsa y-gc-crfmkhta ms zarc, emas ck x-hxms.88. Hky pcmkt mk tna xy-pehka, wncsa x-gc-crfmkhta ms zarc, emas ck x-hxms.82. Vna crm`mk nhs gccrfmkhtas (4, 4).87. Vna crfmkhtas cl hee pcmkts ck h ncrmzckthe emka wnmgn ms phrheeae tc x-hxms hra aquhe m.a. y 9 gcksthkt 9 2.

86. Vna hbsgmssh cl hee pcmkts ck h vartmghe emka wnmgn ms h emka phrheeae tc y-hxms hra aquhe m.a. x-gcksthkt 9

Lcur Xuhfrhkts cl h Gccrfmkhta pehka ?Vna ragthk`uehr hxas ^C^ hkf ]C] fmvmfa tna pehka mktc lcur quhfrhkts hs baecw ?

81. Hky pcmkt mk tna M quhfrhkt nhs (+ va hbsgmssh, + va crfmkhta).8. Hky pcmkt mk tna M\ quhfrhkt nhs (+ va hbsgmssh, - va crfmkhta).


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FMWVHKGA LCRIQEH

Vna fmsthkga batwaak twc pcmkts (x 8 , y 8 ) hkf (x 2 , y 22

822

82 )()( y y x x + ) mk h ragthk`uehr gccrfmkhta systai ms aquhe tc

Trccl ? ^,C^ hkf ]C] hra tna ragthk`uehr gccrfmkhta hxas. T(x 8 , y 8) hkf X (x 2 , y 2

) hra tna `mvak pcmkts. Pafrhw TH hkf XB parpakfmguehrs ck tna x-hxms ? TG hkf XF parpakfmguehr ck tna y-hxms,

Kcw, GT (prcfugaf) iaats BX mk R hkf TR BX.Pa lmkf TR 9 HB 9 CB CH 9 (x 2 x 8 )hkf XR 9 BX BR 9 BX HT 9 GF CG 9 (y 2 y 8

)

Mk rm`nt TRX, usmk` Tytnh`crhs tnacrai, wa nhva2

822

822

282

82222 )()()()( y y x xTX y y x xXRTRTX +9+9+9

Pa nhva thjak tna pcsmtmva squhra rccl vheua baghusa fmsthkga batwaak twc pcmkts ms h kck-ka`htmva quhktmty.

Fmsthkga cl h Tcmkt lrci Crm`mk ?

Vna fmsthkga cl h pcmkt (x, y) lcri crm`mk ms22

y x + .

Trccl ? Eat us thja h pcmkt T (x,y) mk tna `mvak pehka cl hxas C^ hkf ]C] hs sncwk mk tna lm`. Nara, tna pcmkt T(x, y) ms mk tna lmrst quhfrhkt but mt ghk ba thjak hkywnara mk hee tna lcur quhfrhkts. Pa nhva tc lmkf tna fmsthkga CT,m.a., tna fmsthkga cl tna pcmkt T lrci tna crm`mk C.

Lcri tna pcmkt T, frhw TI C^ hkf TE C]. Vnak wa nhvaCI 9 xIT 9 CE 9 yCT 2 9 CI 2 + IC 2 9 x 2 + y 2


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Vnaralcra, CT 9 22 y x +

Vast Lcr @aciatrmghe Lmùras ?(h) Lcr hk mscsgaeas ? Trcva tnht twc smfas hra aquhe.(b) Lcr hk aqumehtarhe trmhkèa ? Trcva tnht tnraa smfas hra aquhe.

(g) Lcr h rm`nt-hkèaf trmhkèa ? Trcva tnht tna sui cl tna squhras cl twc smfas ms aquhe tc tnasquhra cl tna tnmrf smfa.(f) Lcr h squhra ? Trcva tnht hee smfas hra aquhe hkf fmh`ckhes hra aquhe.(a) Lcr h rncibus ? Trcva tnht hee smfas hra aquhe hkf fmh`ckhes hra kct aquhe.(l) Lcr h ragthkèa ? Trcva tnht tna cppcsmtas smfas hra aquhe hkf fmh`ckhes hra hesc

aquhe.(`) Lcr h phrheeaec`rhi ? Trcva tnht tna cppcsmta smfas hra aquhe mk eak`tn hkf fmh`ckhes

hra kct aquhe

Ax.8 Lmkf tna fmsthkga batwaak tna lceecwmk` phmrs cl pcmkts?(h) (2,7), (6,8) (b) (-1, 0), (-8,7) (g) (h,b), (-8, - b) ZKGARVY

Wce. (h) Vna `mvak pcmkts hra ? H (2,7), B (6,8).

Raqumraf fmsthkga 2822

82 )()( y y x x BH HB +999

22>66)2()2()78()26( 2222 99+9+9+9 HB ukmts(b) Fmsthkga batwaak T(- 1, 0) hkf X(-8, 7) ms `mvak by

TX 9 XT 9 2822

82 )()( y y x x +

9 72


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Ax.7 Pnht pcmkt ck y-hxms ms aqumfmsthkt lrci tna pcmkts (7, 8) hkf (8, 1)

Wce. Wmkga tna raqumraf pcmkt T(shy) ms ck tna y-hxms, mts hbsgmssh (x-gc-crfmkhta) wmee ba zarc. Eat tna crfmkhta (y-gc-crfmkhta) cl tna pcmkt ba y.

Vnaralcra gc-crfmkhtas cl tna pcmkt T hra ? (4, y)

m.a., T (4, y)

Eat H hkf B fakcta tna pcmkts (7, 8) hkf (8, 1) raspagtmvaey.TH 9 TB (`mvak)

Wquhrmk` wa at ?TH2 9 TB

2

2222 )1()84()8()74( +9+ y y

y y y y 8421828=22

++9+

28842


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66)24()88( 22 99+9+++

BG 9 22>66)42()88( 22 99+9+

GF 9 22>66)24()87( 22 99+9++

FH 9 22>66)42()78( 22 99+9++

HB 9 BG 9 GF 9 FH

Vna smfas cl tna quhfrmehtarhe hra aquhe (8)

HG 9 68


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BG 9 1722828=


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Vnaralcra, tna gccrfmkhtas cl tna pcmkt T hra

++

++

28

8228

28

8228 ,ii

yi yiii

xi xi

Raihrj ? Vc raiaibar tna sagtmck lcriueh, tna fmh`rhi `mvak baecw ms naeplue?

Tcmkt Fmvmfmk` h Emka Waìakt mk tna Rhtmc j ? 8 ?

Ml T

++

++

28

8228

28

8228 ,ii

yi yiii

xi xi fmvmfas tna emka-saìakt, ocmkmk` H(x8 , y 8) hkf B(x 2 , y 2) mktarkheey mk tna rhtm

i 8 ? i 2

+

+

+ 8,

82

8

822

8

2

8

22

8

ii

y yii

ii

xii

wa ghk axprass mt hs baecw ?

T (By fmvmfmk` tna kuiarhtcr hkf tna fakcimkhtcr by i 2

2

8

ii

)

Tuttmk` 9 j, tna rhtmc bagcias j ? 8 hkf tna gccrfmkhtas cl T hra axprassaf mk tna lcri

T

+++

8,

88282

j yjy

j xjx

Vnaralcra, tna gccrfmkhtas cl tna pcmkt T, wnmgn fmvmfas tna emka-saìakt ocmkmk` H (x 8 , y 8) hkf B(x 2 , y 2

++

++

88

,8

8 22j

yjyj

xjx) mktarkhee

mk tna rhtmc j ? 8, hra `mvak by

Imf-pcmkt Lcriueh ?

Gccrfmkhtas cl tna imf-pcmkt cl tna emka-saìakt ocmkmk` (x 8 , y 8 ) hkf (x 2 , y 2

++

2,

22828 y y x x ) hra .

Vna imf-pcmkt I (x,y) cl tna emka-saìakt ocmkmk` H (x 8 , y 8) hkf B(x 2 , y 2) fmvmfas tna emka-saìakt HB mk tna rhtmc

8 ? 8. Tuttmk` i 8 i 2

++

2,

22828 y y x x 9 8mk tna sagtmck lcriueh, wa àt tna gccrfmkhtas cl tna imf-pcmkt hs

Gceemkahrmty cl tnraa pcmkts ?Vnraa mvak pcmkts H(x 8 , y 8), B(x 2 , y 2), G(x 7 , y 7) hra shmf tc ba gceemkahr ml cka cl tnai iust fmvmfa tna emkasaìakt ocmkmk` tna ctnar twc pcmkts mk tna shia rhtmc.

Raihrj ? Vnraa pcmkts hra gheeaf kck-gceemkahr ml cka cl tnai fmvmfas tna emka saìakt ocmkmk` tna ctnar twc pcmktsmk fmllarakt rhtmcs.

H

J ? 8B

(^ 8, ] 8)

G(^ 7, ] 7)

(^ 2, ] 2)


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GCITAVMVMCK PMKFCP

WAGVMCK LCRIQEH LCR A^VARKHE FM\MWMCKVna gc-crfmkhtas cl emka pcmkt wnmgn fmvmfas tna emka saìakt ocmkmk` tna pcmkt (x8 , y 8) hkf (x 2 , y 2

kikyiy

kikxix 8282 ,

) axtarkheey mk tn

rhtmc i ? k hra

a.`., Mk tna lceecwmk` ghsa, G fmvmfas HB axtarkheey mk rhtmc HG ? BG

Eat H 9 (8, 4), B 9 (6, 4) hkf 1 ? 2 ba tna rhtmc mk wnmgn G fmvmfas HB axtarkheey. Vnak gc-crfmkhtas cl G hra ?

Y4, Lmkf tna gc-crfmkhtas cl tna pcmkt wnmgn fmvmfas tna ocmk cl (-8, 0) hkf (6, -7) mk tna rhtmc 2 ? 7, ZKGARVY Wce. Eat T(x, y) fmvmfas tna emka saìakt HB ocmkmk` H(-8, 0) hkf B(6, -7) mk tna rhtmc 2 ? 7. Vak by usmk` sagtmck lcriueh

emka tna gc-crfmkhtas cl T hra `mvak by ?

9

+9

++

+

181

,11

12872

07)7(2,

72)8(762

TT 9 T(8, 7)

Nakga tna raqumraf pcmkt cl fmvmsmck wnmgn fmvmfas tna emka saìakt ocmkmk` H(-8, 0) hkf (6, -7) mk tna rhtmc 2 ? 7 mT(8, 7).

Wce.

Mt ms mvak tnht HB ms fmvmfaf mktc lcur aquhe phrts ? HT 9 TX 9 XR 9 RB

X ms tna imf-pcmkt cl HB, tnak gc-crfmkhtas cl X hra ? )1,4(2

84,

24

2>2

,2

22 9

9

++

T ms tna imf-pcmkt cl HX, tnak gc-crfmkhtas cl T hra ?

9

9

++

20

,820

,22

212

,2

42

Hesc, R ms tna imf-pcmkt cl XB, tnak gc-crfmkhtas cl R hra ?

9

9

++

2

87,8

2

87,

2

2

2

>1,

2

24

Nakga, raqumraf gc-crfmkhtas cl tna pcmkts hra ?

)2

87,8(),1,4(),

20

8( RXT Ax.84 Ml tna pcmkt G(-8, 2) fmvmfas tna emkas saìakt HB mk tna rhtmc 7 ? 6, wnara tna gc-crfmkhtas cl H hra (2, 1), lmkf tna

gccrfmkhtas cl B.Wce. Eat G (-8, 2) fmvmfas tna emka ocmkmk` H (2, 1) hkf B (x, y) mk tna rhtmc 7 ? 6. Vnak.

(-2, 2) (2, >)H X R BT

GH

B


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84

)2,8(0

247,

0>7 9

++

G y x

G

80

>7 9+ x & 20

247 9+ y 0>7 9+ x & 86247 9+ y 19 x & 29 y Vna gccrfmkhtas cl B hra ? B (-1, -2)

Ax.88 Lmkf tna rhtmc mk wnmgn tna emka sa`iakt ocmkmk` tna pcmkts (8, -0) hkf (


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Ax.87 Vna twc cppcsmta vartmgas cl h squhra hra (-8, 2) hkf (7, 2). Lmkf tna gc-crfmkhtas cl tna ctnar twc vartmgas.Wce. Eat HBGF ba h squhra hkf twc cppcsmta vartmgas cl mt hra H(-8, 2) hkf G(7, 2) HBGF ms h squhra.

HB 9 BG HB 2 9 BG

2

2222 )2()7()2()8( +9++ y x y x =8=> 99 x x HBGF ms rm`ntht B, tnakHG2 9 HB 2 + BG 2

(Tytnh`crhs tnacrai) 222222 )2()7()2()()22()87( 9+++9++ y x y x

222 )78()88()2(28< +++9 y 98< >>8


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82

Hrah cl HBG 9 )()()(28

287872728 y y x y y x y y x ++

Gckfmtmck cl gceemkahrmty cl tnraa pcmkts ?Vna mvak pcmkts H(x8 ,y 8 ), B(x 2 ,y 2) hkf G(x 7 ,y 7

) wmee ba gceemkahr ml tna hrah cl tna trmhk`ea lcriaf by tnai iust ba zarc baghusa trmhk`ea ghk kct ba lcriaf.

hrah cl HBG 9 4

4)()()(28

287872728 9++ y y x y y x y y x

4)()()( 287872728 9++ y y x y y x y y x

ms tna raqumraf gckfmtmck lcr tnraa pcmkts tc ba gceemkahr.Ax.86 Vna gc-crfmkhtas cl tna HBG hra H(6, 8), F(7, 2) hkf G(4, J). @mvak tnht tna hrah cl HBG ms 82 ukmt2

. Lmkf tnavheua cl j.

Wce. Hrah cl HBG lcriaf by tna mvak-pcmkts H(6, 8), B(-7, 2) hkf G(4, j) ms9 )28(4)8)(7()2(6

28 ++ j j 9 )088(

28

7768>28

j j j 9+

But hrah cl HBG 9 82 ukmt 2

26)088(28

28 99 j

..(`mvak)

26)088( 9 j

26)088( 9 j 26)088(26088 99 j cr j

8788260 99 j 0

879j

cr 26)088( 9 j 26088 9+ j

7188260 9+9j 1071

99j Ax.81 Lmkf tna hrah cl tna quhfrmehtarhe wncsa vartmgas thjak mk crfar hra (-6, -2), (-7, -1), (7, -2), hkf (2, 7).

ZKGARVYWce. Ocmk H hkf G

Vna mvak pcmkts hra H(-6, -2), B(-6, -2), G(-6, -2) hkf F(2, 7)Hrah cl HBG

9 )12(7)22(7)21()6(28 ++++

9 812428 + 9 1.84

228 sq. ukmts

Hrah cl HGF

9 )12(7)22(7)72()6(28 ++++

9 1.802

718124

28 99+ sq. ukmts

Hrah cl quhfrmehtarhe HBGF 9 hr. ( HBG) + hr. ( HGF) 9 (84.1 + 80.1) sq. ukmts 9 2> sq. ukmts


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Ax.8< Lmkf tna vheua cl p lcr wnmgn tna pcmkts (-8, 7), (2, p), (1, -8) hra gceemkahr.

Wce. Vna `mvak pcmkts H (8, 7), B(2, p), G(1, -8) hra gceemkahr.

Hrah cl HBG lcriaf by tnasa pcmkts sncuef ba zarc. Vna hrah cl HBG 9 4

)()()(28 287872728 y y x y y x y y x ++ 9 4

)7(1)78(2)8(8 p p +++ 9 4 p p 181>8 + 9 4 8


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86

Vast Lcr @aciatrmghe Lmùras ?

(h) Lcr hk mscsgaeas trmhkèa ? Trcva tnht ht eahst twc smfas hra aquhe(b) Lcr hk aqumehtarhe trmhkèa ? Trcva tnht tnraa smfas hra aquhe(g) Lcr h rm`nt-hkèaf trmhkèa ? Trcva tnht tna sui cl tna squhras cl twc smfas ms aquhe tc tna

squhra cl tna tnmrf smfa.(f) Lcr h squhra ? Trcva tnht hee smfas hra aquhe hkf fmh`ckhes hra aquhe.(a) Lcr h rncibus ? Trcva tnht hee smfas hra aquhe hkf fmh`ckhes hra kct aquhe.(l) Lcr h ragthkèa ? Trcva tnht tna cppcsmta smfas hra aquhe hkf fmh`ckhes hra hesc

aquhe.(`) Lcr h phrheeaec`rhi ? Trcva tnht tna cppcsmta smfas hra aquhe mk eak`tn hkf fmh`ckhes

hra kct aquhe.

Gceemkahrmty cl tnraa pcmkts ? Eat H, B hkf G tnara `mvak pcmkts. Tcmkt H, B hkf G wmee ba gceemkahr ml tna sui, cleak`tns cl hky twc emka-saìakt ms aquhe tc tna eak`tn cl tna tnmrf emka-saìakt.Mk tna hfocmkmk` lm`. tnara hra tnraa pcmkt H, B hkf G.Vnraa pcmkts H, B hkf G hra gceemkahr ml hkf ckey ml(m) HB + BG 9 HG cr (mm) HB + HG 9 BG cr (mmm) HG + BG 9 HB

Wagtmck Lcriueh ? Gccrfmkhtas cl tna pcmkt, fmvmfmk` tna emka-saìakt ocmkmk` tna pcmkts (x 8 , y 8) hkf (x 2 , y 2)

mktarkheey mk tna rhtmc ? i 8 i 2

++

++

28

8228

28

8228 , ii yi yi

ii xi xi

hra `mvak by

Imf-pcmkt Lcriueh ? Gccrfmkhtas cl tna imf-pcmkt cl tna emka-saìakt ocmkmk` (x 8 , y 8) hkf (x 2 , y 2

++

2,

22828 y y x x

) hra

Hrah cl trmhkèa ? Hrah cl )()()(28

287872728 y y y y y x y y x HBG ++9

Gckfmtmck cl gceemkahrmty cl tnraa pcmkts ? Vna mvak pcmkts H(x8 , y 8), B(x 2 , y 2) hkf G(x 7 , y 7

) wmee ba gceemkahrtna hrah cl tna trmhkèa lcriaf by tnai iust ba zarc baghusa trmhkèa ghk kct ba lcriaf.

hrah cl 49 HBG

9++ 4)()()(2

8287872728 y y y y y x y y x 4)()()( 287872728 9++ y y x y y x y y x

ms tna raqumraf gckfmtmck lcr tnraa pcmkts tc ba gceemkahr.


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A^ARGMWA 8 (LCR WGNCCE/BCHRF A^HIW)

CBOAGVM\A V]TA XQAWVMCKW

GNCCWA VNA GCRRAGV C TVMCK MK AHGN CL VNA LCEECPMK@

8. Vna fmsthkga batwaak tna pcmkts (h, b) hkf (-h, - b) ms ?

(H) 22 bh + (B) 22 bh + (G) 4 (F) 222 bh +

2. Vna fmsthkga batwaak pcmkts (h + b, b + g) hkf (h b, g - b) ms ?

(H) 222 bh + (B) 222 gh + (G) b2.2 (F) 22 gh

7. Vna fmsthkga batwaak pcmkts H(8, 7) hkf B(x, 0) ms 1. Vna vheua cl x ; 4 ms ? (H) 6 (B) 2 (G) 8 (F) 7.

6. Vna fmsthkga batwaak tna pcmkts (h gcs 24 4 + b smk 244,4) hkf (h smk 24 4 - b gcs 24 4

22 bh

) ms ?

(H) (h + b) (B) (h b) (G) (F) 22 bh +

1. Imf-pcmkt cl tna emka-sa`iakt ocmkmk` tna pcmkts (-1, 6) hkf (=, - >) ms ?(H) (0,


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8(),7,2(),8,8( G B H (mm) )7,8(),8,8(),2,6( G B H . Lmkf tna vheua cl j, ml tna pcmkt (2, 7) ms aqumfmsthkt lrci tna pcmkts H(j, 8) hkf B(0, j).

=. Lmkf tna vheua cl j lcr wnmgn tna fmsthkga batwaak tna pcmkt H(7j, 6) hkf B(2, j) ms 21 ukmts.84. Lmkf tna gc-crfmkhtas cl tna pcmkt wnmgn fmvmfas tna emka sa iakt ocmkmk` tna pcmkts (8, -7) hkf (-7, =) mk tna rhtmc 8

7 mktarkheey.88. Lmkf tna imf-pcmkt cl HB wnara H hkf B hra tna pcmkts (-1, 88) hkf (0, 7) raspagtmvaey.82. Vna imf-pcmkt cl h emka saìakt ms (1, >). Ml cka akf pcmkts ms (7, 1), lmkf tna sagckf akf pcmkt.87. Vna vartmgas cl h trmhkèa hra H(7, 6), (0, 2) hkf G(-2, -1). Lmkf tna eak`tn cl tna iafmhk tnrcu`n tna vartax H.86. Vna gc-crfmkhtas cl H hkf B hra (8, 2) hkf (2, 7) raspagtmvaey. Lmkf tna gc-crfmkhtas cl R ck emka saìakt HB sc

tnht769

RB HR

.

81. Lmkf tna gc-crfmkhtas cl tna gaktra cl h gmrgea, tna gc-crfmkhtas cl tna akf pcmkts cl h fmhiatar bamk` (-7, >) hkf (1,) hra gceemkahr.24. Lmkf tna fmsthkga cl tna pcmkt (8, 2) lrci tna imf-pcmkt cl tna emka saìakt ocmkmk` tna pcmkts () hkf (2, 6).28. Wncw tnht tna imf-pcmkt cl tna emka saìakt ocmkmk` tna pcmkts (1, 0) hkf (7, =) ms hesc tna imf-pcmkt cl tna emka

saìakt ocmkmk` tna pcmkts (>,


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E CK@ HKWPAR V]TA X QAWVMCK

8. Trcva tnht tna pcmkts (8, 6), (7,


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GC-CRFMKHVA @ACIAVR] AÂRGMWA-2 (^)-

HKWPAR JA]

82. Vna emka saìakt ocmkmk` tna pcmkts (7, -8) hkf (8, 2) ms trmsagtaf ht tna pcmkts T hkf X. Ml tna gc-crfmkhtas cl T hkf

X hra (p, -2) hkf

q,21

raspagtmvaey, lmkf tna vheuas cl T hkf X.

87. Pnht wmee ba tna vheua cl y ml tna pcmkt

y,1

27, fmvmfas tna emka saìakt ocmkmk` tna pcmkts (1, 0) hkf (6, 1) mk tna

rhtmc 2 ? 7 mktarkheey.86. Lmkf tna gc-crfmkhtas cl tna pcmkts wnmgn fmvmfa tna emka sa iakt ocmkmk` tna pcmkts (-6, 4) hkf (4, , 6) hkf (


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24


TRA\MCQW ]AHRW BCHRF (GBWA) XQAWVMCKW

8. Wncw tnht tna pcmkt H(1,


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28

8. Vna emka ocmkmk` tna pcmkts (2, 8) hkf (1, >) ms trmsagtaf ht tna pcmkts p hkf X. Ml pcmkt T emas ck tna emka 2x y +

4, lmkf tna vheua cl j. ZHe-2441Y 8=. Wncw tnht tna pcmkts (4, 8): (2, 8): (4, 7) hkf (2, 8) hra tna vartmgas cl h squhra.

CR

Lmkf tna vheua cl J sugn tnht tna pcmkt (4, 2) ms aqumfmsthkt lrci tna pcmkts (7, J) hkf (J, 1). ZLcram`k-2441Y 24. Vna bhsa BG cl hk aqumehtarhe HBG emas ck y-hxms. Vna gccrfmkhtas cl pcmkt G hra (4, 7). Ml tna crm`mk ms tna im

pcmkt cl tna bhsa BG, lmkf tna gccrfmkhtas cl tna pcmkts H hkf B. ZLcram`k-2441Y

28. Lmkf tna gccrfmkhtas cl tna pcmkt aqumfmsthkt lrci tna pcmkts H(8, 2), B(7, 6) hkf G(1,


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72. Ml tna pcmkt G(8, 2) fmvmfas tna emka saìakt HB mk tna rhtmc 7 ? 6 wnara tna gccrfmkhtas cl H hra (2, 1), lmkf tnagccrfmkhtas cl B. ZHe-2440Y

77. Lcr wnht vheua cl p, hra tna pcmkts (2, 8), (p, 8) hkf (8, 7) gceemkahr3 ZFaenm-244>Y76. Fatarimka tna rhtmc mk wnmgn tna emka 7x + 6y = 9 4 fmvmfas ocmkmk` tna pcmkts (8, 7) hkf (2, 0). ZFaenm-244>Y71. Ml tna fmsthkgas cl T(x, y) lrci tna pcmkts H(7, Y7Y70. Lcr wnht vheua cl j, hra tna pcmkts (8, 8), (7, j) hkf (8, 6) hra gceemkahr3 ZFaenm-244>Y

CRLmkf tna hrah cl tna HBG wmtn vartmgas H(1, 0), B(6, 1) hkf G(6, 1) ZHe-244>Y

7>. Ml tna pcmkt T(x, y) ms aqumfmsthkt lrci tna pcmkts H(7, Y

7=. Vna pcmkt R fmvmfas tna emka saìakt HB, wnara H(6, 4) hkf B(4, Y64. Vna gc-crfmkhtas cl H hkf B hra (8, 2) hkf (2, 7) raspagtmvaey. Ml T emas ck HB lmkf gc-crfmkhtas cl T sugn tnht

679

TB HT

. ZHe-244>Y

68. Ml H(6, >), B(7, Y

62. Lmkf tna vheua cl j ml tna pcmkts (j, 7), (Y

67. Ml T fmvmfas tna ocmk cl H(2, 2) hkf B(2, 6) sugn tnht079

HB HT

, lmkf tna gc-crfmkhtas cl T. ZLcram`k-244>Y

66. Vna imf pcmkts cl tna smfas cl h trmhkèa hra (7, 6), (6, Y

61. Wncw tnht H(7, 2), B(1, 1), G(2, 7) hkf F(6, 6) hra tna vartmgas cl h rncibus. ZLcram`k-244>Y6Y

60. Lmkf tna fmsthkga batwaak tna pcmkts

2,

122,

1> hkf . ZFaenm-244=Y

6>. Lmkf tna pcmkt ck y-hxms wnmgn ms aqumfmsthkt lrci tna pcmkts (1, 2) hkf (7, 2).CR

Vna emka saìakt ocmkmk` tna pcmkts H(2, 8) hkf B(1, >) ms trmsagtaf ht tna pcmkts T hkf X sugn tnht T ms kahrar tc H.Ml T hesc emas ck tna emka `mvak by 2x y + j 9 4 , lmkf tna vheua cl j. ZFaenm-244=Y

6=. Ml T (x, y) ms hky pcmkt ck tna emka ocmkmk` tna pcmkts H(h, 4) hkf B(4, b), tnak sncw tnht 89+b y

h x

. ZFaenm-244=Y

14. Lmkf tna pcmkt ck x-hxms wnmgn ms aqumfmsthkt lrci tna pcmkts (2, 1) hkf ( 2, =) ZFaenm-244=YCR

Vna emka saìakt ocmkmk` tna pcmkts T(7, 7), X(


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GC-CRFMKHVA @ACIAVR] A^ARGMWA-2 (^)-

HKWPAR JA]

(4, 8), (2, 8) hkf (4, 7). ZHe-244=Y11. Ml tna imf-pcmkt cl tna emka sa`iakt ocmkmk` tna pcmkts T( 18. 2 12. 6 ? 8, sq. ukmts 16. 8 sq. ukmts 11. > 1. x + 7y 9 0 1=. h 9 8, b 9 4


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26

GCITAVMVMCK PMKFCP

TRCTARVMAW CL VRMHK@EAWEat H(x 8 , y 8), B(x 2 , y 2), G(x 7 , y 7 ) ba tna vartmgas cl hky HBG, tnak

8. Gcktrcmf ? Mt ms tna pcmkt cl mktarsagtmck cl tna gafmhks. Mt fmvmfas tna iafmhk mk tna rhtmc cl 2 ? 8.

Gc-crfmkhtas cl gcktrcmf ? @ ++++

7,

7728728 y y x x x

2. Mkgaktra ? Mt ms tna pcmkt cl mktarsagtmck cl mktarkhe bmsagtcrs cl tna hkèa. Hesc mt ms tna gaktra cl tna gmrgea tcugnmhee tna smfas cl h trmhkèa.

Gc-crfmkhtas cl mkgaktra

++

++

++

++

gbh

gybyhy

gbh

gxbxhx 728728 , .Pnara h, b, g hra tna eak`tns cl tna smfas cl trmhkèa.

Vna rhfmus cl mkgmrgea.

sr

9

Pnara ms tna hrah clVrmhkèa hkf

2gbh

W ++9

Raihrj ? Hk hkèa bmsagtcr cl h trmhkèa fmvmfas tna cppcsmta smfa mk tna rhtmc cl raihmkmk` smfas.

A.`. HF fmvmfas BG mk tna rhtmc HG HB

FG BF 9 .

7. Gmrguigaktra ? Mt ms tna pcmkt cl mktarsagtmck cl parpakfmguehr bmsagtcrs cl tna smfas cl h trmhkèa. Mt ms hesc tnagaktra cl tna gmrgea phssmk` tnrcu`n tna vartmgas cl h trmhkèa.Ml C ms tna gmrguigaktra cl h trmhkèa HBG, tnak CH 9 CB 9 CG 9 gmrguirhfmus.

6. Crtncgaktra ? Mt ms tna pcmkt cl mktarsagtmck cl hetmtufas cl h trmhkèa.

Raihrj ?

(m) Mk hk aqumehtarhe trmhkèa, tna gaktrcmf, mkgaktra, crtncgaktar hkf gmrguigaktra gcmkgmfa.(mm) Mk hk mscsgaeas trmhkèa, tna gaktrcmf, mkgaktra, crtncgaktar hkf gmrguigaktra hra gceemkahr.(mmm) Mk h rm`nt hkèaf trmhkèa, tna gmrguigaktra ms tna imf pcmkt cl nypctakusa hkf tna crtncgaktar ms tna

pcmkt wnara rm`nt hkèa ms lcriaf.


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(mv) Auear emka ? Vna gmrguigaktra C, tna gaktrcmf @ hkf tna crtncgaktar N cl h trmhk`ea hra gceemkahr,tna emka ck wnmgn tnay ema ms gheeaf Auear emka. Hesc @ fmvmfas NC mk tna rhtmc 2 ? 8.

289

@N C@

VR] CQV VNA LCEECPMK@8. Vwc vartmgas cl h trmhk`ea hra ( 8,


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2) hkf (x, 24) hra gceemkahr ml x 9

(H) 6 (B) 1 (G) < (F) 0

6. Lcr tna trmhk`ea wncsa smfas hra heck` tna emkas x 9 4 , y 9 4 hkf 8>) (B) ( 8, > ) (G) (> , 8) (F) Kcka cl tnasa ) (G) (8 , >) (F) Kcka cl tnasa

8>. Vna hrah cl trmhk`ea lcriaf by tna emkas y 9 x, y 9 2x hkf y 9 7x + 6 ms ?(H) 6 (B) 0 (G) = (F) >


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8=. Vna trmhkèa lcriaf by tna emkas x + y 9 8, 2x + 7y < 9 4 hkf 6x y + 6 9 4 emas mk tna ?(H) Lmrst quhfrhkt (B) Wagckf quhfrhkt (G) Vnmrf quhfrhkt (F) Lcurtn quhfrhkt

24. H emka ms frhwk tnrcu`n tna pcmkts (7, 6) hkf (1, (B) 7< (G) 16 (F) Kcka cl tnasa

26. Ml ,, hra tna rahe rccts cl tna aquhtmck x 7 7px 2

8,

8

8 9 4, tnak tna gaktrcmf cl tna trmhkèa wmtn vartmgas

hkf

8 ms ht tna pcmkt ?

(H) (p, q ) (B) (p/7, q/7) (G) (p + q, p q ) (F) (7p, 7q)21. Vna gc-crfmkhtas cl H, B G hra (71 y xbh

ms tna imf pcmkt cl tna saìakt ?

(H) TX (B) XR (G) RW (F) WV

74. Vna trmhkèa wmtn vartmgas H(2, 0), B(6, y) hkf G(2,


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2>

78. Vna gc-crfmkhtas wmtn cl tna pcmkt wnmgn fmvmfas tna emka sa iakt ocmkmk` ( 7, 6) hkf ( >, 0) axtarkheey mk tnarhtmc 0 ? 1 hra ?

(H)

2 2= 74Hks. H G H B B B B H H B H F G B HXua. 78 72 77 76 71 7< 70 7> 7= 64 68 62Hks. G F G H H G B H G G G H

CBOAGVM\A A^ARGMWA-HKWPAR JA]


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2=


GNCCWA VNA GCRAAGV C KA

8. Vna emkas hx + 2y + 8 9 4, bx + 7y + 8 9 4 hkf gx + 6y + 8 9 4 hra gckgurrakt, tnak ?(H) h, b, g hra mk H.T. (B) h, b, g hra mk @.p. (G) h, b, g hra mk N.T. (F) Kcka cl tnasa

2. Ml tna emkas x + 2hy + h 9 4, x + 7by + b 9 4 hkf x 6gy + g 9 4 hra gckgurrakt, tnak h, b, g hra mk (hbg 4) ?(H) H.T. (B) @.T. (G) N.T. (F) Kcka cl tnasa

7. Ml (4, ) emas ck cr mksmfa tna trmhkèa lcriaf by tna emkas 7x + y + 2 9 4, 7y 2x 1 9 4 hkf 6y + x 86 9 4 tnak ?

(H)70

21

(B)20

71

(G)21

70

(F) Kcka cl tnasa

6. Ml h, x8 , x 2 hra mk @.T. wmtn gciick rhtmc r8 hkf b, y 8 , y 2 hra mk @.T. wmtn gciick rhtmc s wnara s r 9 2, tnak tnahrah cl tna trmhkèa wmtn vartmgas (h, b), (x8 , y 8) hkf (x 2 , y 2

)8( 2 r hb) ms ?

(H) (B) hb (r 2 s 2) (G) hb (s 2 8) (F) hbrs

1. Ml tna gmrguigaktra cl h trmhkèa emas ht tna crm`mk hkf tna gaktrcmf ms tna imffea pcmkt cl tna emka ocmkmk` tna pcmkt(h2 + 8, h 2 + 8

+6

)8(,

6)8( 22 hh

) hkf (2h, 2h), tnak tna gc-crfmkhtas cl tna crtncgaktar hra ?

(H) (B) + 22 )8(67

,)8(67

hh

(G) (7(h + 8) 2, 7 (h 8 )2) (F) Kcka cl tnasa

x 2 (F) Kcka cl tnasa

=. Ml tna gaktrcmf cl tna trmhkèa lcriaf by tna phmr cl emkas 2y 2 + 1xy 7x 2

8>88

,8>8

9 4 hkf x + y 9 j ms , tnak tna

vheua cl j ms ?(H) 8 (B) 4 (G) 8 (F) Kcka cl tnasa

84. Ml x8 , x 2 , x 7 hra tna hbsgmssh cl tna pcmkts H 8 , H 2 , H 7 raspagtmvaey wnara tna emkas y 9 i 8 , x, y 9 i 2x, y 9 i 7x iaattna emka 2x y + 7 9 4 sugn tnht i 8 , i 2 , i 7 , hra mk H.T., tnak x 8 , x 2 , x 7

>1

,8,>,8

hra mk ?(H) H.T. (B) @.T. (G) N.T. (F) Kcka cl tnasa

88. Vna hrah cl tna trmhkèa wmtn vartmgas hkf

>7

2

ms ?

(H) 2 (B)28

(G) 8 (F)27

82. Hk aqumehtarhe trmhkèa wncsa crtncgaktar ms (7, 2), cka smfa ms ck x-hxms tnak vartax cl trmhkèa wnmgn ms kct ckx-hxms ms ?(H) (7,


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74

87. Ml C ms tna crm`mk hkf tna gc-crfmkhtas cl H hkf B hra (x 8 , y 8) hkf (x 2 , y 2 ) raspagtmvaey tnak CH CB gcs HCBms aquhe tc ?(H) x 8y8 + x 2y2 (B) x 8x 2 + y 8y 2 (G) x 8y2 + x 2y8 (F) x 8x 2 y 8y2

86. Ml tna vartmgas cl h trmhkèa nhva mkta`rhe gc-crfmkhtas, tnak tna trmhkèa ms ?(H) Mscsgaeas (B) Kavar aqumehtarhe (G) Aqumehtarhe (F) Kcka cl tnasa

81. Vna gmrguigaktra cl tna trmhkèa lcriaf by tna pcmkts (h gcs , h smk ), (h gcs , smk ), (h gcs , smk ) ms

(H) (4, 4) (B) ++

++

)smksmk(smk7

),gcsgcs(gcs7

hh

(G) (h, 4) (F) Kcka cl tnasa

8 ? 7 (F) 6 ? 7

8>. Vna pcmkt wmtn gc-crfmkhtas (2h, 7h), (7b, 2b) hkf (g, g) hra gceemkahr ?(H) Lcr kc vheua cl h, b, g (B) Lcr hee vheua cl h, b, g

(G) Ml h,1g

, b hra mk N.T. (F) Ml h,12g

, b hra mk N.T.

8=. Ml gc-crfmkhtas cl crtncgaktar hkf gaktrcmf cl h trmhkèa hra (6, 8) hkf (2, 8), tnak gc-crfmkhtas cl h pcmkt wnmgn msaqumfmsthkt lrci tna vartmgas cl tna trmhkèa ms ?(H) (2, 2) (B) (7, 2) (G) (2, 7) (F) Kcka cl tnasa

24. Ml tna emka y 9 ix iaats tna emkas x + 2y 8 9 4 hkf 2x y + 7 9 4 ht tna kuibar cl pcmkts nhvmk` mkta`rhe tc ?(H) 2 (B) 2 (G) 8 (F) 8

28. H trmhkèa ms lcriaf by tna pcmkt C(4, 4), H(4, 28) hkf B(28, 4). Vna kuibar cl pcmkts nhvmk` mkta`rhe gc-crfmkhtas(bctn x hkf y) hkf strmgtey mksmfa tna trmhkèa ms ?(H) 8=4 (B) 741 (G) 8>8 (F) 24Y

(H) Gaktrcmf (B) Mkgaktra (G) Gmrguigaktra (F) Kcka cl tnasa

7=. Ml T(8, 2), X(6, Y

(H) h 9 2, b 9 6 (B) h 9 7, b 9 6 (G) h 9 2, b 9 7 (F) h 9 7, b 9 1

64. Vna mkgaktra cl tna trmhkèa wmtn vartax )7,8( , (4, 4) hkf (2, 4) ms ? ZMM V-2444, HMAAA-2442Y

(H)

27

,8 (B)

7

8,

72

(G)

27

,72

(F)

7

8,8

68. Eat C(4, 4), T(7, 6), X( = 84 88 82 87 86 81

Hks. H G B H F G H G G G B H B B HXua. 8< 80 8> 8= 24 28 22 27 26 21 2< 20 2> 2= 74Hks. G G F F G H H G G H G H H H HXua. 78 72 77 76 71 7< 70 7> 7= 64 68Hks. B G H G G B G B G F G

CBOAGVM\A HKWPAR JA]

Documents

2. Coordinate Geometry