B3C1

- B(. AB. A( ( 3. AA PP 0 a a a 4. ( 5. Ex. ABCDEF Ans8305 Ex. ABCDE Ans00. a+ b a+ b b. a+ b = b + a a b a ( a+ b + c = a+ ( b + c a+ 0= a = 0+a a + ( a = 0 = ( a + a b a b 3. a b = a+ ( b a 4.(P AB = AP + PB ( P uu AB = PB PA(-( Ex3. ( uu BC + CA ( AB AC (3 PB PC (4 AB + BC + CD + DA Ans( uu BA ( CB (3CB (4 0 a r R a r r a. a 0 r>0 r a a a r r<0 r a a a r r=0 r a = 0.= a 0 r a = 0 Ex4. ABC DEF AB BC AC BD = a uu BE u = b a b ( DF ( DE (3 AC (4 AE (5 BF Ans( b ( a + b (3 a + b (4 a + b (5 a + b 3---/4

Ex5. ABCDEF AB = a AC = b a b AD AE AF BF Ans b a b 3 a b a b 3 a Ex6. O ABCD EFG A D H OA OB OC OD E H ( uuu A AB + BC = AE + EF + FG + GC (B AB = EF (C AB BC = DB (D uu AB u + uuu BF O uuu + FE = GC [86 ]AnsABCD F G. a // b r R a =r b a b 0. ( a b a b r a +s b a b ( a b 0 r a +s b = 0 r=s=0 Ex7. ( a b 0 (3x+y a +(x 3y+8 b = 0 xy uu ( ABC 3x AB+(y+ BC +(x 5CA= 0 xy Ans 5 7 B C 3. ABC(O (ABC r R AB = r AC αβ R α+β= O u A = αob + βoc uu ABC =0 (ABC pab+ qac = 0 p=q=0 Ex8. ABC 3 OB =(5t OA +(t+4co t Ans 4. ( A P B AP m BP = n m OP = OA + OB (O ( = n m+ n m+ n P AB OP = ( OA + OB 5. ( A B P AP m BP = OP = n OA + m OB n m n m n Ex9. ABC M AB AM BM =3 P CM CP PM =3 3 AP =x AB+y AC (xyans( 0 4 3---/4

Ex0. ABC D BC BD CD =3E AC 3 DE =x AB+y AC xy Ans( 5 0 Ex. ABCD E CD CD =3 DE AE BD P AP =x AB+y AD (xyans( 4 3 4 Ex. ABC D AB AD BD =3 E AC BE CD P=x AP AB +y AC (xyans( 4 3 8 Ex3. D ABC BC CD = BD G CA GD GD =r AB +s AC rs r+s [83 ]Ans B D A G C 5. uu P ABC lpa + mpb + npc = 0 lmn R PAB PBC PCA= n l m uu Ex4. P ABC PA + 3PB + 4PC = 0 PAB PBC PCA Ans43 uu Ex5. P ABC PA= PB+ 3PC PAB PBC PCA Ans3. a θ (0 θ π a b b a b = a bcosθ ( Bible. a b c k R ( a b = b a ( ( c ( a+ b a = c + c b ( (3 b (k b ka ( = a = a ( kb ( (4 a a = a (5 a b a b =0 (6 ( a+ b ( a+ b = a a+ a b+ b b = a + a b + b Ex6. a =4 3 a + b = 0 a b =Ans 4 3---3/4

Ex7. a = b = a b 60 o OP = a + b OQ = a b PQ =Ans3 Ex8. + a b + c = 0 a b = b c = c a = 3 ( a =( b c θ cosθ=ans 5 Ex9. ABC AB =5BC =7CA=8 AB BC Ans 5 Ex0. ABCD AB =5BC =3 ( AC Ans 66 BD ( AC + BD uu uu Ex. ABC PA + PB + PC = 0 PA = PB = PC = uu ( PA PB (PAB (3ABC (4 AB 3 Ans 7 3 7 4 4 Ex. ABC P BC uu uuu ( AP AB u ( AB AC (3( BC + AP AC (4( BC AP ( AB + AP Ans35 8 Ex3. a = b = a b 0 o ( b a +t b t= a +t b =Ans 3 Ex4. a = b =3 a b 0 o a + b Ans 7 ; 4 Ex5. a =3 b =4 a + b = 3 a b Ans0 o Ex6. a = b = a b 60 o a + b a + b Ans60 o Ex7. + a b +3 c = 0 a b =c a = 3 a Ans5 Ex8. + a b + c = 0 a = b =3 c =4 ( a b ( a b + b c + c a (3 a b θ cosθans( 3 ( 9 (3 4 3---4/4

Ex9. O ABC A=60 o B=45 o uu u OA+ OB + OC Ans6 Ex30. ABC AB =3BC =6CA=7 AB BC Ans u Ex3. MN AC BD AB + CD = MN Ex3. u (x+ u +(x 3 =y( u xyansx=y= Ex33. ABC AP =3 AB + AC AP BC M u 3 AM =x AB+y AC (xyans( 5 5 Ex34. ABC D AB AD BD =3 E BC AE CD P AP PE Ans43 Ex35. ABCD E CD CD =3 DE F AB BE CF P 5 AP =x AB+y AD (xyans( 7 3 7 Ex36. ABCD E AB AF = 3FD BF CE K (CK KE ( ABCD BCK uuu (3BK KF (4CK = xcb + ycd 6 0 Ans(03( (358(4( 5 3 5 3 (xy 3---5/4

-. ( G ABC O OG = ( OA + OB + OC 3 GA + GB + GC = 0 Ex37. O G ABC ( = ( OA OB OC 3 + + OG ( GA + GB + GC = 0 uu u Ex38. ABC DC = 4B DEC = AEFB= AF G DEF AG = αab + βac 7 (αβ[75 ]Ans( 45 8 45 F A E B D Ex39. G ABC GA =3 GB =5 GC =7 (GA GB 5 ABC Ans 45 3 4 C =(. ( ( I ABC BC =aca=bab =co uu a b c OI = OA + OB + OC a+ b+ c a+ b+ c a+ b+ c Ex40. ABC AB =8BC =4CA=6I ABC O uu OI = xoa + yob + zoc (xyzans( 9 3 4 9 3. AB AO = AB O ABC AC AO = AC Ex4. ABC AB =6BC = 7 CA=4O ABC OA BC D ( AB AC uuu ( AO =x A +y (xy uub AC u 4 (3 AD =α AB +β AC (αβans( 9 6 ( 8 3 4. H ABC AB AH = AB AC = AC AH 3---6/4

Ex4. ABC AB =6BC =5CA=7H ABC ( AB AC ( 95 AH =x AB+y AC (xyans30( 44 5 4. ABC DE AB AC DE // BC DE = BC uuu DE = AE AD = AC AB = BC. ABC B=90 o AC AB = + BC uu uu uu uu uu AC = BC BA = BC + BA BC BA = BC + BA BC BA cos B 3. ABCD AC + BD = AB + BC + CD + DA AC = AB + AD = AB + AD + AB AD = AB + AD + AB AD cosa BD = BC+ CD = BC + CD + BC CD= BC + CD + BC CD cosb 4. ABCD AC BD AB BC CD + = + + + DA ABCD uu uu uu = PB PA + PC PB + PD PC + PA PD PC PA PD PB uu uu uu = = PA PB+ PC PD = BA CD =0BA = CD 5. ABC AM AB + AC = ( AM + BM u = ( AM + BC = ( AB + AC + AC AB = ( AB + AC = 6. ( suur suur suur ABC OE OF AC AB D BC OD BC uuu uu 0= E AC ( OC u O = + OA ( OC OA = OC OA OC = OA uu u 0= OF AB = ( OB + OA ( OB OA = OB OA OB = OA u OD BC = ( OC B ( OC OB = OC OB O =0Q.E.D. 3---7/4

7. ( ABC BE CF G G AD ( AD = AB + AC ( AG = AB + AC = AB + AC 3 3 3 3 (3((AD 3 = AG ADG 8. ( suur suur suur ABC AH BH CH suur AH HA BC=0HA HC= HAHB BH suur HB AC=0HB HC = HB HA uu HC AB = HC ( uuu u HB HA = HC HB HC HA = HA HB HA HB = 0CH suur 9. u u u u ( AM + BM = AB + AC = BC = BM AM u = BM 0. ABCD P AP = k AC AP = k AC = k( AB + AD = k AB + k AD PBD k+k= k= P AC P BD. BC = AC AB = AB + AC AB AC = AB + AC AB AC cosa a = b + c bccosa Ex43. ABPQ ACRS uu AB = AQ AS = AC BS CQ [76 ] Ex44. P B Q A S C R Ex45. uu uu Ex46. ABC AB BC = BC CA = CA AB ABC uu uu Ex47. PA PB= PBPC = PCPA P ABC Ans 3---8/4

-3. O P(ab =(ab a x b y. =(ab = a + b 3.A(x y B(x y =(x AB x y y uu uu 4.= a OA u OA u x a (0 θ<π a θ=( a a cosθ a sinθ Ex48. ( a 0 a 50 a π ( AB =4 AB A(38 B 3 Ans(-5 3 5(8+ 3 =(a a a b =(bb a + b =(a +b a +b a b =(a b a b Ex49. A0B5C(74 (ABC ( ABCD D Ans(33(--3(5(33.=(a a a r a =(rara. a =(aa ( a a (a a a a 3. a =(a a =(b b b a // b r R =r a b a b 0 a a = (a b =a b b b Ex50. x>0 AB =( AC =( xx ABC 6 5 xans 30 Ex5. =( a b =(6t R t a + b t [8 ] Ansm= t= 4 Ex5. =( a b =( c =(0 t a //( b +t c t Ans 5 Ex53. A7 3Bk C( k Ans4 3--3-9/4

. A(x y B(x y A-P-B AP : PB = m : n nx + mx ny + my P ( n+ m n+ m. A(x y B(x y P-A-B A-B-P AP : PB = m : n nx mx ny my P ( ( n m n m x + x y + y 3. A(x y B(x y (, 4. A(x y B(x y C(x 3 y 3 x + x + x3 y + y + y3 ABC G(, 3 3 Ex54. ABC A( 8B( 6 C(6 5 suur ( A BC D D Ans(-4 suur ( A BC E E Ans(8-8 x = x0 + a t.p(x 0 y 0 =(ab, t R y = y0 + b t b a.a(x y B(x y suur x = x + ( x - x t ( AB, t R y = y + ( y - y t x = x + ( x - x t ( AB, 0 t y = y + ( y - y t uuur x = x` + ( x - x t (3 AB, t 0 y = y + ( y - y t Ex55. A(3 50B(477( AB Ans uuur (P(xy AB AP 5 x Ans9 x<3 Ex56. A(B( P(xy AB (3x 4y+6(x +y 4AnsM=6m=M=m= 3 Ex57. A(B(3C( +t+tt R ( C ( ABC Ansx y+=0 + 5 3--3-0/4

Ex58. a =(b =(6 PQ = a 3 b ( PQ =( Q(38 P Ans((-4-6((74 Ex59. A(54B(7 AB AB Ans3 35 o Ex60. AB =(43BC =(0 6ABC Ans6 Ex6. xy O OA = AB = BC = 3 OA AB BC 30 45 60 C Ans( + 33 Ex6. =(3 a a 3 Ans 9 3 (, 0 0 Ex63. A( B( C(4 6 3 (AB Ans( 5 4 5 ( CD // AB CD =0 D Ans(0-4or(- Ex64. C (34 5 C AA A 3 A 4A 5A 6 A =(00 uu u uu u uu AA + AA + AA + AA + AA = 3 4 5 6 Ans(84 Ex65. A( 3B(4 7P suur AB AB = 5AP P Ans(0(-5 Ex66. A( B(3 P Ans(65 ( Q Ans( 3 7 3 suur suur Ex67. A(B(3C(0 ( AB (C (3C su AB AB ur suur (4C AB Ans((( 0(3 (4( Ex68. (0 0 x 3y=0 AB AB Ans4 Ex69. A(mn Lx+y= m +n +m+4n 5 Ans- 3--3-/4

-4 a =(aa b =(b b a b θ. a b = a b cosθa b +a b a b.cosθ a b Ex70. A(0B(3C( ( AB AC (sin BACAns45. a b a b =0. a a a // b = (a b =a b b b Ex7. a =(=(34 b a ( a +k b k Ans. ax+by+c=0 (ab. (b a θ π θ.l a x+b y+c =0L a x+b y+c =0 θ aa + bb cosθ= ± a + b a + b. L L x m m θ(θ π/ m m tanθ= ± + m m Ex7. 3 x y+=0 x+y+7=0 Ans75 o 05 o Ex73. (3 x+ 3 y 3=0 60 o Ansx 3 y 3+ 3 =0orx=3 a b θ. a b ( c os b a b a θ = ( b b 0 b b. a b a b a cosθ = b Ex74. a =(4=( 3 b ( b a ( b a Ans5 ( 3--4-/4

Ex75. =( 3 a Lx y=5 Ans( Ex76. ABC A(5B(5C(37P BC 6 AP AB ( 5 8 5 P [84 ]Ans( 5 5 8 5 Ex77. A(aB(bC(34 O OA OB OC a b [87 ]Ans3a 4b= Ex78. OA + OB + OC = 0 OA = OB = OC = OA OB Ans 3. AB =(ab AC =(cd ABC = AB AC ( AC AB = ad bc u = ( a, b = ( c, d ad bc Ex79. A(04B( C(3 (ABC (AB AC Ans. a b a b a b a // b a.a a b b R ( a + a + ( b b ( ab + ab a = b b Ex80. ab (a+ 4 b (b+ 9 a Ans5 Ex8. =(xy a b =(pq a =3 b =4 a b Ans- nπ 4 9 Ex8. θ n Z + Ans5 sin θ cos θ Ex83. x>0y>0 x+y=6 4 9 x + y Ans 5 6 3--4-3/4

Ex84. θ x 3y+3=0 4x+3y =0 tanθans3 x = Ex85. + t y = + 3t (t Ry θ sinθ[76 ]Ans 3 Ex86. L (3 y=x+6 45 o L Ans3x+y 6=0orx 3y+8=0 Ex87. x+y+3=0 4x+ay+5=0 45 o 4 a Ansor 3 Ex88. D ABC BC ABD ADC B 3 (05C (70 D [85 ]Ans( 4 5 3 Ex89. =(x a b =(y a b = x +y Ans 4 5 Ex90. xy R 3x+y= 6 ( x 4 + ( y Ans4 3 Ex9. 3x+4y 7=0 4x+3y+=0 Ans x y+9=0 Ex9. x+y =0x y 7=0x+y+=0 Ans( 3 3 Ex93. L 5x y =0L 4x+3y+=0 L L [79 ]Ans3x+y+7=0 x 3y+9=0 Ex94. (45 3x 4y 7=0 x 5y+6=0 Ans9x 7y =0or7x+9y 73=0 3--4-4/4