MATEMATIKA SA ZBIRKOM ZADATAKA
za III razred sredwe {kole
2
1 A2 ...An
2
1
A1 ,A2 ,...,An A1 A2 ...An A1 A2 ...An ABCD
AB BC CD DE EA A1 A2 ,A2 A3 ,...,An A1 A1 A2 An A1 A2 ...An ω1 ,ω2 ,...,ωn
K A B C D A B C D K
ABB A ABCD BCC B DCC D A B C D ADD A AB BC CD DA AA BB CC DD A B B C C D D A K n
S (ABCD )= ab
◦ =10 S (ABCD )= S (PQRS ) 45◦ , S (PQRS )=10√2
1 B1 C1
2s =6+8+10 s =12 r = 24 12 =2 SOM SON SOP
1 S
ACC1 A1 d =8√2 d1 =2√2 H =4 1 2 (d + d1 )H = 1 2 10√2 · 4=20√2
◦
m n 2 n · 180◦
◦ m n 2 n 180◦ < 360◦ mn 2m< 2n 1 n + 1 m > 1 2 n 3 m< 6 m 3 n< 6
n =3 m =4 s =8 i =12 t =6
n =3 m =5 s =20 i =30 t =12
n =4 m =3 s =4 i =12 t =8
n =5 m =3 s =12 i =30 t =20
S (Ψ)= B + M. M = 1 2 ph p h a h n M = n 2 ah
( SAC )= S ( SBC )= 1 2 8 25=100
SAB 1 2 40 17=340 2 100+340=540
)=24 K a b c a b c
(K )= abc
V (Q)=1 Q e K a b c A
e e p a q b r c pe a< (p +1)e,qe b< (q +1)e,re c< (r +1)e. E K pqr p q r K
(p +1)(q +1)(r +1) pqre 3 V (K ) < (p +1)(q +1)(r +1)e 3 pqre 3 abc< (p +1)(q +1)(r +1)e 3 ,
|V (K ) abc| < (p +1)(q +1)(r +1)e 3 pqre 3
|V (K ) abc| < (pq + pr + qr + p + q + r +1)e 3 (ab + ac + bc)e +(a + b + c)e 2 + e 3 , pe a qe b re c n Q e =1/n n e e 2 e 3
V (K )= abc.
K a b c V (K ) = abc a b B = ab H c V (K ) = abc
(K ) = BH.
(Σ)= BH.
ABCDA B C D H ABCD BB ABB A Φ Σ= ABEDA B E D Σ1 =
BCEB C E Σ Σ2 = ADFA D F K = ABEFA B E F
Σ1 K Σ Σ2 V (Φ)= V (Σ)+ V (Σ1 ),V (K )= V (Σ)+ V (Σ2 ) Σ1 Σ2 V (Σ1 )= V (Σ2 ) V (Φ)= V (K ) B ABCD H K
V (K )= BH V (Φ)= BH. Φ K
V (Σ)= V (Σ ) V (Π)= V (Σ)+ V (Σ ) V (Π)=2V (Σ).
(Π)=2S ( ABC )H,
(Σ)= S ( ABC )H, V (Σ)= BH,
V (Σ)=V (Σ1 )+ V (Σ2 )+ + V (Σn 2 ), =B1 H + B2 H + + Bn 2 H, =(B1 + B2 + + Bn 2 )H, =BH,
ADB1 BK
BK AB1 α
DB1 B H ABB1 |AB1 | =5 |BK | =2
|AK | = p |B1 K | = q |BK | = h pq = h2 p q
p + q =5,pq =4. (1, 4) (4, 1)
A B C D |PQ| = |P Q | ABC A B C h AB A B a |AB | = |A B | p a AC BC P Q A C B C P Q |PQ| = |P Q | ABC PQ p A B C P Q
A1 A2 ...An A1 A2 ...An k |A1 A2 | = k |A1 A2 | SM Ψ SM Ψ S
A1 A2 ...An
S (A1 A2 ...An )
S (A1 A2 ...An ) = |SM |2 |SM |2 . A1 A2 ...An A1 A2 ...An k
S (A1 A2 ...An )= k 2 S (A1 A2 ...An ). SMA1 SM A1 k |SM | = k |SM | Ψ1 Ψ2 S1 S2 A1 A2 ...An B1 B2 ...Bm σ1 Ψ1 Ψ1 A1 A2 ...An σ2 Ψ2 Ψ2 B1 B2 ...Bm S1 σ1
S2 σ2
S (A1 A2 ...An )
S (B1 B2 ...Bm ) = S (A1 A2 ...An )
S (B1 B2 ...Bm ) . Ψ1 Ψ2 h S1 σ1 S2 σ2
S (A1 A2 ...An )
S (A1 A2 ...An ) = H 2 h2 , S (B1 B2 ...Bm )
S (B1 B2 ...Bm ) = H 2 h2 , Ψ1 Ψ2 σ1 σ2 Ψ1 Ψ2
H V (ABCA1 B1 C1 )= BH
1 B1 C1
H + x x
(Ω) B H + h B1 h V (Ω)= 1 3 B (H + x) 1 3 B1 x = 1 3 BH + 1 3 (B B1 )x.
|AB | = x d
AOM
x 2 = H 2 +(2d)2 ,d2 = R2 x 2 2 H =2 R =7 x 2 =4+4d2 ,d2 =49 1 4 x 2 , x 2 =4+4 49 1 4 x 2 , x =10
|ON | N
a = |AB | h = |SD | h = |SN | + |ND | |SN |2 = |SO |2 −|ON |2 =202 122 =256 |SN | =16 h =16+ |ND | x = |ND | d = |OD | NOD SOD
(K )= V (Σ)
(K )= BH V (Σ)= BH V (Σ)= πr 2 H.
πr H M =2πrH
= πr 2 2B + M =2πr 2 +2πrH =2πr (r + H ). P 2R H P =2RH M =2RπH M = Pπ
x Kh x 2 = r 2 (r h)2 =2rh h2
S (Kh )= π (2rh h2 ) Ph πr 2 r h
S (Ph )= πr 2 π (r h)2 = π (2rh h2 ).
V (Λ )= V (T ).
V (T )= V (Σ) V (Ψ)= πr 2 r 1 3 πr 2 r = 2 3 πr 3 Λ Λ r V (Λ)= 4 3 πr 3 . Λh h (0 h r ) Th Σh h r Ωh h r r h
V (Σh )= πr 2 h V (Ωh )= π 3 (r 2 + r (r h)+(r h)2 )h, V (Λh )= πh2 r h 3 h = r Λh Λ h = r
V (Λ1 )= 2 3 πr 3
r h h<r
V (F )= V1 + V2 , V1 h V2
V (F ) F
V1 = 1 3 πh2 (3r h),V2 = 1 3 πa 2 (r h) OAS a 2 = r 2 (r h)2 , a 2 =2rh h2 ,
V2 = 1 3 π (2rh h2 )(r h)= 1 3 πh(2r h)(r h)
V (F )= V1 + V2 = 1 3 πh(3rh h2 +2r 2 2rh rh + h2 )= 2 3 πr 2 h. h>r
V (F )= V1 V2 Λ r
S (Λ)=4πr 2 , K h
S (K )=2πrh.
V = 1 3 BH = 1 3 a 2 H,
a = r √2=6√2 H = R + x R x AOS x 2 =122 62 =108 x =6√3 H =12+6√3 V = 1 3 a 2 H = 1 3 72(12+6√3)=144(2+ √3)
ABCD
s EBC
R r
2R +2r =2s R + r = s
H 2 = s 2 (R r )2 , H 2 =(R + r )2 (R r )2 ,
H 2 =4Rr, H =2√Rr. p H 2 = √Rr
P =4p 2 π =4Rrπ. M = πs(R + r )= π (R + r )2
P : M =4Rr :(R + r )2
1 x + b1 y =
2
+
2 y = c2 , a1 ,b1 ,c1 ,a2 ,b2 ,c2 a1 ,b1 ,a2 ,b2 a1 = b1 = a2 = b2 =0 a1 = b1 = a2 = b2 =0
)
)
+2y =5
x + y =4
= 12 21 = 3,D (x)= 52 41 = 3,D (y )= 15 24 = 6, (x,y )=(1, 2)
+2y =3 ∧ 2x +4y =6 x +2y =3 ∧ 2x +4y =5 D = 12 24 =0 (x,y )=(3 2t,t) t
y (x,y )=(t, 1 t) t
x
y =
x +7y =4 7x 2y = 3 ∧ x + y = 1 3x +4y = 5 ∧ 6x +8y =4 15x 14y = 13 ∧ 7x 6y = 5 3x +2y =1 ∧ 9x +6y =3 ab cd ac bd ad bc ab cd = cd ab , ba dc = ab cd , k k kab kcd = k ab cd , ab kckd = k ab cd ,
kakb =0, kakb
=0,
+ kbb c + kdd = ab cd , a + kcb + kd cd = ab cd ,
+ kdd
456 123 01 1 4 12 078 123 507 065 1 12 x 2 x 1 y 2 y 1 z 2 z 1 xyz zxy yzx abc xyz yzzxxy a1 b1 c1
axxa byyb czzc = yzzxxy 111 bccaab axbycz a 2 b2 c 2 111 = xyz abc bccaab xyz abc a 2 b2 c 2 = bcxcayabz 111 abc
87423 45187 59283
964723 974321 984925 1578 1789 28912 8910 456 919293 17467 20498 23529 17467 20498 235212 1001172 1002184 1003197
y + zz + xx + y xyz 111 b cc aa b cab bca
x + yu + vs + t
2x +3y 2u +3v 2s +3t 4x +5y 4u +5v 4s +5t y zy + zy z xz + xz x yx + yx
(x 1)2 x 2 +1 x
(y 1)2 y 2 +1 y
(z 1)2 z 2 +1 z
xyz x 2 y 2 z 2 =(y z )(z x)(x y )
x + y 7x +8y 9x +11y u + v 7u +8v 9u +11v s + t 7s +8t 9s +11t
xD =
a1 xb1 c1
a2 xb2 c2
a3 xb3 c3 =
a1 x + b1 yb1 c1
a2 x + b2 yb2 c2
a3 x + b3 yb3 c3
a1 x + b1 y + c1 zb1 c1
a2 x + b2 y + c2 zb2 c2
a3 x + b3 y + c3 zb3 c3
d1 b1 c1
=
d2 b2 c2
d3 b3 c3 =D (x) yD = D (y ) zD = D (z ) xD = D (x),yD = D (y ),zD = D (z ), D =0 x = D (x) D ,y = D (y ) D ,z = D (z ) D .
D = 111 2 11 1 22 = 6,D (x)= 611 3 11 3 22 = 6 D (y )= 161 231 132 = 12,D (z )= 116 2 13 1 23 = 18, D =0 (x,y,z )= D (x) D , D (y ) D , D (x) D = 6 6 , 12 6 , 18 6 =(1, 2, 3)
a1 x + b1 y + c1 z =0 ∧ a2 x + b2 y + c2 z =0 ∧ a3 x + b3 y + c3 z =0,
(x,y,z )=(0, 0, 0)
D =0 D =0
x + y + z =1, 2x +4y 3z =9, 5x 4y + z =0
2x + y + z = 1, x + y +2z =3, 3x +2y +4z =1
x + y 4z =0, x y +3z =0, 5x y + z =0
2x y + z =7, 3x + y 5z =13, x + y + z =5
x 3y + z =1, 2x 5y +2z =3, 5x 9y +5z =10
2x y + z =7, 3x + y 5z =13, 5x 4z =20
x + y + z =6 2x y + z =3 x 2y +2z =3
3y + z =9 3y z =3
x + y + z =6
3y + z =9 3y z =3 ,
D =0
x + y + z =6, 5x y +2z =9, 3x +6y 5z =0
2x y + z =7, 2x + y 5z =13, 2x 6y +14z =21
2z =6, x + y + z =6 3y + z =9 2z =6, z y z z =3 z 3y +3=9 3y =6 y =2 y z z +2+3=6 x =1 (1, 2, 3)
2x + y z =7 5x 4y +7z =1 7x 3y +6z =8
2x + y z =7 13y 19z =33 13y 19z =33 y
2x + y z =7 13y 19z =33 0=0 y = 19z +33 13 y 2x + 19z +33 13 z =7 x = 29 3z 13 , z x = 29 3t 13 ,y = 19t +33 13 ,z = t t
x + y + z =3 2x 3y +2z =1 3x 2y +3z =7
x + y + z =3
5y =5 5y =2 , y 5y =5 5y =2
x + y + z =3 5y =5 0=3
x + y +3z =4 y +7z =5
4y +5z =11 y +16z =3;
x + y +3z =4, 2x + y z =3, 3x y +4z =1, x +2y 13z =1
x + y +3z =4 y +7z =5 23z =9 23z =8;
x +2y +3z +4u =5 2x +2y z +2u =6 3x +2y 5z = λ,
x +2y +3z +4u =5
2y +7z +6u =4 4y +14z +12u =15 λ;
x + y +3z =4 y +7z =5 23z =9 0=1
x +2y +3z +4u =5 2y +7z +6u =4 0=7 λ.
3x +2y + z =5
2x +3y + z =1
2x + y +3z =11
x +2y +3z u =4
2x 2y + z 4u = 7
x 3y + z +3u =5
x + y 2z 3u = 6
x + y 2z =0
x 3y +2z =0
4x y 3z =0
x +2y 3u +2v =1
x y 3z + u 3v =2
2x 3y +4z 5u +2v =7
9x 9y +6z 16u +2v =25
2x +6y +(a +6)z =0, x +7y +5z =0,ax +5y +13z =0 a
ax + y + z =1
x + ay +2z =2
2x + y + z =0
x + y 2z =3
3x +10y z =2
4x +3ay +3z =1
6x +13y 5z =2a +1.
ax + y z = a x + ay z =1 x y az = a.
2x +3y + z +2u =3
4x +6y +3z +4u =5
6x +9y +5z +6u =7
8x +12y +7z + au =9
# « AB =(2, 7, 1) B =(1, 1, 3) A #« a =(1, 1, 3) #« b =(2, 1, 5) #« c =( 3, 1, 1) | #« a + #« b + #« c |
x1 ,y
#« i #« j #« k #« i × #« i = #« o, #« i × #« j = #« k, #« i × #« k = #« j #« i #« j #« k
i #« j #« k
i #«
a =(x1 ,y1 ,z1 ) #« b =(x2 ,y2 ,z2 )
a × #« b =(x1 #« i + y1 #« j + z1 #« k ) × (x2 #« i + y2 #« j + z2 #« k ), #« a × #« b =(x1 #« i ) × (x2 #« i )+(x1 #« i ) × (y2 #« j )+(x1 #« i ) × (z2 #« k ) +(y1 #« j ) × (x2 #« i )+(y1 #« j ) × (y2 #« j )+(y1 #« j ) × (z2 #« k ) +(z1 #« k ) × (x2 #« i )+(z1 #« k ) × (y2 #« j )+(z1 #« k ) × (z2 #« k ) = x1 x2 #« i × #« i + x1 y2 #« i × #« j + x1 z2 #« i × #« k + y1 x2 #« j × #« i + y1 y2 #« j × #« j + y1 z2 #« j × #« k + z1 x2 #« k × #« i + z1 y2 #« k × #« j + z1 z2 #« k × #« k, #« a × #« b =(y1 z2 z1 y2 ) #« i (x1 z2 z1 x2 ) #« j +(x1 y2 y1 x2 ) #« k. #« a =(x1 ,y1 ,z1 ) #« b =(x2 ,y2 ,z2 ) #« a × #« b =(y1 z2 z1 y2 , (x1 z2 z1 x2 ),x1 y2 y1 x2 ) #« a =(x1 ,y1 ,z1 ) #« b =(x2 ,y2 ,z2 ) #« a × #« b = #« i #« j #« k x1 y1 z1 x2 y2 z2 #« a #« b #« a × #« b = #« b × #« a #« o | #« a | =0 | #« b | =0 x1 = kx2 y1 = ky2 z1 = kz2 #« a #« b #« a #« b P | #« a || #« b | ( #« a, #« b )
#« a #« b #« c #« a #« b #« c [ #« a, #« b, #« c ]=0 #« a #« b #« c [ #« a, #« b, #« c ] #« a #« b #« c #« a × #« b #« c #« a #« b [ #« a, #« b, #« c ]=( #« a × #« b ) #« c =0. ( #« a × #« b ) · #« c =0 #« c = #« o #« a × #« b = #« o #« a × #« b #« c #« a × #« b = #« o #« a #« b #« a #« b #« c #« a × #« b #« c #« a × #« b #« a #« b #« a #« b #« c #« a × #« b #« a #« b #« c #« a =(x1 ,y1 ,z1 ) #« b =(x2 ,y2 ,z2 ) #« c =(x3 ,y3 ,z3 ) ( #« a × #« b ) · #« c = #« i #« j
k
y1 z1 y2 z2
i x1 z1 x2 z2
j + x1 y1 x2 y2
k (x3 #« i + y3 #« j + z3 #« k )
y1 z1 y2 z2 x3 x1 z1 x2 z2 y3 + x1 y1 x2 y2 z3 = x3 y3 z3 x1 y1 z1 x2 y2 z2 = x1 y1 z1 x3 y3 z3 x2 y2 z2 = x1 y1 z1 x2 y2 z2 x3 y3 z3 , [ #« a, #« b, #« c ]= x1 y1 z1 x2 y2 z2 x3 y3 z3 .
a =(x1 ,y1 ,z1 ) #« b =(x2 ,y2 ,z2 ) #« c =(x3 ,y3 ,z3 ) x1 y1 z1 x2 y2 z2 x3 y3 z3 =0
b, #« c ]+[ #« a 2 , #« b, #« c ] ABCD A =(2, 4, 5) B =( 1, 3, 4) C =(5, 5, 1) D =(1, 2, 2) AH V ABCD 1 3 Sh BCD h h = |AH | S = 1 2 | # « BC × # « BD | ABCD W # « BC # « BD # « BA V = 1 6 W = 1 6 [ # « BC, # « BD, # « BA], # « BC =(6, 8, 5) # « BD =(2, 1, 2) # « BA =(3, 1, 1) [ # « BC, # « BD, # « BA]= 68 5 21 2 3 11 = 45, = 45 6 = 15 2 = 1 6
b =(1, 4, 4) #« c =(3, 5, 7)
=(1, 1, 1) B =(4, 4, 4) C =(3, 5, 5) D =(2, 4, 7)
(x,y, 0) (x,y )
b =(x2 ,y2 ) (x1 ,y1 ) × (x2 ,y2 )=(x1 ,y1 , 0) × (x2 ,y2 , 0)=
,y
,y
AB | = |y2 y1 |, A B y1 = y2 A B x |AB | = |x2 x1 | x1 = x2 y1 = y2 A x B y C =(x2 ,y1 ) |BC | = |y2 y1 | |AC | = |x2 x1 |
)
, A B
AB | = (x2 x1 )
+(y2 y1 )2 x1 = x2 |AB | = (y2 y1 )2 = |y2 y1 |, y1 = y2 |AB | = (x2 x1 )2 = |x2 x1 |
=(x1 ,y1 )
AOB )=
= 1 2 |OA|·|OB |( α2 α1 α1 α2 ) = 1 2 (|OA| α1 |OB | α2 −|OA| α1 |OB | α2 ) = 1 2 (a1 b2 a2 b1 ) A B
Π( AOB )= 1 2 (a2 b1 a1 b2 ) A1 A2
=(x2 ,y2 ) A3 =(x3 ,y3 )
Π( A1 A2 A3 )=Π( OA1 A2 )+Π( OA2 A3 ) Π( OA1 A3 ) = 1 2 (y2 x1 y1 x2 )+ 1 2 (y3 x2 y2 x3 ) 1 2 (y3 x1 y1 x3 ) Π( A1 A2 A3 )= 1 2 x1 (y2 y3 )+ 1 2 x2 (y3 y1 )+ 1 2 x3 (y1 y2 ) Π( A1 A2 A3 )= 1 2 x1 y1 1 x2 y2 1 x3 y3 1 A1 A2 A3 Π( A1 A2 A3 )=Π( OA3 A2 )+Π( OA2 A1 ) Π( OA3 A1 ), A1 A2 A3
Π( A1 A2 A3 )= 1 2 x1 (y2 y3 )+ x2 (y3 y1 )+ x3 (y1 y2 ) ,
Π( A1 A2 A3 )= 1 2 |D |, D = x1 y1 1 x2 y2 1 x3 y3 1 A1 A2 A3 A1 A2 A3 A1 A2 A3
A1 A2 A3 A1 A2 A3 A1 A2 A3 x1 (y2 y3 )+ x2 (y3 y1 )+ x3 (y1 y2 )=0 A1 A2 A3 A =(1, 2)
B =(4, 6) C =(3, 7)
Π( ABC )= 1 2 1 · (6 7)+4 · (7 2)+3 · (2 6) = 1 2 |− 1+20 12| = 7 2 ABCD
A =(1, 1) B =(3, 1) C =(2, 4) D =( 1, 3) Π ABC ACD Π( ABC )= 1 2 |1(1 4)+3(4+1)+2( 1 1)| =4, Π( ACD )= 1 2 |1(4 3)+2(3+1) 1( 1 4)| =7, Π=4+7=11
A =(x1 ,y1 ) B =(x2 ,y2 ) #« a = # « OA #« b = # « OB C =(x,y ) A B |AC | : |CB | = λ #« c = # « OC λ> 0
x2 x1
j
2 y1
x3 x1 y3 y1
=((x2 x1 )(y3 y1 ) (y2 y1 )(x3 x1 )) #« k =(x1 (y2 y3 )+ x2 (y3 y1 )+ x3 (y1 y2 )) #« k, Π= 1 2 x1 (y2 y3 )+ x2 (y3 y1 )+ x3 (y1 y2 ) , Π= 1 2 |D |, D = x1 y1 1 x2 y2 1 x3 y3 1 .
A =( 4, 2) B =(3, 5)
A =( 3, 4) B =(0, 0)
A =(a 1,b +2) B =(a 4,b +5)
A =(a t,a t) B =(a t,a t)
ABC A =(2, 7) B =(5, 7) C =(5, 11)
A =(3, 5) B =( 7, 4)
A =( 2, 4) B =(6, 8)
A =(2, 2) B =( 5, 1) C =(3, 5)
ABC A =(1, 1) B =(2, 5) C =( 6, 7)
ABCD A =(3, 0) C =( 4, 1) B D A =(x1 ,y1 ) B =(x2 ,y2 ) ABC C x1 = y2 =0 x2 = y1 = a M AB m : n
A =( 1, 4) B =(3, 1) m : n =1:2
A =( 1, 4) B =(3, 1) m : n = 1:2
A =( 1, 4) B =(3, 1) m : n =2:1
A =(a +5,b 6) B =(a +7,b +4) m : n = 2:5
ABC A =( 7, 3) B =(1, 5)
C =(9, 5) M =(3, 1) AB B A =(1, 1) A =(3, 1) B =(2, 1) M A B
A =(3, 5) B =(5, 3) C =( 1, 3) ABCD D B
A =( 3, 5) B =(1, 7) (1, 1)
M = 2, 1 2 N =(5, 1) P = 4, 5 2 AB BC CA ABC A =(2, 5) B =(1, 2) C =(4, 7) ABC B AC
AB A =(1, 3) B =(4, 3) C D C D
AB M =(2, 2) N =(1, 5) A B
ABC A =( 2, 0) B =(4, 8) C =(12, 2)
ABC A =( 2, 2) B =(4, 8) 5√2 A = (6, 3) B = (9, 6) AC D = ( 3, 0)
ABC
ABC C A = (3, 6) B = ( 1, 3) C = (2, 1)
ABC A = (4, 2) B = (3, 5) x
A B C
A =( 3, 0) B =(1, 2) C =(5, 4)
A =(3, 2) B =(4, 1) C =(6, 4) ABC
A =(3, 1) B =(1, 3) ABC x C t (2t +1,t) (2, 3+4t) ( 4t +4, 1) F (x,y ) x y F (x,y )=0 (x,y ) S = (x,y )|F (x,y )=0 , (x,y ) S ( ) ( ) ( ) ( ) M =(x,y ) ( ) M ∈ ( ) ⇔ F (x,y )=0 ( ) ( ) ( ) ( ) ( ) A =( 1, 1) B =(1, 1) M =(x,y ) ∈ ( ) |MA| = |MB |,
( ) F (x,y )=0 x
F (x,y )=0 ⇒ F (x, y )=0; y F (0,y )=0 ( ) ( 1 ) F1 (x,y )=0
F (x,y )=0 ∧ F1 (x,y )=0 A =(x1 ,y1 ) B =(x2 ,y2 )
M =(x,y )
xy 1 x1 y1 1 x2 y2 1
, (y1 y2 )x +(x2 x1 )y +(x1 y2 x2 y1 )=0 M =(x,y )
M A B A =(x1 ,y1 ) B =(x2 ,y2 ) a = y1 y2 ,b = x2 x1 ,c = x1 y2 x2 y1 ,
ax + by + c =0(a 2 + b2 > 0), a 2 + b2 > 0 a b a = b =0 y1 = y2 x1 = x2 A
A =(1, 2) B =(2, 1) AB xy 1 121 211 =0, x + y 3=0 A B x1 = x2 AB y (y1 y2 )x + x1 y2 x1 y1 =0, (y1 y2 )x (y1 y2 )x1 =0. x1 = x2 y1 = y2 A = B y x = x1 . x1 = x2 AB y x2 x1 y = y2 y1 x2 x1 x + x2 y1 x1 y2 x2 x1 , y = kx + n, k = y2 y1 x2 x1 ,n = x2 y1 x1 y2 x2 x1 .
C =(x2 ,y1 )
CAB = y2 y1 x2 x1 k AB x n y (0,n)
y = kx + n ∧ x =0 y (0,n)
x = p (p ∈ R) y y = kx + n (k,n ∈ R) y x k =0 y = n (n ∈ R)
a b c a b a b b =0 y = a c x c b k = a b ,n = c b y y = 2 3 x +4 3y =2x +12 2x 3y +6=0 x y =0 ∧ 2x 3y +6=0, 2x +6=0 x = 3 ( 3, 0) x y x =0 ∧ 2x 3y +6=0, 3y +6=0 y =2 (0, 2) y ax + by + c =0 a b y =0 x =0 x = c/a y = c/b ( c/a, 0) (0, c/b) x1 = x2 AB y y y1 = y2 y1 x2 x1 (x x1 )
(x1 ,y1 )
(x2 ,y2 )
x m + y n =1(m =0,n =0) x =0 y = n y =0
x = m
(0,n) (m, 0) m n x y
5x 3y + 15 = 0
5 · ( 3) = 15 x 3 + y 5 =1 x 3 y x m + y n =1, M =(m, 0) N =(0,n)
y A y1 = kx1 + n.
y y1 = k (x x1 ). (x1 ,y1 ) k A
x = x1 y
(x1 ,y1 )
( 3, 0)
x 2π /3 x√3+ y +3√3=0 ( 3, 0) y = k (x +3)
2π 3 = k = √3 y = √3(x +3)
#« n =(a,b)= a #« i + b #« j p P =(x,y ) #« r = # « OP =(x,y )= x #« i + y #« j
n #« r + c =0, ax + by + c =0 #« n #« r = ax + by A B P # « AB # « AP # « AB × # « AP = #« o
=(x1 ,y1 ) B =(x2 ,y2 ) P =(x,y )
x +2y 8=0 4 1 3x 2y +1=0 1 1/2 A =(1, 2) B =(2, 3)
2x + y 2=0
ax + by + c =0 ax by + c =0(b =0) x ax + by + c =0
+ by c =0 2x +3y =6 2x +3y 9=0 y =2x 4 m mx +4y 8=0 x 3x +2my 6=0 3x + my =12 m n (m 3n 2)x +(2m +4n 1)y =3m n +2 x y 2
A =(3, 2) ϕ = π 4 A =( 4, 1) ϕ = 3π 4 (2, 3) x y 4 (15, 8) (5, 0) (0, 2) (15, 4) ( 5, 4)
A =(3, 4) B =(5, 1) x 2y +1=0 (4, 3) 12x 5y +26=0 x +
⎟ ⎟ ⎠ ;
2 3 y +3= 2 3 (x 4)
2x 3y 17=0 P =(2, 3) Q =( 1, 0) Q PQ PQ 0 3 1 2 =1 1 y = 1(x +1) x + y +1=0 x y 1=0 x +2y 4=0 x y +3=0 2x + y =0 x 2y 2=0 y =2 x 4 + y 2 x =2 x +5y 7=0 3x 2y 4=0 7x + y +19=0 2x 3y 1=0 4x 5y +5=0 3x y +2=0 m mx +(2m +3)y +(m +6)=0 (2m +1)x +(m 1)y +(m 2)=0 y y =3x +2 2y = x +2 3x + y 2=0, 2x y +5=0 x√3+ y √2 2=0 x√6 3y +3=0 (3+ √2)x +(√6 √3)y +7=0 x√2 y √3 5=0 x 3=0 x y +3=0 x√3 y 4=0 y 2=0 x +3=0 y 2=0 3x +4y =1 x 7y =17 7x + y +31=0 ax + by =1 3x 4y =7 (5, 2) a b m 2x 3y +4=0 5x 4my +6=0 π /4 3x + y +4=0 3x 5y +34=0 3x 2y +1=0 2x 3y +5=0 3x +2y =7 (2, 3) 7x y +8=0 ( 4, 5) T S =(x0 ,y0 ) ax + by + c =0
21x 20y =29
(5, 3) 9x +40y =123
(1, 2) ( 1, 1) ( 2, 3)
( 2, 5)
M =(4, 3) N =(5, 6)
( 2, 2) (4, 2) (3, 0) A =( 3, 4) B =(4, 2) C =( 1, 3) A1 B1
, 5)
( 7, 3) (11, 15) M (8, 9)
(3, 4) ( 1, 2)
( 3, 1) (2, 2) (3, 0) S =(x0 ,y0 )
S =(x0 ,y0 ) x = x0 y y0 = k (x x0 )
+ by + c
=(x0 ,y0 ) a1 x0 + b1 y0 + c1 =0 a2 x0 + b2 y0 + c2 =0
y =1 4x y = 3
x +2y +6=0 6x 2y =1 4x y +3=0
x 2y 1+ λ(4x y +3)=0, (6+4λ)x (2+ λ)y +(3λ 1)=0
x + 2y + 6 = 0
λ 2+
=
= 22/13 (6x 2y 1) 22 13 (4x y +3)=0, 10x +4y +79=0 3x +4y =4 9x +5y +4=0 x 4y +3=0 7x 2y 1=0 x y 2x 3y +6=0 5x +4y 1=0 ( 4, 5) 2x y +4=0 2x y +4=0 π /4 x +2y 1=0 5x +4y 17=0 x 4y +11=0 α(2x +5y +4)+ β (3x 2y +25)=0
α(2x 3y +20)+ β (3x +5y 27)=0 x +7y 16=0
(t +4)x +(3t 1)y +(t 2)=0 t ( 1 ) ( 2 ) P
( 1 ) P ( 2 ) S ( 2 ) d ( 1 ) ( 2 ) |PS |
( 1 ) ( 2 ) x ϕ1 + y ϕ1 p1 =0,x
2 + y ϕ2 p2 =0, p1 p2 ( 1 ) ( 2 )
( 1 ) ( 2 )
d = p1 + p2 ( 1 ) ( 2 )
d = |p1 p2 | 3x 4y =10 6x 8y +5=0 3 5 x 4 5 y 2=0, 3 5 x + 4 5 y 1 2 =0 d 2+ 1 2 = 5 2 M ( 1 ) M
( 2 ) ( 1 ) M ( 1 ) ( 1 ) ( 2 )
M =(x0 ,y0 ) ( 1 )
x ϕ + y ϕ p =0 ( 2 ) ( 1 )
x ϕ + y ϕ p =0
( 1 ) ( 2 ) 2 ) d M ( 1
d = |p p| M x0 ϕ + y0 ϕ p =0
p = x0 ϕ + y0 ϕ.
d = |x0 ϕ + y0 ϕ p|.
( 1 ) ( 2 ) ( 2 )
x ϕ y ϕ ( p )=0, ( 1 ) ( 2 ) d M =(x0 ,y0 ) ( 1 )
d = p + p.
M x0 ϕ y0 ϕ = p
d = x0 ϕ y0 ϕ + p. d> 0 d M x ϕ + y ϕ p =0
d = |x0 ϕ + y0 ϕ p| M
ax + by + c =0,
ax + by + c
( c)√a2 + b2 =0,
d M =(x0 ,y0 )
d = x0 + by0 + c √a2 + b2 .
x +2y +3=0 √5
x 3y +13=0
M =(x0 ,y0 ) M x 3y +13=0
x0 3y0 +13=0
x0 +2y0 +3 √5 = √5, x0 +2y0 +3= ±5. x0 3y0 +13=0 ∧ x0 +2y0 +3=5, x0 = 4 y0 =3 x0 3y0 +13=0 ∧ x0 +2y0 +3= 5, x0 = 10 y0 =1
x 3y +13=0 √5 x +2y +3=0 ( 4, 3) ( 10, 1) 5x 12y +26=0 5x 12y 13=0 2x +3y =6 4x +6y +17=0
5x 2y =1 5x 2y =9 5x 2y +7=0
M ( )
M =(1, 2) ( ): x y 4=0
M =( 3, 4) ( ):4x =3y 12y =5x +46 (1, 1)
A =(1, 3) B =(2, 2)
C =(3, 5) x + y 8=0 2x y 4=0 3x y =0 (7, 1)
A =(2, 5) x 2y 7=0
(2, 5) (5, 1)
x 3y +5=0 3x y 2=0 x y =0 x + y =0 3x 4y =0 x
3x +4y =36 5x +12y =20 24x +7y =21
(3, 4) (0, 8) (0, 0)
ABC
AB y =2x AC 2x + y +8=0 C
2x 7y +24=0
ABC AB 5x 3y +2=0 ha 4x 3y +1=0 hb 7x +2y =22
( 1, 1) ( 3, 5) (7, 11) H T S |HS | =3|TS | x 2x 3y +6=0 ( 4, 1) (0, 1) x + y =8 (2, 8) x 3y +2=0 2x y +5=0 2x y +10=0 √10 ( 1, 1) x +2y 6=0 ϕ ϕ =1/2
(5, 2) (1, 4)
(3, 0) y = x +1
3x 4y 8=0 5x +12y 48=0 3:2
4x +3y +10=0
= (p,q ) r> 0 (K ) C r M =(x,y ) (K ) |CM | = r (x p)2 +(y q )2 = r, (x p)2 +(y q )2 = r 2 . x y M =(x,y )
|CM | = (x p)2 +(y q )2 = r, M (K ) (p,q ) r> 0 p = q =0 (0, 0) x 2 + y 2 = r 2 p =0 q =0 y p =0 q =0 x |p| = r y |q | = r x |p| = |q | = r AB A =(3, 7)
B =(1, 5) p = 3+1 2 =2 q = 7+5 2 =6 r A r = (7 6)2 +(3 2)2 = √2 (x 2)2 +(y 6)2 =2 P =(x0 ,y0 ) (x0 p)2 + (y0 q )2 = r 2 P
, 1)
,y
,y
BC S S A A =(7, 1) B =(5, 5) C =( 2, 4) A B C (7 p)2 +(1 q )2 = r 2 ∧ (5 p)2 +(5 q )
(p,q )=(2, 1) p q p q r r 2 =25 (x 2)2 +(y 1)2 =25 AB BC S |SA| x 2 + y 2 = 49/4 x 2 + y 2 = c (c > 0) C = ( 4, 6) r =
x + y = 1 P P
AB A =(5, 1)
B =( 3, 7) x (6, 0)
(9, 9) y x x y r =3 x +2y =7 3x y =0
(5, 0) P x 2 + y 2 6x 8y =75
P =(1, 7) P =(9, 12) P =( 7, 8)
x 2 + y 2 2x +4y 20=0
x 2 + y 2 2x +4y +14=0
x 2 + y 2 +4x 2y +5=0 x 2 + y 2 +6x 4y +14=0
x 2 + y 2 + x =0 3x 2 +3y 2 +3x +3y 1=0
400x 2 +400y 2 320x 600y +189=0 (10, 4)
(17, 3)
(11, 2) (7, 2) y =3x 19
(1, 1) (4, 2) (2, 4) ( 4, 2) ( 4, 6) (2, 2)
D =( 1, 4)
ABCD A =(6, 5) B =(8, 1) C =(0, 3)
ABC A(7, 1) B =(5, 5) C =( 2, 4) A B C H (3, 4) (4, 5) x 2 + y 2 =50
( 4, 1) (3, 0) y
r
(1+ k 2 ) (kp q + n)
)
2 (1+ k 2 ) (kp q + n)2 > 0
2 (1+ k 2 ) (kp q + n)2 < 0 r 2 (1+ k 2 ) (kp q + n)2 =0
2 (1+ k 2 ) (kp q + n)2 =0
x 2 + y 2 = r 2
p = q =0
r 2 (1+ k 2 ) n 2 =0 y x = m, x = m ∧ (x p)2 +(y q )2 = r 2 .
r 2 (m p)2 > 0
r 2 (m p)2 < 0
r 2 (m p)2 =0 k y = kx +10
x 2 + y 2 =20 k 20(1+ k 2 ) 100=0,k 2 =4, k = ±2 y = 2x +10 y =2x +10 a 3x 4y + a =0 x 2 + y 2 10y =0
M =(x1 ,y1 ) CM C =
(p,q ) K = y1 q x1 p CM k = 1 K = x1 p y1 q . M y1 = kx1 + n
n = y1 kx1
n = y1 + x1 p y1 q x1
(x1 ,y1 ) y = x1 p y1 q x + y1 + x1 p y1 q x1 , (x1 p)(x x1 )+(y1 q )(y y1 )=0.
(x p + p x1 )(x1 p)+(y q + q y1 )(y1 q )=0,
(x p)(x1 p)+(y q )(y1 q ) (x1 p)2 (y1 q )2 =0 (x p)(x1 p)+(y q )(y1 q )= r 2 (x1 ,y1 ) (x1 p)2 +(y1 q )2 = r 2 (x1 ,y1 ) x 2 + y 2 = r 2
(x1 ,y1 ) xx1 + yy1 = r 2
y 2
B
=4
10y +29=0, y 2 10y +21=0, y =3 y =7 A = (4, 3) B =(4, 7)
4x +3y 3(x +4) 5(y +3)+29=0 x 2y +2=0, B 4x +7y 3(x +4) 5(y +7)+29=0, x +2y 18=0 (x1 ,y1 ) y k n k n
(x1 ,y1 )
x = x1
(k1 ,n1 ) (k2 ,n2 )
y = k1 x + n1 , y = k2 x + n2 (x1 ,y1 ) y (x1 ,y1 )
(1, 3) x 2 + y 2 =5
5(1+ k 2 )= n 2 ∧ 3= k + n, n =3 k
2k 2 +3k 2=0, k = 2 1 2 n 5 2 (1, 3)
y +2x 5=0 2y = x +5
x 2 + y 2 = r 2 P =(x1 ,y1 )
A =(a1 ,b1 ) B =(a2 ,b2 ) A B
xa1 + yb1 = r 2 ,xa2 + yb2 = r 2 P =(x1 ,y1 ) x1 a1 + y1 b1 = r 2 ,x1 a2 + y1 b2 = r 2 . AB xx1 + yy1 = r 2 A B A B (x p)2 +(y q )2 = r 2
P =(x1 ,y1 ) P (x p)(x1 p)+(y q )(y1 q )= r 2 .
B
B
=(x1 ,y1 )
x 2 + y 2 =5 P =( 1, 3)
x ( 1)+ y 3=5 3y = x +5 M N x 2 + y 2 =5 ∧ 3y = x +5 M =(1, 2) N =( 2, 1) M x +2y =5 N 2x + y =5 ( ) (K ) ( ) (K ) ( ) (K ) 3x y =1 x 2 + y 2 +4x 6y 7=0 3x y =1 ∧ x 2 + y 2 +4x 6y 7=0
(0, 1) (2, 5) (0, 1)
x 0+ y ( 1)+2(x +0) 3(y 1) 7=0, x 2y 2=0 1/2 ϕ ϕ = 3 1 2 1+3 1 2 =1, ϕ =45◦ x 2 + y 2 =5 3x + y 5=0 3x + y 5√2=0 3x + y =9
x 2 + y 2 =10
2x + y =2 (1, 3)
k y = kx x 2 + y 2 10x +16=0
√10 (4, 3) x 3y 15=0 3x +4y 35=0 3x 4y 35=0 x =1
x 2 + y 2 +6x +4y +3=0 y +3x =9
x 2 + y 2 +2x +4y +1=0 3x 4y =0 5x +12y +4=0
3x 4y 5=0
√5 x 2y = 1 (5, 6) (1, 0) 2x + y +2=0 2x + y 18=0 5x 7y =8
2x y =0 x 2y 6=0
ABC x +2y =5 x =3y y =2x ABC x 2 +(y 2)2 =5 M =(2, 3) x 2 + y 2 +6x +4y =0 M =(0, 0) ( 1, 5) x 2 + y 2 =13 ( 2, 5) (x 5)2 +(y 6)2 =25 x 2 + y 2 2x 2y 8=0 (4, 2) ( 2, 2) (2, 2) x 2 + y 2 =25 ( 4, 3) (3, 4) x 2 + y 2 2x 4=0 x 3y 6=0 x 2 + y 2 +8x +6y 15=0 (1, 2) (2, 3) (x 1)2 +(y +5)2 =4
x 2 + y 2 + x +2y 5=0 M
x 2 + y 2 =1
(2, 4)
( ) y =2x 4
B M ( ) AB P P
( ) x 2 + y 2 =1 P
x 2 + y 2 = r 2 A B PAB 1 2 r 2 P x 2 + y 2 =2r 2 P x 2 + y 2 = r 2 A B |AB | =2b b P (r 2 b2 )(x 2 + y 2 )= r 4
x 2 + y 2 = r 2
45◦ P x 2 + y 2 =2r 2 (2+ √2) x 2 + y 2 =2r 2 (2 √2)
y +3=0 x 2 + y 2 =10
7x 17y +169=0 x 2 + y 2 =169
(14, 2) x 2 + y 2 2x 4y 8=0
45◦ (K1 ) (K2 )
r1 r2 (K1 ) (K2 )
(K1 ) (K2 )
|C1 C2 | >r1 + r2 |C1 C2 | < |r1 r2 | (K1 ) (K2 )
|C1 C2 | = r1 + r2
|C1 C2 | = |r1 r2 | x 2 + y 2 =5 x 2 + y 2 8x +8y + 30=0 (0, 0) (4, 4) r1 = √5 r2 = √2 |C1 C2 | √16+16=4√2 4√2 > √5+ √2 |C1 C2 |2 = r 2 1 + r 2 2 , C1 C2 r1 r2 y ( 2, 3) (x +7)2 +(y 4)2 =10 x 2 +(y q )2 = r 2 ( 2, 3) 4+( 3 q )2 = r 2 r 2 q 2 6q 13=0.
72 +(4 q )2 =10+ r 2 r 2 q 2 +8q 55=0 q r q =3 r = √40 x 2 +(y 3)2 =40 (x p)2 +(y q )2 = r 2 , (x P )2 +(y Q)2 = R2 , y = kx + n, k n (x +5)2 + y 2 =
10 (x 5)2 + y 2 =40 10(1+ k 2 )=( 5k + n)2 ∧ 40(1+ k 2 )=(5k + n)2 5k + n 5k + n 2 =4, 5k + n 5k + n = ±2, k = 1 15 n k = 3 5 n 15k = n ∧ 10(1+ k 2 )=( 5k + n)2 5k =3n ∧ 10(1+ k 2 )=( 5k + n)2 , (k,n) (3, 5) ( 3, 5) 1 3 , 5 1 3 , 5 y =3x +5 y = 3x 5 3y = x +15 3y = x 15 x 2 + y 2 +10x +4y +19=0 x 2 + y 2 8x 2y 23=0 (1, 3) x x 2 + y 2 +6x +2y 22=0 (4, 1) (10, 1) x 2 + y 2 =5
x 2 + y 2 26x +2y +145=0 x 2+y 2 16x 20y+115=0 x 2+y 2+8x 10y+5=0 √17
x 2 + y 2 +2x 2y 32=0 (2, 6) 45◦ x 2 + y 2 + ax + by + c =0
x 2 + y 2 + Ax + By + C =0
x 2 + y 2 +2x 3y +1=0 x 2 + y 2 +3x 2y +2=0 x 2 + y 2 + x 5y 1=0 x 2 + y 2 =29 (x +2)2 +(y 5)2 =58 (x 3)2 +(y 1)2 =8 (x 2)2 +(y +2)2 =2 x 2 + y 2 2mx 2ny m 2 + n 2 =0 x 2 + y 2 2nx +2my + m 2 n 2 =0 m n (2, 2) x 2 + y 2 =4 (x 3)2 + y 2 =1 ( 4, 3) ( 2, 3) (x 3)2 + y 2 =16 x 2 + y 2 =45 x 2 + y 2 20x 25=0 (1, 1) (6, 6)
A =( 1, 1) B =(5, 3) ( 1, 4) (4, 3) (5, 1) 5x +12y 4=0 12x 5y +10=0 (x 1)2 +(y +3)2 =25 x 2 + y 2 = r 2 (r, 0) x 2 + y 2 =8 (4, 6) (x +3)2 +(y 2)2 =25 3y =4x +18 (2, 2) x + y 10+ λ(x y )=0 x 2 + y 2 =5
N M ( ) M =(x,y ) (P ) FM NM |FM | = x p 2 2 + y 2 ; |NM | = x + p 2 |FM | = |NM | x p 2 2 + y 2 = x + p 2 , x 2 px + p2 4 + y 2 = x 2 + px + p2 4 , x y M =(x,y ) ( ) x = p 2 F = p 2 , 0 M (P ) |FM | = x p 2 2 + y 2 ; |NM | = x + p 2 . y 2 =2px |FM | = x p 2 2 +2px = x + p 2 2 = x + p 2 = |NM | M F M ( ) M (P ) (P )
(P )
y 2 =2px
y 2 0 p> 0 x 0
y 2 =2px (0, 0) y 2 =2px x > 0 y x y p y 2 = 2px (p> 0) x 2 =2py
(p> 0) x 2 = 2py (p> 0)
y = 1+ D 4a y = 1 4 x 2 + x +1. y 2 =10x 2p =10 p =5 F 5 2 , 0 x = 5 2 (P ) y p = 1 2 x 2 =2py p = 1 2 x 2 = y 0, 1 4 y = 1 4 x ( 1, 3) ( 1, 3) x y 2 = 2px ( 1, 3) 9=2p p = 9 2 9 4 , 0 x = 9 4 y 2 = 9x y 2 =4x 4y 2 =5x y 2 = 8x x 2 =10y x 2 = 12y y =2√x y = √ x y = √ 6x x = 2y
M =(x,y ) (E ) |FM |2 =(x ae)2 + y 2 , |MN |2 = a e x 2 , |FM |2 = e 2 |MN |2 (x ae)2 + y 2 = e 2 a e x 2 =(a ex)2 ,
, M =(x,y ) (E ) (E ) M1 =(x1 ,y1 ) (E ) M2 =( x1 ,y1 ) M3 =( x1 , y1 ) M4 =(x1 , y1 ) (E ) M1 ∈ (E )
+ ( y1 )
2 =1, M2 ,M3 ,M4 ∈ (E ) A =(a, 0) A =( a, 0) B =(0,b) B =(0, b) x = a x = a y = b y = b A A (E ) B B (E ) a = 1 2 |A A| b = a 1 e2 = 1 2 |BB | F =( ae, 0) ( ) x = a e F ( ) F F M =(x,y ) (E ) N M ( ) N M ( ) |FM | = e|MN |, |F M | = e|MN |, |FM | + |F M | = e(|MN | + |MN |) |MN | + |MN | ( ) ( ) 2 · a e |FM | + |F M | = e 2 a e =2a, M (E ) |F F | F F
e < 1 F =
( ae, 0) F =(ae, 0) ( ) x = a e ( ) x = a e b2 = a 2 (1 e 2 ) a 2 e 2 = a 2 b2 e = 1 a a2 b2 F =( a2 b2 , 0) F =
( a2 b2 , 0)
( ) x = a2 √a2 + b2 ( ) x = a2 √a2 + b2 a>b> 0
a>a 1 e2 e< 1
(E ) F =( a2 b2 , 0)
F =( a2 b2 , 0) x b>a> 0
F =(0, b2 a2 ) F =(0, b2 a2 ) y e 0 <e< 1 e
e =0 ⇒ a = b x 2 + y 2 = a 2 e e =1 16x 2 +25y 2 =400 x2 25 + y 2 16 =1, a =5 b =4
e = 1 a a2 b2 = 1 5 52 42 = 3 5 F =( 3, 0)
F =(3, 0)
9x 2 +25y 2 =225 x2 25 + y 2 9 =1 a =5 b =3 x = 25 4 x = 25 4 (E ) ( √5, 2) (E ) a2 b2 = 4 e = 4 5
= a 1 t2
,b
=(a t b t x2 a2 + y 2 b2 =1 t F ( ) (H ) M N M ( ) M (H ) |FM | = e|MN | (e> 1). K F ( ) A A |A F | = e|A K |, |FA| = e|AK | (H ) O A A |A A| =2a O Ox OK y OK |A O | = |OA| = a |A F | = |A O | + |OF | |A K | = |A O | + |OK | |A O | + |OF | = e(|A O | + |OK |), a + |OF | = e(|OK | + a). a −|OF | = e(|OK |− a) ∧ |OF | |OK | |OF | = ae |OK | =
F =(ae, 0) ( ) x =
M =(x,y ) (H ) x2 a2 y 2 b2 =1(b2 = a 2 (e 2 1)). M =(x,y ) (H ) |FM |2 =(x ae)2 + y 2 , |MN |2 = a e x 2 ,
b2 = a 2 (e 2 1) a 2 e 2 = a 2 + b2 e = 1 a a2 + b2 F =( a2 + b2 , 0) F =( a2 + b2 , 0) x = a2
2 + b2 16x 2 25y 2 =400 x2 25 y 2 16 =1, a =5 b =4 a 2 + b2 =25+16=41 F = ( √41, 0) F =(√41, 0) ( 5, 0) (5, 0) x2 25 y 2 16 =0, x 5 y 4 x 5 + y 4 =0, 5y =4x 5y +4x =0. y = ± 3 4 x 64 5 b a = 3 4 ∧
Q P =(2, 1) Q =(10, 7) P =
(1, 3)
√3, 0)
3y ± 4x =0 4y ± 3x =0 x = ± 32 5 ( 5, 3)
2 (2, 3) x = ± 1 2
x2 100 + y 2 64 =1
y = kx + n
y 2 =2px (p> 0). y = kx + n ∧ y 2 =2px. y (kx + n)2 =2px k 2 x 2 +2(kn p)x + n 2 =0, 4(kn p)2 4k 2 n 2 4p(p 2kn) p> 0
(x1 ,y1 )
p 2kn> 0 x1 x2 M1 M2
p 2kn< 0
p 2kn =0 x1 M M
k =0 k =0 y = n
M =(x1 ,y1 ) M x1 = 2(kn p) 2k 2 = 1 k 2 (p kn)= 1 k 2 (2kn kn)= n k M
y1 = kx1 + n = k n k + n =2n. n = 1 2 y1 k = n x1 = y1 /2x1
y = y1 2x1 x + y1 2
y1 y 2 1 =2px1 (x1 ,y1 ) yy1 = 2px1 2x1 x + 2px1 2 , yy1 = p(x + x1 )
(0, 0) x =0 y y = kx + n y 2 = 2px
(p> 0) x 2 =2py (p =0) x 2 =2py (x1 ,y1 ) xx1 = p(y + y1 )
P =(x1 ,y1 )
y 2 = 4x ( 2, 1)
y = kx + n y = kx + n ∧ y 2 = 4x, x1,2 = nk 2 ± √4nk +4 k 2 , y1,2 = 2 ± √4nk +4 k ( 2, 1) 2= 1 2 (x1 + x2 ), 1= 1 2 (y1 + y2 ), 2= 2nk 4 2k 2 , 1= 4 2k , k =2 n =3 y =2x +3
(2, 5) y 2 =8x 4=2kn ∧ 5=2k + n. y = kx + n y 2 =8x (2, 5) y = kx + n (k1 ,n1 )=(2, 1) (k2 ,n2 )= 1 2 , 4 (2, 5) y 2 =8x y =2x +1 y = 1 2 x +4 y =2x +1 ∧ y 2 =8x, y = 1 2 x +4 ∧ y 2 =8x. 1 2 , 2 (8, 8) y = 4 5 x + 8 5 M1 =(a1 ,b1 ) M2 =(a2 ,b2 ) (2, 5) y 2 =8x b1 y =4(x + a1 ),b2 y =4(x + a2 ) (2, 5)
5b1 =4(2+ a1 ), 5b2 =4(2+ a2 ) M1 M2 5y =4(2+ x), 5y =4x +8,
M1 M2
5y =4x +8 ∧ y 2 =8x (2, 5)
y 2 =8x 1 2 , 2 (8, 8) y =2x +1
y = 1 2 x +4
2x y 6=0 x 2 =2py 2x y 6=0 ∧ x 2 =2py x 2 =2p(2x 6) x 2 4px +12p =0
16p 2 =48p p =3 p =3 x 2 12x +36=0
(6, 6) x2 a2 + y 2 b2 =1, y = kx + n ∧ b2 x 2 + a 2 y 2 = a 2 b2 . y b2 x 2 + a 2 (kx + n)2 = a 2 b2 (a 2 k 2 + b2 )x 2 +2kna2 x + a 2 n 2 a 2 b2 =0, 4k 2 n 2 a 4 4(a 2 k 2 + b2 )(a 2 n 2 a 2 b2 ) 4a 2 b2 (a 2 k 2 + b2 n 2 ) 4a 2 b2 > 0 a 2 k 2 +b2 n 2 > 0 y = kx+n a 2 k 2 + b2 n 2 < 0 a 2 k 2 + b2 n 2 =0 M (x1 ,y1 ) x1 x1 = kna2 a2 k 2 + b2 = kna2 n2 = ka2 n y1 y = kx + n y1 = kx1 + n = k 2 a2 n + n = a2 k 2 + n2 n = b2 n
n = b2 y1 k = nx1 a2 = b2 x1 a2 y1 (x1 ,y1 ) y = b2 x1 a2 y1 x + b2 y1 , b2 xx1 + a 2 yy1 = a 2 b2 , xx1 a2 + yy1 b2 =1. x 2 +4y 2 =20 (4, 1)
4x +4( 1)y =20 x y 5=0 (7, 2)
9x 2 +16y 2 =144 y = kx + n 16k 2 +9= n 2 (7, 2) 2=7k + n 16k 2 +9= n 2 ∧−2=7k + n (k1 ,n1 )=( 1, 5) (k2 ,n2 )= 5 33 , 101 33 y = x +5 33y =5x 101
9x 2 +16y 2 =144 ∧ y = x +5 9x 2 +16y 2 =144 ∧ 33y =5x 101 16 5 , 9 5 80 101 , 297 101 (7, 2)
9 7 x + 16( 2)y =144 63x 32y =144 x y 63x 32y =144 (7, 2) (7, 2) 63x 32y =144 ∧ 9x 2 +16y 2 =144 16 5 , 9 5 , 80 101 , 297 101 , 9 · 16 5 · x +16 · 9 5 · y =144, x + y =5,
· y =144, 5x 33y =101 x + y 2=0 x 2 +3y 2 =12 x 2 +3y 2 =12 ∧ x + y 2=0 (0, 2) (3, 1) 6y =12 3x 3y =12 y =2 x y =4 k1 =0 k2 = k = 1
◦
◦ x 2 +3y 2 =12
+3y +16=0
24x 2 +40y 2 =960 x 2 +8y 2 =8 8x 2 + y 2 =8 y = kx + n 8k 2 +1= n 2 ∧ k 2 +8= n 2 , (k1 ,n1 )=(1, 3) (k2 ,n2 )=(1, 3) (k3 ,n3 )= ( 1, 3) (k4 ,n4 )=( 1, 3) y = x +3 y = x 3 y = x +3 y = x 3 x2 a2 y 2 b2 =1
y ∧
(b2 a 2 k 2 )x 2 2a 2 knx (a 2 n 2 + a 2 b2 )=0.
n 2 + b2 a 2 k 2 > 0
n 2 + b2 a 2 k 2 < 0
n 2 + b2 a 2 k 2 =0
(x1 ,y1 )
b2 xx1 a 2 yy1 = a 2 b2 , xx1 a2 yy1 b2 =1.
b2 a 2 k 2 =0
b2 a 2 k 2 =0 k = ± b a kn =0
y = ± b a x + n y = ± b a x x 2
b2 + a 2 k 2
25x 2 9y 2 =225 5, 20 3 b2 xx1 a 2 yy1 = a 2 b2
25 · 5 · x 9 20 3 y =225 25x +12y 45=0
(1, 10) x2 8 y 2 32 =1
(1, 10)
8 + y · 10
=1 2x +5y 16=0 2x +5y 16=0 ∧ 4x 2 y 2 =32
3 , 14 3 (3, 2) 4 11 3 x 14 3 y =32 4 · 3 · x 2 · y =32 22x +7y +48=0 6x y 16=0 8x 2 16y 2 = 128 2x +4y 5=0
2 =16 b2 =
a b
+2y =4 x +2y +4=0
+2y 4
5 =0, x +2y +4
5 =0
· 4 √5 = 8 √5 2y ± x =
5x 6y 8=0 a b b a = 1 2 ∧ a 2 25 36 b2 = 16 9 , a =2 b =1 x 2 4y 2 =4 k y = kx +2 y 2 =4x 2x + y 12=0 3x y 3=0 y 2 =2px (1, 2) (1, 2) (9, 6) y 2 =4x M (P ) M =(5, 7) (P ) y 2 =8x M =( 3, 4) (P ) x 2 =4y y 2 =4x 2x + y 4=0 y 2 =4x 2x + y =12
y 2 =16x ( 4, 2) y 2 =8x 2x +2y =3 2x +2y =3 x 2y +6=0 y 2 =2px y 2 =12x x 3y =4 45◦ n y = x + n 5x 2 +20y 2 =100 (E ) M
M =( 2, 7) (E ) x 2 +4y 2 =100 M =(0, 4) (E ) 3x 2 +16y 2 =192 x 2 +2y 2 =162 (6, 15) 3x 2 +4y 2 =48 x 2y 12=0 x 2 +4y 2 =16 3x 2y +18=0 x +4y 25=0 4x +9y 75=0 x +4y =10 (2,q ) 4x 2 +5y 2 =20 5x 2 +4y 2 =20 4x 2 +9y 2 =36 x 2 + y 2 =5 x +2y =7 x 2 +4y 2 =25 3x +10y =25 4x 2 +25y 2 =100 3x + y 9=0 3x 2 y 2 =3 M
4x 2 5y 2 =20 M =(5, 4) x 2 y 2 =8 M =(3, 1) ax 3y =24 a x 2 y 2 =36 M
3x 2 4y 2 =12 M =(4,q ) q> 0 x 2 4y 2 =36 M =(10, 4) x 2 y 2 =16 ( 1, 7) (1, 5) 5x 2 3y 2 =1
Ax2 + Bxy + Cy 2 + Dx + Ey + F =0(A,B,C,D,E,F ∈ R). ax + by + c =0(a 2 + b2 > 0), x 2 + y 2 + ax + by + c =0(a 2 + b2 4c> 0) x2 a2 + y 2 b2 =1
F1 =( c, 0) F2 =(c, 0)
F1 =(1, 1) F2 =(3, 3) 2a (E ) M |F1 M | + |F2 M | =2a a =2
1 F2 A = B = C =0 D 2 + E 2 > 0 A = C =0 B =0 D 2 + E 2 4AF> 0
y 2 x 2y 1=0 (x,y ) y 2 2y +1 (x +2)=0, (y 1)2 = x +2 X Y X = x +2,Y = y 1 x = X 2 y = Y +1) Y 2 = X,
(x,y ) (X,Y ) ( 2, 1) (0, 0)
(a,b)
P (x,y )
(X,Y )
2 + Bxy + Cy 2 + Dx + Ey + F =0
2 + BXY + CY 2 + F1 =0
(X + a)2 + B (X + a)(Y + b)+ C (Y + b)2 + D (X + a)+ E (Y + b)+ F =0,
2 + BXY + CY 2 +(2Aa + Bb + D )X +(Ba +2Cb + E )Y +(Aa2 + Bab + Cb2 + Da + Eb + F )=0
Aa + Bb + D =0 ∧ Ba +2Cb + E =0 = B 2 4AC =0
x 2 +2y 2 2x +8y +8=0 O =(1, 2)
3x 2 +4y 2 24x 16y +59=0 O =(4, 2)
y 2 =2x O =( 5, 1)
3x +4y =0 O =(2, 3)
x 3 3x 2 + y 2 +3x +2y 2=0 O =(1, 1)
4x 2 +12xy +8y 2 4x 6y 3=0
x 2 +4xy + y 2 6x 3=0
4x 2 +8xy +2y 2 +20x +12y 15=0
x 2 + xy 3x y +1=0
ϕ = 3 4
4x + y 1=0
52x 2 72xy +73y 2 =225
9x 2 24xy +16y 2 40x 30y 25=0 60x 2 +119xy 60y 2 230x 597y 414=0 (3, 2)
12x 2 +12xy 4y 2 =1
6x 2 4xy +3y 2 =2 2xy =1
xy
ax 2 + bxy + cy 2 + dx + ey + f =0, a b c d e f a b c d e f a = b = c = d = e = f =0 0=0 (x,y )
a = c = f =1 b = d = e =0 x 2 + y 2 +1=0
a = c =1 b = d = e = f =0 x 2 + y 2 =0 (0, 0)
a = c =1 b = 2 d = e = f =0 x 2 2xy + y 2 =
0 (x y )2 =0 y = x a = b = c = f =0
d = e =0
a =1 c = 1 b = d = e = f =0 x 2 y 2 =0
(x y )(x + y )=0 y = x y = x
a = c =1 b = d = e =0 f = 1 x 2 + y 2 1=0
a =4 c =9 b = d = e =0 f = 36 4x 2 +9y 2 36=0
a =4 c = 9 b = d = e =0 f = 36 4x 2 9y 2 36=0
a = b = e = f =0 c =1 d = 1 x = y 2
AX 2 + CY 2 + DX + EY + f =0, X Y A C D
a b c d e
A = C =0
DX + EY + f =0
D 2 + E 2 > 0
(∅) D = E =0 f =0 (R2 ) D = E = f =0
A =0 C =0
CY 2 + DX + EY + f =0
D =0 D =0 E 2 4Cf> 0 D =0 E 2 4Cf =0
(∅)
A =0 C =0
D =0 E 2 4Cf< 0
X 2 + DX + EY + f =0
E =0 E =0 D 2 4Af> 0 E =0 D 2 4Af =0
(∅) E =0 D 2 4Af< 0
A =0 C =0 A X + D 2A 2 + C Y + E 2C 2 + F =0
A = C AF< 0
A = C AC> 0 AF< 0
AC> 0 F =0
(∅)
AC> 0 AF> 0
AC< 0 F =0
AC< 0 F =0
(R2 ) (∅) b2 4ac =0
b2 4ac =0
A1 X 2 + C1 Y 2 + F1 =0 A1 X 2 + D1 X + E1 Y + F1 = 0 A =( 2, 2) B =(1, 4) C =(4, 5) M N P MNP ABC
A =( 5, 6) ϕ = ( 2) x x y A =( a2 b2 , 0) B =( a2 b2 , 0) x a θ + y b θ =1 θ
A =(x2 ,y2 )
A =(x1 ,y1 ) x α y α = p x2 =2p α x1 2α y1 2α y2 =2p α x1 2α + y1 2α
A =(a + b,a b +1) B =(a + b +2,a b +3) C =(a + b +5,a b +2) D =(a + b +3,a b) a b
+2)x (a 1)y 2a 3=0
ABC A1 =(4, 5) B1 =(0, 2) C1 =(4, 2)
ABC A1 B1 C1
ABC A =( 1, 2) B =(3, 5) C =(0, 6) ABC x y =0 (0, 2) (3, 2) (6, 8) x 9 2 x 4 3 y =8 x 4x 2 y
( 1, 0) x +3y 5=0 x y √3 2=0 (2, 1) (1, 9) (10, 6) (8, 5) ( 4, 1) (x 2)2 +(y 3)2 =10 x + y 1=0 k n y = kx + n y =2x +3 π 4 x 2 + y 2 =4 (15, 5) x 2 + y 2 =50
ay +15a 2 =0
x 2 + y 2 2ax +2ay 18a 2 =0 x 2 + y 2 8x
x 2 + y 2 +2gx =0 x 2 + y 2 +2fy =0 A =( 2, 0) B =(2, 0) C =(0, 4)
x 2 + y 2 10y +16=0 y = 4 x 7y +50=0 ( 1, 7)
x 2 + y 2 =25 x 2 +5y 2 =25
b2 x 2 + a 2 y 2 = a 2 b2 (0, 4) (0, 4) 9x 2 +16y 2 =144 A B 4x 2 +25y 2 =100 A B AB b2 x 2 + a 2 y 2 = a 2 b2 9x 2 +16y 2 =288
D =(4,q ) q> 0 3x y +6=0 x 2 +8y 2 =72 π 4 x 2 y 2 = a 2
x 2 y 2 = a 2 (H ) 4y ± 3x =0 5x 4y =16 ( 3, 4) (H ) 3x 2 4y 2 =12 (4, 3) x y = kx + b (y n)2 =2(x m) by = x 2 +2ax + a 2 + b2 a b y 2 =2px p π 4 x + y =0 x 2 + y 2 +4y =0 x
2x y +2=0
y 2 =2px r =2p 3x 2 y 2 =12 y 2 =16x
x 2y +8=0
y 2 =2px (1, 4)
b2 x 2 + a 2 y 2 = a 2 b2 y 2 =2px
x + y +1=0
b2 x 2 a 2 y 2 = a 2 b2 √7 y 2 =2px (0, 2)
3x + y =11 3x 2 4y 2 =11 x
3x 5y +25=0
b2 x 2 + a 2 y 2 = a 2 b2 y 2 =2px
x +7y 9=0 (2,q ) (2, 0) (6, 4) 3x 2 +2xy 3y 2 2x +10y =8 4x 2 xy + 7y 2 10x +17y =29
Ax2 + Bxy + Cy 2 + Dx + Ey + F =0
2ABD
B 2CE
DE 2F =0
ax + by + c =0,
a b c y = a b x c b (b =0) (L)
a b c
T =(x,y ) (L) x y
T
(A) (B )
(L) T =(x,y )
(A) (A)
P =(x,y ) y> a b x c b b> 0
b< 0
ax + by + c> 0
ax + by + c< 0 (B )
Q =(x,y )
y< a b x c b
ax + by + c< 0
ax + by + c> 0 b> 0 b< 0
(A) (B )
ax + by + c< 0
P =(x,y )
ax + by + c> 0
ax + by + c 0
ax + by + c =0
> 0 0 < 0 x y
ax + by + c =0
ax + by + c> 0
P1 =(x1 ,y1 ) P2 =(x2 ,y2 )
ax + by + c< 0
ax1 + by1 + c> 0 ax2 + by2 + c> 0
ax1 + by1 + c< 0 ax2 + by2 + c< 0. P1 P2 P1 P2 P1 P2
(L) ax + by + c =0
T =(u,v ) ∈ P1 P2
(1 λ) 0 λ 1
P1 P2
P1 P2 λ :
u = λx1 +(1 λ)x2 ,v = λ1 y1 +(1 λ)y2 , (0 λ 1) P1 P2
ax + by + c =0
ax1 + by1 + c> 0 x2 + by2 + c< 0. f :[0, 1] → R
f (λ)= au + bv + c = a(λx1 +(1 λ)x2 )+ b(λy1 +(1 λ)y2 )+ c, u v
f (1)= ax1 + by1 + c> 0 f (0)= ax2 + by2 + c< 0
f (λ)=(ax1 ax2 + by1 by2 )λ + ax2 + by2 + c λ f (1) > 0 f (0) < 0 (0,f (0)) (1,f (1)) λ λ0 ∈ (0, 1) f (λ0 )=0
a(λ0 x1 +(1 λ0 )x2 )+ b(λ0 y1 +(1 λ0 )y2 )+ c =0
T =(u0 ,v0 ) P1 P2 u0 = λ0 x1 +(1 λ0 )x2 , v0 = λ0 y1 +(1 λ0 )y2 , (L) ax+by +c =0
P1 P2
x y
ax + by + c> 0
T =(u,v )
3x y +2 > 0
ax + by + c< 0.
3x y+2 > 0
T =(u,v ) 3u v +2 > 0
3x y +2=0 T =(u,v ) u =1 v =0 T =(1, 0)
T =(1, 0)
,... T =(x,y )
T =(x,y )
x y +1 0 x + y 1 0 y +5 0
(L2 ) x + y 1=0 (L3 ) y +5=0
T =(0, 0)
(L1 )
x y +1 0
T =(x,y )
(L1 ) x y +1=0
x y n n ∈ N ρ > >
a1 x1 + a2 x2 + + an xn + bρ0 n xi (i =1,...,n) ai ∈ R
(i =1,...,n) n x1 + x2 + x3 1 0,x1 0,x2 0,x3 0
x + y +1 0 x 2y 2 0 2x y 4 0
x + y 0 x 2y 0 2x y 0
2x y 0 4x +2y 0 x + y 0
2x y 0 x + y 0 3x + y 0
x + y 1 0 x y +2 0 2x +2y +3 0
F1 F2 F3 Fi
pi (i =1, 2, 3)
M1 M2 M3 M4 Mj
qj (j =1, 2, 3, 4)
Mj (j =1, 2, 3, 4)
Fi (i =1, 2, 3)
Fi (i =1, 2, 3)
Mj (j =1, 2, 3, 4) xij Fi Mj aij xij aij
F1 F2 F3
M1 x11 x21 x31
M2 x12 x22 x32
M3 x13 x23 x33
M4 x14 x24 x34
F1 F2 F3
M1 a11 a21 a31
M2 a12 a22 a32
M3 a13 a23 a33
M4 a14 a24 a34
pi qi aij i =1, 2, 3 j =1, 2, 3, 4 xij Fi Mj
F = 3 i=1 4 j =1 aij xij
ij
xij
xij pi qi aij xij 0(i =1, 2, 3; j =1, 2, 3, 4);
ij = pi i =1, 2, 3
=1
, 3, 4
xij xij F xij
1 =40 p2 =60 M1 M2 M3 q1 =20 q2 =30 q3 =50
ij
1 F1 M1
2 M3 xij i =1, 2 j =1, 2, 3
=120x11 +110x12 +90x13 +110x21 +95x22 +130x23
ij 0(i =1, 2; j =1, 2, 3) x11 + x12 + x13 =40 x21 + x22 + x33 =60 , x11 + x21 =20 x12 + x22 =30 x13 + x23 =50
i Mj
c1 0 c2 0, ··· ,cn 0 aij 0 i =1,...,m j =1,...,n a11 a12 a1n a21 a22 a2n am1 am2 ··· amn
F = c1 x1 + c2 x2 + + cn xn , xk (k =1,...,n) xk 0(k =1,...,n),
a11 x1 + a12 x2 + ··· + a1n xn b1 a21 x1 + a22 x2 + ··· + a2n xn b2 am1 x1 + am2 x2 + ··· + amn xn bm , b1 0 b2 0,...,bm 0 x1 ,x2 ,...,xn
ck xk (k =1,...,n)
aij 0
bk (k =1,...,m)
aij i =1,...,m j =1,...,n aij
x 0 1 ,...,x 0 n x 0 1 ,x 0 2 ,...,x 0 n
x1 ,...,xn
ck bk
F =2x +3y
x 0,y 0
x + y 1, 3x y 3, x + y 1
F = x +2y +3z
x 0,y 0,z 0
2x y + z 15
x y + z 6
x + y z =0
z = x + y F =4x +5y
x 0 y 0 x + y 0
x 5 x 3 D
F =2x + y,
3x + y 300,x + y 150,x 0,y 0 x
3x + y =300,x + y =150,x =0,y =0
L1 L2 L3 L4
L3 L4 y x (0, 0)
3x + y 300
L1 x + y 150
x 0 y 0
L4
L2
L3
2x + y =
(x,y ) ∈ D
L1 L2 L3 L4 O C O L3 L4
x =0,y =0, O =(0, 0)
F =2 · 0+0=0 C L1 L2
3x + y =300,x + y =150 x =75 y =75
C =(75, 75) F =2 75+75=225. OACB
F =0 F =2x + y F =225 (x,y ) O (x,y ) ∈ D F = x + y x + y 1,x + y 10,x 0,y 0. x + y =1,x + y =10,x =0,y =0 L1 L2 L3 L4
1 x + y =1 (x, 1 x) x ∈ [0, 1]
= x +(1 x)=1 RQ (x, 10 x)
,
+ y = F
2x + y =2,x y =1 x =1 y =0 B =(1, 0) F F =1+0=1 (x,y ) 1 F = x + y,
x21 =20 x11 ,x22 =30 x12 , x23 =50 x13 x11 + x12 + x13 =40 x13 =40 x11 x12 F =50x11 +45x12 +9950, x11 0,x12 0,x11 + x12 40
L1 L2 L3
x12 =0,x11 =0, x11 + x12 =40 L
50x11 +45x12 = F 9950
O =(0, 0)
x11 =0 x12 =0 x13 =40 x21 =20 x22 =30 x23 =10 F =9950 F = 2x +3y, x + y< 2, x + y 1,x 0,y 0 F1 =3x +2,F2 =2x +4y x + y 2, x + y 1,x 0,y 0. F = 1 2 x +37y x y 3,x +2y 11,x y 2,x 4,x 0,y 0 F =2x 1 2 y x y 1 2 , 2x + y 12,y 2x 4,y 5,x + y 1,x 0,y 0. F =5x 9y 3x +2y 4,y x 4,y x 5,x 0,y 0 5x + y 5, 5x + y 10,x 0,y 0.
+14
+12x
1 + x2 3
x1 +2x2
= 6x1 4x2
1 0 x2 0 x1 1 x2 1 F =2x1 +3x2 F = ax1 +(1+ a)x2 a> 0 x1 0 x2 0 x1 2 x2 3 a
a1 ∈ S 2 a2 ∈ S nan ∈ S
(a1 ,a2 ,...,an ,...)
(an ) an n
(an )=(a1 ,a2 ,...,an ,...)
a1 a2
(a1 ,a2 ) =(a2 ,a1 )
(a1 ,a2 ,a3 )
(a1 ,a2 )
{a1 ,a2 } = {a2 ,a1 }
(a1 ,a2 ,...,an ,...) S = R N R
n =1 n =2 n =3
n ∈ N n =4 1+2+3+4=10 1 2 (2 43 11 42 +23 4 12)=16 an = n 2 + n +41 n =1, 2,... n =1, 2,..., 39 n =40 a40 =402 +40+41=402 +2 40+1= 412 Fn =22n +1 n =0, 1,... F0 =3 F1 =5 F2 =17 F3 =257 F4 =65537 Fn F5 =4294967297 4294967297=641 6700417 F5
n ∈ N
a1 = F (1)
F (n)+ an+1 = F (n +1) φn
(φn ) n (φn+1 )
(φn ) n an+1
a1 + a2 + + an + an+1 = F (n)+ an+1 ,
a1 + a2 + ··· + an + an+1 = F (n +1), (φn+1 ) (φn ) n φ1
(φn ) ⇒ (φn+1 ) n ∈ N
(φn ) n ∈ N n =1 (φ1 )
(φ2 ) (φ1 )
(φ2 ) n =2 (φ2 ) (φ3 )
(φ2 ) (φ3 ) (φn ) n
(φn ) n ∈ N
(φn )
(p ∧ (p ⇒ q )) ⇒ q
n ∈ N
2n >n (n ∈ N)
(φn ) (φ1 ) 21 > 1 n n ∈ N
2n+1 > 2n. n> 1
2n = n + n>n +1, n> 1
2n+1 >n +1, (φn+1 ) (φn ) n (φn+1 ) (φn+1 ) (φn ) (φ2 ) (φ1 )
(φ1 ) (φ2 ) (φ3 )
(φ2 ) (φ2 )
(φ3 ) (φn ) n
(T1 ,T2 ,...,Tn ,...) T1 Tn Tn+1 Tn n ∈ N n 1+2+ + n = 1 2 n(n +1) n =1 1= 1 2 1 2 n +1 1+2+ + n +(n +1)= 1 2 n(n +1)+(n +1).
1 2 n(n +1)+(n +1)=(n +1) 1 2 n +1 = 1 2 (n +1)(n +2) 1+2+ +(n +1)= 1 2 (n +1)(n +2), n n +1 n ∈ N n n +1 n +2 S (n)= n 3 +(n +1)3 +(n +2)3 n =1 S (1)=13 +23 +33 =36 S (1) S (n) S (n)=9m m ∈ N S (n +1)=(n +1)3 +(n +2)3 +(n +3)3 =(n +1)3 +(n +2)3 + n 3 +9n 2 +27n +27 =S (n)+9n 2 +27n +27 =9m +9(n 2 +3n +3)=9(m + n 2 +3n +3) S (n) S (n +1) S (1) S (n) n ∈ N n 6n 5n +4 f (n)=6n 5n +4 f (1)=6 5+4 f (1) f (n) f (n +1) f (n)=6n+1 5(n +1)+4 (6n 5n +4) =6n+1 6n 5=6n (6 1) 5=5(6n 1), f (n +1) f (n) f (n) f (n +1) f (n) f (n +1) f (n) n ∈ N h ∈ R h> 1 n ∈ N (1+ h)n 1+ nh.
n =1 1+ h 1+ h h> 1 1+ h> 0 1+ h (1+ h)n+1 (1+ nh)(1+ h)=1+(n +1)h + nh2 n ∈ N n> 0 h2 > 0 nh2 0 1+(n +1)+ nh2 1+(n +1)h, (1+ h)n+1 1+(n +1)h.
N 1, 1 2 , 1 3 ,... Q
(T1 T2 , . . . , Tn , . . .)
T1 n ∈ N Tn Tn+1 (T1, T2, , Tn, )
n Tn p Tp p> 1 p 1 <p Tp 1 Tp Tp 1
Tn a b (a,b)= n a = b n =1 T1 a b
(a,b)=1 a = b (=1) Tn n a b (a,b)= n +1 c = a 1 d = b 1 (c,d)= n Tn c = d a = b Tn+1
Tn n ∈ N
n ∈ N
1+3+ +(2n 1)= n 2
12 +22 + ··· + n 2 = 1 6 n(n +1)(2n +1)
13 +23 + + n 3 = 1 4 n 2 (n +1)2
1+2+22 + +2n =2n+1 1
1 3 + 1 15 + ··· + 1 (2n 1)(2n +1) = n 2n +1
1 6 + 1 12 + 1 20 + + 1 n2 +3n +2 = n 2n +4 1 1 · 4 + 1 4 · 7 + ··· + 1 (3n 2)(3n +1) = n 3n +1 1 3 + 2 32 + 3 33
12 22 +32 +( 1)n 1 n 2 =( 1)n
(n +1) 2 4n +3 2 π + x = x 4n +1 2 π + x = x 4n +2 2 π + x = x 4n +3 2 π + x = x x + 3x + + (2n 1)x = 2 nx x x + 3x + ··· + (2n 1)x = 2nx 2 x f (n) p
f (n)= n 3 +11n p =6
f (n)= n 3 +5n p =6
f (n)= n(n +2)(25n 2 1) p =24
f (n)= n(n +1)(2n +1) p =6
f (n)=9n+1 +26n+1 p =11
f (n)=92n+2 64n+1 p =17
f (n)=7 52n 1 +23n+1 p =17
f (n)= n 4n+1 (n +1)4n +1 p =9
f (n)=74n+3 4 · 165n+1 +1215n+1 p =100
n2 +1 n2 n n +1 n
2n+4 > (n +4)2
2n+9 > (n +9)3 1 2 3 4 5 6 2n 1 2n < 1 √3n +1
,
n + 1 an an+1 b a1 , , an+1 , b a1 , , an 1 b ,
, 2,...,n
+1
n 2 1 22 + 1 32 + ··· + 1 n2 < n 1 n (n 2) 1+ 1 4 + + 1 n2 < 2(n 2) n 2 < 1+ 1 2 + + 1 2n 1 <n (n 2)
2n >n2 (n 5)
2n >n3 (n 10) (Pn ) n Pn P1 P2 (Pn ) (Pn )
2 = f (P1 ),
P1 f P2 n =2 P3 = f (P2 ),
P2 f P3
(Pn ) m P1 ,P2 ,...,Pm Pn+m = f (Pn ,Pn+1 ,...,Pn+m 1 )(n =1, 2,...) Pm+1 n =1 Pm+2 n =2
∈ R x 1 = x
n+1 = xn x (n =1, 2,...)
n n ∈ N x 1 x n =1 x 2 = x x x 2 n =2 x 3 = x 2 · x x 2 x x 3 n x1 ,x2 ,...xn n k =1 xk n k =1 xk = x1 + x2 + ··· + xn 1 k =1 xk = x1 n+1 k =1 xk = n k =1 xk + xn+1 (n =1, 2,...)
a1 ,a2 ,...,an a1 ,a2 (a1 ,a2 ) (a1 ,a2 ,...,an )= (a1 ,a2 ,...,an 1 ),an (n =3, 4,...) n =3 (a1 ,a2 ,a3 )= (a1 ,a2 ),a3 , n =4 (a1 ,a2 ,a3 ,a4 )= (a1 ,a2 ,a3 ),a4 , n (a1 ,a2 ,...,an )=(b1 ,b2 ,...,bn ) ⇐⇒
(a1 ,a1 + d,a1 +2d,a1 +3d,...,a1 +(n 1)d,...) a1 d a1 d (an )
(an ) a5 =19 a10 =39 n ∈ N a5 = a1 +4d =19,a10 = a1 +9d =39 a1 +4d =19 ∧ a1 +9d =39, 5d =20 d =4 a1 =19 16=3 an =3+4(n 1)(n =1, 2,...), an =4n 1(n =1, 2,...) (3, 7, 11,...) (an ) (an ) a3 =4a1 a6 =17 a3 = a1 +2d =4a1 ,a6 = a1 +5d =17, 3a1 2d =0,a1 +5d =17, a1 =2 d =3 an =2+3(n 1)=3n 1(n =1, 2,...), (2, 5, 8,...) (n)=(1, 2, 3,...,n,...) 1+2+ + n = 1 2 n(n +1), n n (1, 2, 3,...) n (an ) Sn Sn = a1 + a2 + a3 + + an . Sn = a1 +(a1 + d)+(a1 +2d)+ +(a1 +(n 1)d)
( 9, 6, 3,...)
9 d =3 n
(an ) n 2 (2a1 +(n 1)d) n n 2 ( 18+3(n 1))=66, n 2 7n 44=0. 4 a b c b = a + c 2 a b c d c b = d,b a = d,
c = b + d,a = b d,
a + c =2b a b x
a x b x = b + a 2 a b m x1 ,x2 ,...,xm a,x1 ,x2 ,...,xm ,b
a,x1 ,x2 ,...,xm ,b m +2 (a1 ,a2 ,...) a1 = a,am+2 = b = a +(m +1)d, a1 = a,d = b a m +1 (a1 ,a2 ,...) an = a + n 1 m +1 (b a)(n =1, 2,...)
xk = ak +1 (k =1, 2,...,m)
xk = a + k m +1 (b a)(k =1, 2,...,m)
=7
x :2+5+8+ ··· + x =155 p q r a b c (q r )a +(r p)b +(p q )c =0
(an ) (An ) sn Sn n sn Sn = 7n +1 4n +27 , a11 /A11
x19 , 39 4 (an )=(a1 ,a2 ,...,an ,...)
,x
(0, 0,..., 0,...)
,...,x
,...,x
,
a1 d
1+ a + a 2 + + ax =(1+ a)(1+ a 2 )(1+ a 4 )(1+ a 8 ) a b a a +2b 2a + b (b +1)2 ab +5 (a +1)2 p a q b r c aq r br p cp q =1 (an ) an+1 an = d (n =1, 2,...), d (an ) an+1 = qan (n =1, 2,...), q (=0) (an ) an+1 = f (an )(n =1, 2,...)
F (n,an ,an+1 )=0(n =1, 2,...);
F (n,an ,an+1 ,an+2 )=0(n =1, 2,...) (an )
1
an = C + nd C
(an )
an+1 = C +(n +1)d,an = C + nd,
an+1 an = d (an ) a2 = qa1 ,a3 = qa2 ,...,an = qan 1 , an = qan 1 = q 2 an 2 = = q n 1 a1 . a1
a1 /q an = q n 1 a1 =(a1 /q )q n
an = Cq n C (an )
an+1 = Cq n+1
an = Cq n an+1 = qan
an+1 qan = d, q d q =1 d =0 q =1 an = C + nd C q =1 (bn )
bn = an d 1 q (n =1, 2,...), an = bn + d 1 q (n =1, 2,...).
bn+1 + d 1 q qbn qd 1 q = d,
bn+1 qbn =0 f f (t)= qt + d.
β q =0
an+2 + pan+1 =0 an+1 + pan =0
β (bn ) bn = α n an
an = αn bn . αn+2 bn+2 (α + β )αn+1 bn+1 + αβαn bn =0, αn+2 (bn+2 bn+1 ) β α (bn+1 bn )=0
(cn )
cn = bn+1 bn ,
cn+1
cn = Kγ n γ = β α K (bn )
α cn =0,
bn+1 bn = Kγ n
b2 b1 = Kγ,b3 b2 = Kγ 2 ,...,bn bn 1 = Kγ n 1 ,
bn b1 = K (γ + γ 2 + + γ n 1 ) α = β γ =1 bn = b1 + K (n 1)=(b1 K )+ Kn. b1 b1 K C
bn = C + Kn C K (an )
an =(C + Kn)αn C K
(an ) α = β α = β γ =1
bn = b1 + Kγ 1 γ n 1 γ = b1 + Kγ 1 γ (1 γ n )= b1 + Kγ 1 γ Kγ 1 γ γ n .
(an ) a1 = 1 an = bn+1 bn 2,
n+2
n+1 2 +3 bn+1 bn 2 +4=0,
n+2 bn + bn+1 bn 2=0
n+2 + bn+1 2bn =0
2 + t 2=0 2 bn = C + D ( 2)n C D an = C + D ( 2)n+1 C + D ( 2)n 2= 1+ K ( 2)n+1 1+ K ( 2)n 2 K = D /C
a1 = 1 K =0 a1 = 1
an = 1 an+1 +2an =3 a1 =0 an+1 4an =7 a1 =2 an+2 2an+1 5an =0 a1 =0 a2 =1 an+2 4an+1 +4an =0 a1 =1 a2 =3 an+2 6an+1 +8an =0 a1 =3 a2 =2
n+2 5an+1 +6an =8 an = bn +4
n+2 3an+1 +2an = 7 an = bn +7n an+2 2an+1 + an =6
an = bn +3n 2
an )=(a1 ,a2 ,...)
an )= 1, 1 2 , 1 3 ,..., 1 n ,... n an = 1 n an > 0 n 1 n n an = 1 n an n> 1000 an n an n> 1000000 1 n n an
n0 (an ) S
(an ) S
n0 (an ) an0 +1 an n = n0 +1 n0 +2,... S (an ) S ( 2, 2 1, 1, 0, 1, 2, 3,...,n,n +1,...)
ε<an a<ε n>n0
|an a| <ε n>n0 (an ) ε a n0 |an a| <ε n>n0 a (an ) ε a a (an ) ε> 0 n0 |an a| <ε n>n0
(an ) +∞ M> 0 n0 an >M n>n0
(an ) −∞ P< 0 n0 an <P n>n0 (an ) K
|an | <K n =1, 2,...
(an ) (bn ) an = n +1 n ,bn =( 1)n+1 n +1 n = n +1 n < 3, ( 1)n+1 =1 < 2, (an ) (bn ) M an <M n ∈ N (an ) M m an >m n ∈ N (an ) m (an )
m<an <M (n =1, 2,...)
|an | < (|m|, |M |)(n =1, 2,...). (n 2 )
(an ) n→∞ an = a ε =1 n0 |an a| < 1 n>n0 a 1 <an <a +1 n>n0 , |an | < |a| +1 n>n0 . M = (|a| +1, |a1 |,..., |an0 |) |an | <M, n ∈ N (an ) n→∞ an = n→∞ bn = a an xn bn n ∈ N (xn ) n→∞ xn = a
ε> 0 n1 n2
|an a| <ε n>n1 |bn a| <ε n>n2
a ε<an <a + ε,a ε<bn <a + ε, n
n0 = (n1 ,n2 )
a ε<an xn bn <a + ε n>n0 , a ε<xn <a + ε n>n0 n→∞ xn = a
→∞ an = a n→∞ bn = b an bn n ∈ N a b b<a ε = 1 2 (a b) n1 n2 |an a| <ε n>n1 |bn b| <ε n>n2
a ε<an n>n1 bn <b + ε n>n2
n0 = (n1 ,n2 )
bn <b + ε = b + 1 2 (a b)= a 1 2 (a b)= a ε<an , n>n0 an bn b<a a b
an bn an <bn a b a<b
c (an ) (can ) (an ) (bn )
(an + bn ), (an bn ), (an bn ), an bn , bn =0 n ∈ N
n→∞ an = a n→∞ bn = b
n→∞ can = ca (c ∈ R)
n→∞ (an + bn )= a + b
n→∞ (an bn )= a b
n→∞ (an bn )= ab
n→∞ an bn = a b (b =0)
c =0
c =0 ε> 0 n0 |an a| < ε |c|
n>n0
|can ca| = |c||an a| < |c| ε |c| = ε n>n0 , ε> 0 n1 n2
|an a| < ε 2 n>n1 |bn b| < ε 2 n>n2 |an + bn (a + b)| = |(an a)+(bn b)| |an a| + |bn b| < ε 2 + ε 2 = ε
n>n0 = (n1 ,n2 )
n→∞ (an bn )= n→∞ (an +( 1)bn )= n→∞ an + n→∞ ( 1)bn = n→∞ an +( 1) n→∞ bn = a +( 1)b = a b.
(bn ) M ∈ R
|bn | <M M |a| <M ε> 0 n1 n2
|an a| < ε 2M n>n1 |bn b| < ε 2M n>n2
|an bn ab| =|an bn abn + abn
| = |bn (an a)+ a(bn b)| |bn (an a)| + |a(bn b)| = |bn ||an a| + |a||bn b| <M ε 2M + M ε 2M = ε n>n0 = (n1 ,n2 )
→∞ bn = b =0
n |bn | >K K> 0 n>n ε> 0 n |bn b| <ε n>n |bn | |b|− ε = K n>n 1 |bn | < 1 K n>n ε> 0 n1 n2 |an a| < Kε 2 n>n1 ) |a||bn b| < K |b|ε 2 n>n2 an bn a b = an b abn bn b = an b ab + ab abn bn b |b||an a| + |
n>n0 = (n ,n1 ,n2 ) (an ) (bn ) (an ) (bn ) an = 1 n bn = 1 n an <bn n→∞ an =
an < n→∞
k> 1 n→∞ 1 nk =0 k> 1 n k >n 0 < 1 nk < 1 n (0, 0,..., 0,...) 1 n n→∞ 1 nk =0 (an ) an
(an )
a1 <a2 < <an < , an <an+1 (n =1, 2,...), (an ) (an )
an an+1 (n =1, 2,...)
an+1 <an (n =1, 2,...)
an+1 an (n =1, 2,...)
(an ) a1 <a2 < ··· <an < ··· <K n ∈ N
an <an+1 ,an <K. a K ε
(an ) a K n→∞ an = a K (an ) K K K1 a (an ) a = n→∞ an
ε> 0 (a ε,a) an0 (an ) a (an ) a ε a (an ) an0+1 ,an0+2 ,... a ε,a (an ) ε a
1 1 n n = 1 e (K ) (Mn )
A1 ,A2 ,...,An pn An+1 (K ) A1 ,A2 ,...,An
A1 An |A1 An+1 | + |An An+1 | > |A1 An | pn+1 (Mn+1 )
A1 ,A2 ,...,An An+1 pn (Mn ) (K )
(Mn ) pn (pn ) pn (K ) n→∞ pn (K )
(K ) (K ) (K1 ) (K2 ) d1 d2 r1 r2 d1 d2 = r1 r2 d1 /d2 >r1 /r2 = λ d1 λd2 > 0 d1 (K1 ) ε> 0 (K1 ) p p>d1 ε d1 λd2 > 0
(M1 ) (K1 ) p1 p1 >d1 (d1 λd2 ) p1 >λd2 (M2 ) (K2 )
(M1 ) p2 (M2 ) p1 = λp2 λp2 >λd2 p2 >d2
d1 /d2 >r1 /r2
d1 /d2 <r1 /r2 d1 r1 = d2 r2 , 2π
r 2πr a b a<b x a x b [a,b]
[a,b]= {x| a x b}.
([a1 ,b1 ], [a2 ,b2 ],..., [an ,bn ],...)
[ak +1 ,bk +1 ] ⊆ [ak ,bk ] k = 1, 2,...
an <bn (n =1, 2,...)
an an+1 bn bn+1 (n =1, 2,...)
[an+1 ,bn+1 ] ⊆ [an ,bn ];
n→∞ (bn an )=0
c [a1 ,b1 ] [a2 ,b2 ] (an ) an <bn b1
n→∞ an = c an c (bn ) a1 an <bn n→∞ bn = b bn b n→∞ (bn an )= n→∞ bn n→∞ an = b c =0,
b = c c (an ) (bn )
an c bn ,
c [an ,bn ] n =1, 2,...
c
c = c [an ,bn ] an c bn
bn c an , an bn c c bn an ,
bn an [c c ].
c = c |c c| = k> 0 bn an k> 0
n→∞ (bn an ) k> 0
(xn ) a1 b1 a1 <xn <b1
[a1 ,b1 ] (xn )
[a2 ,b2 ]
[a2 ,b2 ] (xn )
b2 a2 = 1 2 (b1 a1 )
[a2 ,b2 ]
(xn ) [a3 ,b3 ]
[a3 ,b3 ] (xn )
b3 a3 = 1 2 (b2 a2 )= 1 22 (b1 a1 )
([a1 ,b1 ], [a2 ,b2 ]
..., [an ,bn ],...)
bn an = 1 2n 1 (b1 a1 ), n→∞ (bn an )=0 c c (xn ) (xn )
an )
d1 < 0
d1 1
x<c + d1 +1 10 c + 1+1 10 = c, d1 0 d1 9 102 x c d1 10 d2 d2 102 x c d1 10 <d2 +1,
c + d1 10 + d2 102 x<c + d1 10 + d2 +1 102 , 0 d2 9
0 d1 9 c,d1 ,d2
dn ,... 0 dk 9(k =1, 2,...) n ∈ N c + d1 10 + + dn 10n x<c + d1 10 + + dn +1 10n . (an ) (bn )
an = c + d1 10 + d2 102 + + dn 10n (n =1, 2,...), bn = c + d1 10 + d2 102 + ··· + dn +1 10n (n = 1, 2, )
+1
x x a1 ,a2 ,... x (an ) (bn ) an x<bn , x = n→∞ an = n→∞ bn (an ) (bn ) x = c + d1 10 + d2 102 + ··· + dn 10n + ··· = c + ∞ k =1 dk 10k x c x c =[x] dn ∈{0, 1,... 9} n x n =1, 2,... x x = c,d1 d2 ...dn ... r
0 <r< 1
r =0,d1 d2 d3 ..., dk ∈{0, 1,... 9} k =1, 2,... r p q p,q ∈ N p<q r< 1 r = p q p q dn n>k p q 1 8 1 8 =0, 12500 = 1, 125 p q p<q p q q 1 q 1 q k k
1 8 =0, 12500 1 6 =0, 1666 ... 1 11 =0, 090909 1 7 =0, 142857142857
a b (xn ) (yn ) x1 = a y1 = b xn+1 = √xn yn yn+1 = 1 2 (xn + yn )(n =1, 2,...)
(xn ) (yn )
→∞ xn n→∞ yn
→∞ xn = n→∞ yn = M (a,b) M (a,b) a b (an ) a1 =1 an+1 = k + an (k> 0) a a t2 t k =0
S =1+2+4+8+16+32+ ··· S> 0
2S =2+4+8+16+32+ = S 1 S = 1 1 > 0
p p √2
x,y,z ∈ A
x x; x y ∧ y x ⇒ x = y ; x y ∧ y z ⇒ x z. B A
m ∈ A x m x ∈ B m B B m B m m s B s B s m s B s = B B A A Q R Q R S = {r | r ∈ Q r 2 < 2} Q s s ∈ Q s = S S R R S √2
x,y,z ∈ R
R + R + R R
x + y = y + x
x +(y + z )=(x + y )+ z
0 ∈ R x +0= x
x ∈ R x(∈ R) x +( x)=0
xy = yx
x(yz )=(xy )z
1 ∈ R x 1= x
x ∈ R x =0 x 1 xx 1 =1
x(y + z )= xy + xz
x< 0 x =0 x> 0
x> 0 x< 0 x< 0 x> 0
x> 0 y> 0 x + y> 0 xy> 0 S R (+) ( ) ( ) R + · R\Q √2(/ ∈ Q)
r | r ∈ Q r 2 < 2 ,
x> 0 y n nx>y.
x> 0 y ∈ R n ∈ N n ∈ N nx y y S = {nx| n ∈ N} s = S (n +1)x ∈ S nx =(n +1)x x s x, s x S x> 0 s x<s s S n ∈ N nx<y
n ∈ N
S (x, y) S z S S
z ∈ S (x, y )
z = x ∗ y S ∗ N (x, y )
z = x + y S +
S = { , ⊥} S ∨ ∧
(, ) x,y,z,u ∈ S ∗ x ∗ y,x ∗ (y ∗ z ), ((x ∗ y ) ∗ z ) ∗ u, (x ∗ y ) ∗ (z ∗ u), x ∗ (y ∗ ((y ∗ y ) ∗ u))
(S, O)
O = {∗1 , ∗2 , , ∗n }
(S, {∗1 , ∗2 , . . . , ∗n }) (S, ∗1 , ∗2 , . . . , ∗n ) (N, +) (Z, +, ) (R, +, ) (N, {+}) (Z, {+, ·}) (R, {+, ·})
S, O ) S
S (S, ∗) x,y ∈ S x ∗ y ∈ S
(N, +) (N, ) N + (N, ) 3 ∈ N 5 ∈ N 3 5/ ∈ N S Z (S, +) S = {−3, 5, 6} 3+5/ ∈ S S ⊂ Z (S, +)
S = {0} S = Z
,
= x| x =3n,n
N ,D
|
Z (A, +) (A, ) (A, ·) (B, +),..., (D, ·) S = {a,b,c} ∗ ∗ abc a bca b abc c cab
(
) =(
)
(S, ∗)
3 ({0, 1}, ) S = {0, 1} Z (S, ·) S = {1, 1,i, i} + (S, +) (S, ·) S ⊂ R (S, ·) · (S, ∗)
(S, ∗)
x,y,z ∈ S x ∗ (y ∗ z )=(x ∗ y ) ∗ z,
(S, ∗)
(S, ∗) S ∗ +
(S, ∗) e ∈ S x ∈ S
(S, ∗)
+ e
N0 = N ∪{0} (N0 , +) x ∈ N0 x +0=0+ x = x (N, +)
(Z, ) x ∈ Z
x 0= x 0 x = x (Z, )
(S, ∗) e e e ∗ e = e , e ∗ e = e, e = e (S, ∗)
(N0 , +) (N, +) (Z, ) (N, ) S ∗ (S, ∗)
x,y ∈ S x ∗ y ∈ S (S, ∗) x,y,z ∈ S x ∗ (y ∗ z )=(x ∗ y ) ∗ z ;
e ∈ S x ∈ S x ∗ e = e ∗ x = x; x ∈ S x x ∗ x = x ∗ x = e. (S, ∗) (Z, +) (Z, +)
x x x +( x)=( x)+ x =0 (Z, ) x 2 x = x 2=1 x = 1 2 1 2 / ∈ Z (Q, +) (Q, ·) x =0 x 1 = 1 x x 1 x = 1 x x =1. (Q, ) (Q, ) Q = Q\{0} Q (Q , ·) ∗ x,y ∈ S x ∗ y = y ∗ x, (S, ∗)
x + y = y + x x y = y x x y (S, ∗) S ∗ + (S, ∗)
S = {(x,y )| x,y ∈ R}
(x,y ) ∗ (u,v )=(x + u,yv ) (S, ∗)
S = {x + y √2 | x,y ∈ Q x 2 + y 2 > 0} (S, +) (S, ) S = {0, 1}
S = {1, 2, 3,...,p 1} p ∗
p (S, ∗) p =7
S = {a,b,c,d,e,f,g,h} ∗ ∗ abcdefgh a abcdefgh b badcfehg c cdbahgef d dcabghfe e efghbadc f fehgabcd
g ghfecdba h hgefdcab
(S, ∗)
S ∗ ◦
(S, ∗, ◦)
(S, ∗)
(S, ◦) x,y,z ∈ S x ◦ (y ∗ z )=(x ◦ y ) ∗ (x ◦ z ), (y ∗ z )
x =(y ◦ x) ∗ (z ◦ x). S
(Z, +, ) (Z, , +) (Z, ) (S, ∗) e x x (S, ∗, ◦)
(x ◦ y ) ∗ (x ◦ y )=(x ∗ x) ◦ y = e ◦ y = e, x ◦ y x ◦ y
◦ y =(x ◦ y )
◦ y =(x ◦ y )
(S, ∗, ◦) + · (S, +) x x + · (S, +) (S, ) x (y + z )= x y + x z (y + z ) · x = y · x + z · x (x,y,z ∈ S ) x 0=0 x =0 ( x) · y = x · ( y )= (x · y ) ( x) ( y )= x y
+2
+
+2. (S, ∗, ◦)
∈ Z x ∗ y x
y ∈ Z (Z, ∗) (Z, ◦) x ∗ (y ∗ z )= x ∗ (y + z +2)= x +(y + z +2)+2= x + y + z +4, (x ∗ y ) ∗ z =(x + y +2)
=(x + y +2)+ z +2= x + y + z +4,
∗ (y ∗ z )=(x ∗ y ) ∗ z. x ◦ (y ◦ z )= x ◦ (2y +2z + yz +2) =2x +2(2y +2z + yz +2)+ x(2y +2z + yz +2)+2 =4x +4y +4z +2yz +2xz +2xy + xyz +6, (x ◦ y ) ◦ z =(2x +2y + xy +2) ◦ z =2(2x +2y + xy +2)+2z +(2x +2y + xy +2)z +2 =4x +4y +4z +2yz +2xy +2zx + xyz +6, x ◦ (y ◦ z )=(x ◦ y ) ◦ z.
(Z, ∗) (Z, ◦) (Z, ∗) x ∗ y = x + y +2= y + x +2= y ∗ x. e x ∗ e = x, x + e +2= x, e = 2 ∈ Z x ∈ Z x ∗−2= x x ∈ Z x
x = x x ∗ x = e x ◦ (y ∗ z )= x ◦ (y + z +2)=2x +2(y + z +2)+ x(y + z +2)+2 =2x +2y +2z +4+ xy + xz +2x +2=4x +2y +2z + xy + xz +6; (x ◦ y ) ∗ (x ◦ z )=(2x +2y + xy +2) ∗ (2x +2z + xz +2) =2x +2y + xy +2+2x +2z + xz +2+2=4x +2y +2z + xy + xz +6, x ◦ (y ∗ z )=(x ◦ y ) ∗ (x ◦ z ) (y ∗ z ) ◦ x =(y ◦ x) ∗ (z ◦ x). (Z, ∗) (Z, ◦) (Z, ∗, ◦)
(S, ∗, ◦) (S, ∗) (S , ◦) S = S \{e} e (S, ∗) x,y,z ∈ S x ◦ (y ∗ z )=(x ◦ y ) ∗ (x ◦ z )
(S, ∗, ◦)
(Q, +, ) (R, +, ) ∗ ◦ + (S, +)
(S , ) S = S \{0} S = {0, 1} ∗ ◦
01 0 01 1 10 ◦ 01 0 00 1 01
(S, ∗, ◦) + · S ⊂ R (S, +, ) S (S, +) 1+1=
2 ∈ S 2+1=3 ∈ S S x ∈ S x S 1, 2, S Z ⊂ S x S x =0 S 1 2 1 3 1 n n ∈ Z n =0 S S m n (m,n ∈ Z n =0 S S (S, +, ) Q
S = {a,b,c,d}
(S, ∗, ◦)
S = {(x,y )| x,y ∈ Z}
(x,y ) ∗ (u,v )=(x + u,y + v ), (x,y ) ◦ (u,v )=(xu,yu + xv + yv ) (S, ∗) (S, ◦) (S, ∗, ◦)
S = {0, 1, 2} ∗ ◦ (S, ∗, ◦) (x ∗ y )3 = x 3 ∗ y 3 , x 3 x ◦ x ◦ x
S = {a,b,c,d} ∗ ◦
abcd a bcda b cdab c dabc d abcd
abcd a abcd b bdbd c cbad d dddd
◦ x ∗ y =(x ∩ y ) ∪ (x ∩ y ),x ◦ y = x ∩ y, x x ( E, ∗, ◦)
S = {(x,y )| x,y ∈ R} ∗ ◦ (x,y ) ∗ (u,v )=(x + u,y + v ), (x,y ) ◦ (u,v )=(xu 5yv,xv + yu). (S, ∗, ◦)
S = {x + y √2 | x,y ∈ Q}
x ∈ R x 2 0 x x 2 x 2 +1=0 R i2 = 1 x + iy x,y ∈ R i2 1
C = {(x,y )|x,y ∈ R} (x,y ) (u,v ) x = u y = v C + (x,y )+(u,v )=(x + u,y + v ); (x,y )(u,v )=(xu yv,xv + yu).
z + w = w + z z +(w + t)=(z + w )+ t
z +(0, 0)= z zw = wz z (wt)=(zw )t
z (1, 0)= z z (w + t)= zw + zt
z w t C (C, +)
z =(x,y ) w =(u,v ) z + w =(0, 0) (x,y )+(u,v )=(0, 0)
(x + u,y + v )=(0, 0), x + u =0,y + v =0, u = x,v = y. ( x, y ) (x,y ) (x,y ) z ( x, y ) z (C , ·) C = C\{(0, 0)}
z =(x,y ) =(0, 0) w =(u,v )
(x,y )(u,v )=(1, 0) (xu yv,xv + yu)=(1, 0) xu yv =1,xv + yu =0 x y u v x 2 + y 2 =0 x y x =0 x y x 2 u xyv = x,xyv + y 2 u =0, (x 2 + y 2 )u = x. x 2 + y 2 =0 u = x x2 + y 2 xv = yu = xy x2 + y 2 , x =0 v = y x2 + y 2 x =0 y =0 yv = xu 1= x2 x2 + y 2 1= y 2 x2 + y 2 ,
x 2 + y 2 =0 z =(x,y ) x x2 + y 2 , y x2 + y 2 z 1 1 z x =0 y =0 z =(0, 0) 0=1 0=0 (C, +) (C , ) (C, +, ·) z +( w ) z w z w zw 1 z 1 w z w z w z =(1, 3) w =( 2, 4)
z + w w z w zw w 1 z w z + w =(1, 3)+( 2, 4)=( 1, 1) w =(2, 4)
z w = z +( w )=(1, 3)+(2, 4)=(3, 7) zw =(1, 3)( 2, 4)=(1( 2) ( 3)4, 1 · 4+( 3)( 2))=(10, 10) w 1 = 2 ( 2)2 +42 , 4 ( 2)2 +42 = 1
(x,y ) Oxy x y (x, 0) x x x (x, 0) x (x, 0) x R ⊂ C
(x, 0)+(y, 0)=(x + y, 0);(x, 0)(y, 0)=(xy, 0);
(x, 0) (y, 0)=(x y, 0); (x, 0) (y, 0) = x y , 0 (y =0) K (x, 1) x ∈ R K R (x, 1) x x (x, 1)
(x, 1)+(y, 1)=(x + y, 2), (x, 1) (y, 1) K x y R K
(x, 1)+(y, 1)=(x + y, 1), (0, 0) (1, 0) (0, 1) i i2 = 1 i2 =(0, 1)(0, 1)=( 1, 0)= 1
z =(x,y )
z = x + iy
(x,y )=(x, 0)+(0,y )=(x, 0)+(0, 1)(y, 0)= x + iy. z w z = x + iy w = u + iv (x,y,u,v ∈ R) (x + iy )+(u + iv )=(x + u)+ i(y + v ); (x + iy ) (u + iv )=(x u)+ i(y v ); (x + iy )(u + iv )=(xu yv )+ i(xv + yu); x + iy u + iv = xu + yv u2 + v 2 + i yu xv u2 + v 2 (u,v ) =(0, 0), i i2 1 i3 i i4
z = x + iy x,y ∈ R x z x = z y z
y = z
z = z + i z.
z =0 z z =0 z
z = x + iy (x,y ∈ R) x iy z z z M z M x z = z z z ∈ R+ 0 z = 1 2 (z + z ) z = 1 2i (z z ) z + w = z + w z w = z w zw = z w z w = z w (w =0) z ∈ R z = z z = x + iy w = u + iv x,y,u,v ∈ R z = x + iy z = x iy = x + i( y ) z = x i( y )= x + iy = z z z = x 2 + y 2 > 0 z = z + i z z = z i z z + z =2 z, z = 1 2 (z + z ), z z =2i z, z = 1 2i (z z ). z + w =(x + u)+ i(y + v ) z + w =(x + u) i(y + v ). z + w =(x iy )+(u iv )=(x + u) i(y + v ), z + w = z + w. z ∈ R z =0 z = z z = z z =0 z ∈ R i 2 1 2 i t i t t ∈ R a +2bi (1+ c)+2i a,b,c ∈ R 1 3i 1+ i i 2+ i (1+ i)2 (1+ i)3 i40 i4n 1 i10
|z | < 1 |z | 1 |z | > 1 |z | 2 1 < |z | < 2 π 4 z π 3 |z | 1 z 0 x 2 + y 2 +2x +2y =0
z1 = x1 + iy1 z2 = x2 + iy2 z1 + z2 =(x1 + x2 )+ i(y1 + y2 ) z1 = x1 + iy1 z2 = x2 + iy2 z3 = x3 + iy3
z1 + z2 + z3 =(x1 + x2 + x3 )+ i(y1 + y2 + y3 ). n zk = xk + iyk k =1, 2,...,n
z1 + z2 + + zn =(x1 + x2 + + xn )+ i(y1 + y2 + + yn ). z1 = x1 + iy1 z2 = x2 + iy2 z1 z2 =(x1 x2 y1 y2 )+ i(x1 y2 + y1 x2 ). zk = xk +iyk k =1, 2, 3 z1 z2 z3 =(x1 x2 x3 y1 y2 x3 x1 y2 y3 y1 x2 y3 ) +i(x1 y2 x3 + y1 x2 x3 + x1 x2 y3 y1 y2 y3 ) n n z1 = r1 ( θ1 + i θ1 ) z2 = r2 ( θ2 + i θ2 z1 z2 = r1 ( θ1 + i θ1 )r2 ( θ2 + i θ2 ) = r1 r2 (( θ1 θ2 θ1 θ2 )+ i( θ1 θ2 + θ2 θ1 )),
z1 z2 = r1 r2 ( (θ1 + θ2 )+ i (θ1 + θ2 )) z1 z2
( θ + i θ )
n = r n ( nθ + i nθ )(n ∈ N)
(r ( θ + i θ ))n = r n ( nθ + i nθ ) n =1
( θ + i θ )= r ( θ + i θ ). n ∈ N r ( θ + i θ ) (r ( θ + i θ ))n+1 =(r n ( nθ + i nθ ))r ( θ + i θ ) = r n+1 ( nθ θ nθ θ + i( nθ θ + nθ θ )) = r n+1 ( (n +1)θ + i (n +1)θ ). n 1
p = q p>q p<q p>q p,q ∈{0, 1,...,n 1} p q ∈{0, 1,...,n 1} p q n p<q p q ∈{−1, 2,..., 1 n} p q n
zp = zq p = q
zp = zq
a = r ( θ + i θ ) =0 n z0 ,z1 ,...,zn 1
zk = n √r θ +2kπ n + i θ +2kπ n (k =0, 1,...,n 1). n ∈ N a ∈ C z : z n = a n a (=0) n n n a n √a
n √a = n √r θ +2kπ n + i θ +2kπ n (k =0, 1,...,n 1) n √a n n √r ( θ + i θ ) n √r θ +2π n + i θ +2π n ,..., n √r θ +2(n 1)π n + i θ +2(n 1)π n (x, 0) x +0i x n =3 a =1 1( 0+ i 0) 3 √1= 3 √1 2kπ 3 + i 2kπ 3 (k =0, 1, 2), 3 √1 1= 2kπ 3 + i 2kπ 3 (k =0, 1, 2),
(3x 4yi) ( x +2yi) (2a 3bi)+( a bi)+(4a +2bi) (2a 5bi) 2i + 1 3i 2 1 i +3i 2 n
a0 ,a1 ,...,an P z an z n + an 1 z n 1 + +
a1 z + a0 P C C
P (z )= an z n + an 1 z n 1 + + a1 z + a0 . z
an z n + an 1 z n 1 + ··· + a0 z P (z )
an =0 n P n n = P P (z )=0 z ∈ C P
a0 ,a1 ,...,an P
an an z n ,an 1 z n 1 ,...,a0 P an z n a0 P n a P
P (a)=0 P z = a P P (z )=0
a0 = a1 = = an =0 P
P (z )= 3z 2 +2z +1 P (1)= 3 · 12 +2 · 1+1=0 P (z )=0 F z
F (z )=(1 z )(z i) (1 z )(z i)= z i z 2 + iz,
F (z )= z 2 +(1+ i)z i, F 1 1+ i i z =1 z = i
P Q
U V W
U (z ) = P (z ) + Q(z ); V (z ) = P (z ) Q(z ); W (z ) = P (z )Q(z )
U ( P, Q), V ( P, Q), W = P + Q.
P (z )= z 2 + z +1; Q(z )= z 2 + iz +2 P Q
U (z )=(z 2 + z +1)+( z 2 + iz +2)=(1+ i)z +3
U< ( P Q) V
V (z )=(z 2 + z +1) ( z 2 + iz +2)=2z 2 +(1 i)z 1 V = ( P, Q) W P Q
W (z )=(z 2 + z +1)( z 2 + iz +2), W (z )= z 4 +( 1+ i)z 3 +(1+ i)z 2 +(2+ i)z +2 W = P + Q P Q G
G(z )= P (z ) Q(z ) P Q S P (z )= S (z )Q(z )
P (z )= z 2 +1 Q(z )= z + i S S (z )= z i P (z )= S (z )Q(z ) z 2 +1=(z i)(z + i) P Q
P (z ) Q(z ) = z 2 +1 z + i = z i.
P (z )= z +1 Q(z )= z S P (z )= S (z )Q(z ) P Q G
G(z )= z +1 z ,
P Q P (z )= S (z )Q(z ), S R P (z )= S (z )Q(z )+ R(z ), R R< Q R P Q C p q p/q p q s p = sq s r p = sq + r C C Z +
P (z )= z +1 Q(z )=3z 2 z +2 S (z )=0 R(z )= z +1 R< Q P< Q S (z )=0 R(z )= P (z ) P (z )=3z 2 z +2 Q(z )= z +1 S R P Q (3z 2 z +2):(z +1) z 3z 2 3z (3z 2 z +2):(z +1)=3z z +1 3z (3z 2 z +2):(z +1)=3z. 3z 2 +3z
Q(z )=3z 4 R(z )=6 3z 2 z +2=(3z 4)(z +1)+6, P Q P (z )=2z 5 z 4 +8z 3 4z 2 + z +3,Q(z )= z 2 + z +1 S R P Q (2z 5 z 4 +8z 3 4z 2 + z +3):(z 2 + z +1)=2z 3 3z 2 +9z 10 2z 5 +2z 4 +2z 3 3z 4 +6z 3 4z 2
z 4 3z 3 3z 2
z 3 z 2 + z 9z 3 +9z 2 +9z
z 2 8z +3 10z 2 10z 10 2z +13 S (z )=2z 3 3z 2 +9z 10,R(z )=2z +13. P C Q Q(z )= z a, a S R P (z )= S (z )Q(z )+ R(z ), R =0 R< Q Q =1 R R =0 R(z )= R R P (z )= S (z )(z a)+ R. z = a P (a)= R.
(a)
P (a)=0
P (z )=0 P P (z )=(z a)S (z ), S = P 1 S P
P (z )=2z 3 7z 2 +7z 2 2 2
1, 1, 2, 2 P z +1 z 1 z +2 z 2 P (1)=0 P ( 1) =0 P (2)=0 P ( 2) = 0 P z 1 z 2 Q(z )= z 2 3z +2 2z 3 7z 2 +7z 2=(z 1)(z 2)(2z 1) P Q
P (z )=3z 3 +7z +6 Q(z )= z +1
P (z )=5z 4 +14z 2 +5 Q(z )= z 2
P (z )= z 3 + z 2 + z +1 Q(z )= z +3 a b P Q
P (z )=3z 2 + az 2 7z +6 Q(z )= z 1
P (z )=5z 4 + az 3 +6z 4 Q(z )= z +2
P (z )= z 8 +2z 7 +3z 2 + az + b Q(z )= z 2 + z a P (z )= z 5 + az 2 + b Q(z )=(z +2)(z +3) R(z )= z +1 a b
an z n + an 1 z n 1 + ··· + a1 z + a0 =0, a0 ,a1 ,...,an C P (z )= z 2 +1 Q(z )= z 2 z +1 R z z 2 +1=0,z 2 z +1=0 C i i (1+ i√3)/2, (1 i√3)/2 z 2 +1=(z + i)(z i),
z 2 +3z +2=(z +1)(z +2), z 2 +(2 i)z 2i =(z i)(z +2), 2z 2 +8z 10=2(z 1)(z +5) P n P (z )= an z n + an 1 z n 1 + ··· + a1 z + a0 (an =0) P (z )= an (z α1 )(z α2 ) (z αn ). P C α1 P
P (z )=(z α1 )S1 (z ), S1 = n 1 S1 an S1 C α2 S1 S1 (z )=(z α2 )S2 (z ) S2 = n 2 S2 an
P (z )=(z α1 )S1 (z ) S1 (z )=(z α2 )S2 (z ) S2 (z )=(z α3 )S3 (z ) Sn 2 (z )=(z αn 1 )Sn 1 (z ) Sn 1 (z )=(z αn )an , Sk = n k k =1, 2,...,n 1
P (z )=(z α1 )S1 (z ) =(z α1 )(z α2 )S2 (z ) =(z α1 )(z α2 )(z α3 )S3 (z ) =(z α1 )(z α2 ) (z αn 1 )Sn 1 (z ) = an (z α1 )(z α2 ) (z αn 1 )(z αn ).
P (z )= bn (z β1 )(z β2 ) (z βn )(bn =0). z ∈ C an (z α1 )(z α2 ) (z αn )= bn (z β1 )(z β2 ) (z βn ).
Q(z )= z a
S (z )=(z a)2 a P k P Q(z )=(z a)k S (z )=(z a)k +1 a k P k =2 k =3 P (z )= z 3 3z +2 1 1 2 2 P n α1 ,α2 ,...,αn P n an α1 ,α2 ,...,αm m n α1 k1
)
m P (z )= z 3 3z +2 P (z )=(z 1)2 (z +2) P (z )=16z 4 +32z 3 +24z 2 +8z +1 P (z )=16 z + 1 2 4
P (z )=(z 1)(z +1)(z +3) z =1 z = 1 z = 3 z P (z )= z 3 3z +2 z =1 z = 2 z P z = 1/2 P n n n an P (z )= an (z α1 )(z α2 ) ··· (z αn )(an =0) P z = α1 z = α2 ,...,z = αn n α1 ,α2 ,...,αn n α1 ,α2 ,...,αn P z = α = αi i =1, 2,...,n P (α)= an (α α1 )(α α2 ) (α αn )=0.
αi =0
n =0 P = n P P (z )= an z n + an 1 z n 1 + ··· + a0 a0 = a1 = ··· = an =0 P am 0 m n P
Q(z )= am z m + + a0 m m z am =0 P Q P Q P = Q P Q
P (z )= an z n + an 1 z n 1 + + a0 ,Q(z )= bm z m + bm 1 z m 1 + + b0 n>m P Q z ∈ C P (z )= Q(z ) P (z ) Q(z )=0 S S (z )= an z n + + am+1 z m+1 +(am bm )z m + +(a0 b0 )
=1 b = 6 c =7 d = 1 a x +1 + b x 2 Ax + B (x +1)(x 2) ,
A B a b A = a + b B = b 2a a b 7x 3 (x +1)((x 2) = a x +1 + b x 2 x 1 (x +1)(x 2) 7x 3= a(x 2)+ b(x +1)=(a + b)x + b 2a, a + b =7,b 2a = 3 a =10/3 b =11/3
P (z )= z 5 + az 4 + bz 3 + cz 2 + dz + e
Q(z )= z 3 +1 a = d b = e c =1 P Q
z 5 + az 4 + bz 3 + cz 2 + dz + e =(z 3 +1)(z 2 + pz + q )= z 5 + pz 4 + qz 3 + z 2 + pz + q.
a = p,b = q,c =1,d = p,e = q, a = d b = e c =1
a b c x (x b)(x c) (a b)(a c) + (x c)(x a) (b c)(b a) + (x a)(x b) (c a)(c b) =1 P
P (x)= (x b)(x c) (a b)(a c) + (x c)(x a) (b c)(b a) + (x a)(x b) (c a)(c b) 1
P (a)= P (b)= P (c)=0 P P
P (x)=0 x a b c d x 2 = a(x +2)2 + b(x +2)+ c n 3 = a(n 1)n(n +1)+ b(n 1)n + c z 3 +3z 2 +5z 7= a(z 1)z (z +1)+ b(z 1)z + c(z 1)+ d P z z 1 P n x n x a a
a b c P
P (a) (x b)(x c) (a b)(a c) + P (b) (x c)(x a) (b c)(b a) + P (c) (x a)(x b) (c a)(c b) = P (x)
P (x)= ax 3 + bx2 + cx + d Q(z )= x 2 + p 2 ad = bc
P (z )= z 4 pz + q Q(z )=(z a)2
27p 4 =256q 3 a b c 2x +1 (x 2)(x 3) = a x 2 + b x 3 x2 (x +2)(x2 1) = a x +2 + b x 1 + c x +1 x x3 +1 = a x +1 + bx + c x2 x +1
b 2x 1 (x 1)(x 2)2 =
x 1 + b (x 2)2 ?
b c 2x 1 (x 1)(x 2)2 = a x 1 + b x 2 + c (x 2)2 ? P (z )= an z n + an 1 z n 1 + ··· + a0
P (z )= an (z α1 )(z α2 ) (z αn ), α1 ,α2 ,...,αn z
a0 ,a1 ,...,an a0 ,a1 ,...,an α1 ,α2 ,...,αn
n R z 2 +1=0 R
P (z )= an z n + an 1 z n 1 + + a1 z + a0 (a0 ,a1 ,...,an ∈ R), P (α)=0
i√7)/2 P
P (x)=0 P
4 + x 3 16x 2 4x +48=0
β γ δ αβ =6 αβγδ =48 γδ =8 P (x)= x 4 + x 3 16x 2 4x +48 Q S α
β γ α + β + γ =3,αβ + αγ + βγ = 6,αβγ = 8.
β γ α αk αk 2 β = αk γ = αk 2 α(1+ k + k 2 )=3,α 2 (k + k 2 + k 3 )= 6,α 3 k 3 = 8 αk = 2 α = 2/k 2(1+ k + k 2 )=3k, 2k 2 +5k +2=0, k = 1/2 k = 2 α =4 α =1 α =4 k = 1/2 2 α =1 k = 2 2 x y z x + ay + a 2 z = a 3 ,x + by + b2 z = b3 ,x + cy + c 2 z = c 3 , a b c a b c t t3 zt2 yt x =0 a + b + c = z bc + ca + ab = y abc = x x = abc y = (bc + ca + ab) z = a + b + c α β γ x 3 + ax 2 + bx + c =0 α 2 + β 2 + γ 2 1 α + 1 β + 1 γ a b c
(u,v ) (u,v )
2 y 2 =47 u +4v =20 ∧ 3u v =47, (u,v )=(16, 1) (4, 1) ( 4, 1) (4, 1) ( 4, 1) x 2 +4y 2 =20
x 2 y 2 =47 (4, 1) ( 4, 1) (4, 1) ( 4, 1)
x 2 + y 2 =5 ∧ 4x 2 +9y 2 =25
5x 2 6y 2 =111 ∧ 7x 2 +3y 2 =714
3x 2 +5y 2 =323 ∧ 7x 2 12y 2 =375
x 2 + y 2 =97 ∧ x 2 +9y 2 =225
x 2 5xy +6y 2 =0 ∧ 2x 2 3xy +3y 2 =20
2x 2 +5xy 12y 2 =0 ∧ x 2 y 2 =20
x 2 8xy 20y 2 =0 ∧ 2x 2 +3xy +4y 2 =6a 2
2x 2 2xy y 2 =39 ∧ x 2 +2xy +4y 2 =39
(x +2y )2 +2x(x y )=20 ∧ (x y )2 +4y (x y )=5
(3x +4y )(x 2y )=288 ∧ x(7x 12y )=576
x(ax + by )= a(a 2 y 2 ) ∧ x(bx + ay )= b(b2 y 2 )
x 2 2xy y 2 =2 ∧ xy + y 2 =4
x 2 + xy +4y 2 =16 ∧ 3x 2 +8y 2 =14
3y 2 2xy =160 ∧ y 2 3xy 2x 2 =8
x2 2xy +3y 2
3x2 y 2 9xy + 2 3 =0 ∧ xy =6
(x +3y 2)2 4(x +3y 2) 12=0 ∧ x 2 +2xy +3y 2 =24
9x 2 24xy +16y 2 =54 3(3x 4y ) ∧ x 2 + xy 2y 2 =0
2x + y 3x 2y +4 3x 2y 2x + y =4 ∧ x 2 y 2 =9
(x +2y )2 +(2x y )2 =26 ∧ (x +2y )(2x y )=5
(x + y )2 +5(x + y )=84 ∧ x 2 +3xy +4y 2 =71
(x y )2 + x y =2 ∧ 4x 2 7y 2 =1
x 2 + y 2 + x + y =18 ∧ x 2 y 2 + x y =6
x 2 + y 2 + x 3y =2 ∧ x 2 + y 2 5x y =2
x y y x = 9 20 ∧ x 2 y 2 =9
x + 1 y = a ∧ y + 1 x = b
x2 + y 2 x = a ∧ x2 + y 2 y = b
(2x 3y +1)2 9=0 ∧ x 2 +5y 2 =39
x 2 + xy =1260 ∧ y 2 + xy =504
x 2 xy =2(a +1) ∧ y 2 xy =2(1 a)
(x 6)2 +(y +2)2 =50 ∧ x 2 + y 2 +8x 6y =0
x 2 + y 2 =3x +4y ∧ x + y = a
x 2 + y 2 2y 1=0 ∧ x 2 = ay +1
x 2 + y 2 =4a 2 ∧ x + y =2a
xy = 2a 2 ∧ x + y = a
x + y x = a ∧ 1+ xy a +1 = a 2
x + y = a ∧ x 2 + y 2 = bxy
= x 2 ∧ x 2 + y 2 + px + qy + r =0 y = x 2 ∧ x 4 +(1+ q )x 2 + px + r =0 x x 4 +(1+ q )x 2 + px + r =0 1+ q = a p = b r = c x 4 + ax 2 + bx + c =0 y = x 2 x 2 + y 2 + bx +(a 1)y + c =0. Az 4 + Bz 3 + Cz 2 + Dz + E =0(A =0) z = x B 4A
N | x y x|y z y = xz y x x y z z1 y = xz y = xz1 xz = xz1 z = z1 | N
x y z x|x
x|y y |x x = y
x|y y |z x|z N N x y x y y x N | 2|3 3|2 | + N
x|y x|z x|y + z
x|y x|z x|y z y>z
x|y x|z x|yz x|y xz |yz
x|y x|z u v y = xu z = xv y + z = xu + xv = x(u + v ) x|y + z a b a|b
x ax = b b a b a b : a b/a b a a|b | Z y x =0 x|y
y x y x y : x y /x
Z Z | N b/a b a a|b a b
(q,r ) q,r ∈ N0 q r b = aq + r r<a.
{an | n ∈ N ,an b} b aq am aq b aq<aq + a b<aq + a
b aq<a aq b b aq 0 0 b aq<a.
b aq = r b = aq + r 0 r<a q,r ∈ N0
(q,r ) (q1 ,r1 )
b = aq + r,b = aq1 + r1 , 0 r<a, 0 r1 <a.
aq + r = aq1 + r1 ,
q>q1 ⇔ r<r1 .
q>q1 r r1 aq>aq1 r r1
aq + r>aq1 + r1 q<q1 r>r1
q>q1 r<r1
a(q q1 )= r1 r.
q>q1 q q1 > 0 q q1 1
r1 r a r<a r1 <a
r1 r<a
q<q1 q = q1 r = r1 q r b a
(q,r ) 17=3q + r {3n| n ∈ N 3n 17} q =5 r =2 17=3 5+2 17=3 6 1 17=3 4+5, (6, 1) (4, 5) r<a 5 < 3
{3n| n ∈ N 3n 17}
{4n| n ∈ N 4n 16689} 16689:4 16689=4 4172+1
b a> 0 q r b = aq + r (0 r<a)
( 24, 3) 0 < 3 < 7 165=7 · ( 24)+3 a b c ∈ Z a b c a b c< (|a|, |b|) a b a b (a,b)
(a,b)
A = {1, 1, 2, 2, 3, 3, 6, 6, 9, 9, 18, 18}, B = {1, 1, 2, 2, 3, 3, 5, 5, 6, 6, 10, 10, 15, 15, 30, 30}
A ∩ B = {1, 1, 2, 2, 3, 3, 6, 6}. (18, 30)=6 (18, 30)=6 a b
(a,b)=(b,a)
(a,a)= |a| (a, 1)=1
(a, 0)= |a| a|b (a,b)= |a|
(a,b)=( a,b)
(a,b)=(a, b)
(a,b)=( a, b)
(a,b)=(|a|, |b|)
a> 0 b q r
b = aq + r (0 r<a), (a,b)=(a,r ) x|a x|r x|b (a,r )|a
(a,r )|r (a,r )|b (a,r ) a b
(a,r ) (a,b). r = b aq, (a,b) (a,r ) (a,b)=(a,r ) a b a b a < b a|b (a, b) = a a|b q r
b = aq + r (r < a) (a, b) = (a, r ) (a, b)
(a, r ) r < a r |a (a, r ) = r r |a q1 r1
a = rq1 + r1 (r1 < r ), (a, r ) = (r, r1 ) r r1 q2 r2 r = r1 q2 + r2 (r2 <r1 ) rk 1 = rk qk +1 + rk +1 , rk +1 =0 (a,b)=(a,r )=(r,r1 )= ··· =(rk 1 ,rk )=(rk ,rk +1 )=(rk , 0)= rk , rk =(a,b)
45736=2540 18+16
2540=16 · 158+12 16=12 1+4 12=4 3+0, (45736, 2540)=4
a b a b m n (a,b)= ma + nb. a b a<b a b
b = aq1 + r1 (0 <r1 <a)
a = r1 q2 + r2 (0 <r2 <r1 )
r1 = r2 q3 + r3 (0 <r3 <r2 )
r2 = r3 q4 + r4 (0 <r4 <r3 )
rk = rk +1 qk +2 +0
rk +1 =(a,b)
r1 = q1 a + b, r1
m1 = q n1 =1
m1 a + n1 b m1 n1
r2 = a r1 q2 = a ( q1 a + b)q2 =(1+ q1 q2 )a q2 b,
r2 m2 a + n2 b m2 n2
m2 =1+ q1 q2 n2 = q2
ri (1 i k +1)
mi a + ni b mi
ni m n rk +1 =(a,b)=
ma + nb a b
(a,b)=(|a|, |b|) (45736, 2540)=4 m n
4=45736m +2540n.
16=45736 2540 · 18
12=2540 16 158
=2540 (45736 2540 18) 158
=45736 ( 158)+2540(1+18 158)
=45736( 158)+2540 · 2845
4=16 12 1
=(45736 2540 18) (45736( 158)+2540 2845)
=45736(1+158)+2540( 18 2845)
=45736 · 159+2540 · ( 2863)
m =159 n = 2863
a b c m
(ma,mb)= m(a,b) (ca,cb)= |c|(a,b) c|a c|b a c , b c = (a,b) c a b (a,b)=1 a b m n ma + nb =1 (ac,b)=(c,b) c b|ac b|c a b a (a,b) , b (a,b) =1. (a,b)|a (a,b)|b a (a,b) , b (a,b) = (a,b) (a,b) =1.
45736=2540 18+16, 2540=16 158+12, 16=12 1+4
45736 2540 =18+ 16 2540 =18+ 1 2540/16 2540 16 =158+ 12 16 =158+ 1 16/12 16 12 =1+ 4 12 =1+ 1 12/4 , 45736 2540 =18+ 1 158+ 1 1+ 1 3
t x = Bt x = Bt ABt = By y = At x = b (a,b) t,y = a (a,b) t, t 4x 2y =5 x y
(a,b)|c (x0 ,y0 ) ax0 + by0 = c (a,b)|a (a,b)|b (a,b)|ax0 + by0 (a,b)|c (a,b)|c m n am + bn =(a,b) c/(a,b) a mc (a,b) + b nc (a,b) = c, (x0 ,y0 ) x0 = mc (a,b) ,y0 = nc (a,b)
(a,b)|c (x,y ) ax0 + by0 = c,ax + by = c a(x x0 )+ b(y y0 )=0, aX + bY =0, X = x x0 Y = y y0 X = b (a,b) t,Y = a (a,b) t, t (a,b)|c x = x0 + b (a,b) t,y = y0 a (a,b) t,
x0 y0 t (a,b)|c (a,b)
ax + by = c (a,b)=1 3x +15y =7 (3, 15) 17x +13y = c c m n 17m +13n =1 17=1 · 13+4, 13=3 · 4+1, 4=1 · 4, 1=13 4 3=13 (17 1 13) 3=17 ( 3)+13 4; m = 3 n =4
0 = 3c,y0 =4c. 17x +13y =0 x =13t y = 17t t x = 3c +13t,y =4c 17t, t p> 1 p1 ,p2 ,...,pn
1 ,p2 ,...,pn
= p1 p2 pn +1. a> 1 a>p1 ,...,a>pn a
1 ,p2 ,...,pn
pk (1 k n)
p1 ,p2 ,...,pm
p1 ,p2 ,...,pm +1 p1 ,p2 ,...,pm
2 3+1=7
2 3 7 43+1=1807
2 3 7+1=43
ab p a b p a (a,p)=1 m
n ma + np =1 b mab + npb = b p|ab k ab = pk b = mpk + npb = p(mk + nb), p|b p p|a1 a2 an p|ak k =1, 2,...,n
N = p1 p2 ··· pr
N = q1 q2 qs
p1 p2 pr = q1 q2 qs . p1
p1 |qk k =1, 2,...,s qk
p1 = qk p2 p2 = qj j =1, 2,...,s j = k 1=1
{p1 ,p2 ,...,pr } = {q1 ,q2 ,...,qs }
N = p1 p2 pm , p1 ,p2 ,...,pm
N = p α1 1 p α2 2 p αk k α1 ,α2 ,...,αk
600=2 2 2 3 5 5=23 3 52 p p 1 p +1 k p 1=
4k p +1=4k p =4k +1 p =4k 1 4k +1 (k ∈ Z) = {4k ± 1| k ∈ Z}
b a q r b =
qa + r (0 r<a) b a a Z 2k k ∈ Z
2k +1 k ∈ Z
A0 = {3k | k ∈ Z}
A1 = {3k +1| k ∈ Z}
A2 = {3k +2| k ∈ Z} m
0, 1,... m 1 Z m
A0 = {mk | k ∈ Z} m
A1 = {mk +1| k ∈ Z} m
Am 1 = {mk + m 1| k ∈ Z} m m 1 a b Ak (0 k m 1) m a b m m
≡ b ( m)
b m a = mq1 + r,b = mq2 + r (0 r<m),
a b = mq1 mq2 = m(q1 q2 ),
m|a b m|a b a ≡ b ( m) a ≡ b m ⇔ m|a b. m m>a m>b 10 ≡−3( 13) m
a ≡ a ( m)
a ≡ b ( m) b ≡ a ( m)
a ≡ b ( m) b ≡ c ( m) a ≡ c ( m) m
a ≡ b ( m) c ≡ d ( m)
a + c ≡ b + d ( m) a c ≡ b d ( m)
ac ≡ bd ( m)
a ≡ b ( m) c ≡ d ( m)
m|a b m|c d m|(a b)+(c d)
m|(a + c) (b + d) p ab ≡ 0( p) a ≡ 0( p)
b ≡ 0( p) ab ≡ 0( p) p|ab p p|a p|b p 2 3 ≡ 0( 6) 2 ≡ 0( 6) 3 ≡ 0( 6)
a =0( p) ab ≡ ac ( p) b ≡ c ( p) p ab ≡ ac ( p) p|ab ac p|a(b c) a =0 ( p) p a p b c b ≡ c ( p) p
N = a0 10n + a1 10n 1 + + an 1 10+ an ,
m +1 6m +5 m {6k ± 1|k ∈ N}
(19332, 78696)=36 (75422, 106210)=86 a b (a + b,a b)= c
c ∈{1, 2}
d =(354, 273) m n 354m +
273n = d 21x +3y =15
x =5+ t y = 30 7t t x ∈ Z
x ≡ 0( 5)
x ≡ 3( 5) x +1 ≡ 3( 5) 3x ≡ 2( 5) x =5k x =3+5k x =2+5k x =4+5k k
x 2 ≡ 1( 7) x 4 ≡ 1( 7) ±1 ±6 ±1 x ∈ Z ax ≡ b ( m), ∗ a =0 b m (a,m)|b x (∗) ax b m ax = b + km k x b + km a b + km = na n (∗) n k na km = b
436 7 261
(1, 1, 1) (4, 1, 0). (1, 2, 3) ( 1, 0, 1) t 2 , 7t 2 ,t t (4+4t, 1+13t, 5t) t
(2, 2, 3) (t,t,t) t (1, 1, 1, 2) a =4:( 2t, t,t) t a = 45/7 33t/20 67t/140 t) t a =2 ((a 2) 1 (2a 1)(a 2) 2 (5 4a)(a 2) 2 a =2 a =0 a = ±1 ((a 2 +1)a 1 (1+ a) 1 (1 a)a 1 (1+ a) 1 (1 a)a 1 ) a =0 a = 1 a =1 (t +1, 0,t) t a =5 (1 3t,t, 1 t) t a =5 a =8 (u,v, 1, 2 u 3v /2) u v a =8 (t, 2(2 t)/3, 1, 0) t
(10, 9, 3) ( 5, 18, 6) (5, 18, 6) (0, 1, 10) ( 1, 8, 4) √10 √38 π 3 , π 4 , π 3 1 2 (x1 + x2 ,y1 + y2 ,z1 + z2 )
√5 21 1 2 (| #« a |2 + | #« b |2 + | #« c |2 ) (4, 2, 3) (3 p 2q,p,q ) p,q (1 2t, 2+3t,t) t 1 7 (10, 21, 5) 1
( 22, 9, 12) 1 3
(2t,t +2,t) t (0, 2, 0) #« c × #« b 4 #« j 1 #« i + #« j 49 √323 ± 2 3 , 2 3 , 1 2 √154 (12, 1, 3) #« o D =( 1, 1, 2) 13, 26 √46
(2
2 x + √2 2 y √2=0 √5 3 x 2 3 y 1=0 y kx n ±√1+ k 2 =0 (2, 1) ( 1, 2) (6, 2) (2, 1) (2, 1) ( 2, 5) ( 3, 2) 5 11 , 7 11
a =
b = 3 26 1 4 , 25 4 x 3y +26=0 2x +3y 9=0 5x +3y +8=0 17 3 , 61 9 3x +2y =0 2x 3y 13=0 3x 4y +32=0 4x +3y 24=0 3x 4y +7=0 4x +3y + 1=0 x +7y 31=0 x0 (b2 a2 ) 2a(by0 + c) a2 + b2 , y0 (a2 b2 ) 2b(ax0 + c) a2 + b2 10x 33y +49=0 33x +10y 195=0 2x + y +1=0 x 2y +3=0 3x y +4=0 ( 1, 1) 9x y +23=0 x =1 2x +4y 11=0 10x 4y 1=0 1, 9 4 11x y 28=0 x + y =8 (10, 5) 3x 5y +4=0 3x 5y 22=0 x +7y 16=0 x +7y +10=0 63x 28y +172=0 y = 11 13 x = 5 13 7y 9x 1=0 x +2y 11=0 3x + y 13=0 x 3y +9=0 4x 5y +22=0 4x + y 18=0 2x y +1=0 x + y +5=0 4x +3y 14=0 3x 4y +27=0 3x 4y +2=0 4x +3y +11=0
5 13 , 6 13
8x +12y +5=0
x
y 32=0 √
4x 3y 25=0 3x +4y 25=0 x =2 7x +24y 134=0 4x 4y +3=0 2x +2y 7=0 x =0 y =0 x 3y =0 8x +14y 71=0 187x 209y +227=0 13x +9y 67=0 7 6 , 4 3x +4y 22=0 2x 7y 5=0 3x +5y 23=0 x 3y +26=0 2x +3y 9=0 5x +3y +8=0 x y +4=0 3x y +2=0 y =5 x 3y +8=0 5x +3y 34=0 2x +3y 21=0 5y =12x ( 1, 2) ( 3, 2) (3, 5) ( 37, 45) 3x + y =0 x 3y =0 8x + y +9=0 4x +7y +11=0 2x y =3 y = x 1 4x +3y +20=0 4x +3y =0 3x
y
153x +76y =928 3, √13 2 , 3, √13 2 (±√c, 0) (0 ± √c) (x +4)2 +(y 6)2 =1 (x 4)2 +(y +6)2 =1 (x +6)2 +(
,
,
(x 5)2 +(y +2)2 =20 x 9 5 2 + y 22 5 2 =20 (x 3)2 +(y 1)2 =5 x + 17 6 2 + y + 19 6 2 = 5 4 x 1+ 3 √2 2 + y 2+ 1 √2 2 = 5 2 2x + y =7 3x +2y =0 2x 3y 13=0 3x +2y +13=0 3y =4x +23 4y +3x =14 π 2 (1, 9) (2, 2) 25 (1, 11) (5, 1) ( 4, 4) π 2 5 2 4 3 x +2y +5=0 1 2 , 1 4 π 4 1 3 x 5y =4 x +5y =24 ( 2, 1) (x +2)2 + y 2 =18 (x 6)2 +(y 3)2 =20 (x 5)2 +(y 5)2 =25 (x 29)2 +(y 29)2 =292 20 13 , 95 13 (x 3)2 +(y 2)2 =17 (x 1)2 +(y 10)2 =17 (x 6)2 +(y 7)2 =17 (x +2)2 +(y 5)2 =17 aA + bB =2(c + C ) x 2 + y 2 +2y 4=0 π 4 π 2 (x 2)2 +(y 1)2 =1 x + 13 6 2 + y 5 18 2 = 1745 162 2y = x +15 2y + x +15=0 (x +3)2 +(y +3)2 =72 (x 2)2 +(y +1)2 =13 5x + y =4 (x 5)2 + (y +1)2
(
22 5 , 34 5 2x y 5=0 x 2y +5=0 y = x +2 y = x 2 √2 y =2x 6 y = 2x 6 2x 3y +8=0 3x +2y 14=0 2√5 (1
,
,
|
|
| =5 |n| >
y 5=0 2x 3y +25=0 x +4y 16=0 x 4y +16=0 18 x 2y ± 8=0 2x +3y ± 10=0 x 2 +9y 2 =225 x2 20 + y 2 5 =1 y = ±x ± 3 y = ±x ± 5 2 19 1 8 0 (2, 3) π 4 (7, 12) 1 5 x + y 1=0 3x + y 8=0 ±5 x y 1=0 x + y 7=0 5x 3y 16=0 13x +5y +48=0
71√10
50 (2, 3) (0, 0) ( 5, 10) ( 1, 8) ( 1, 3) (7, 9) x 2 +2y 2 =1 3x 2 +4y 2 =5 (y +1)2 =2(x 5) 3x +4y =6 x 2 + y 2 2=0 4x 2 +12xy +8y 2 4=0 x 2 +4xy + y 2 =0 2x 2 +4xy + y 2 16=0 x 2 + xy 1=0 19x 8y 5=0 x 2 +4y 2 =9 y =2x +1 172 · 72 · xy +13 · 9 · 6 · 10x 17 · 4 · 35 =0 14x 2 6y 2 =1 1 3 2x 2 +7y 2 =2 2 x 2 y 2 =1
(2, 6) (5, 8) ( 4, 1) 7 3 y +2x +4=0 y =2x +4 y +2x =4 b2 5 3 , 1 3 (1, 1) (6, 4) ( 2, 8) (2, 1) (6, 0) (5, 3) 11y 2x =22 32y 49x +211=0 y = 4 x + y =3 4x 3y =80 y = 8 4x 3y =0 4x 3y +80=0 y = 8 4x 3y =0 y =2x y = 2x (0, 0) x =0 y = x (3, 4) x 1 5 2 + y 3 5 2 = 9 5 x + 8 5 2 + y 6 5 2 = 9 5 2+ √3 2 , 1 2 2 √3 2 , 1 2 (2 √3, 1) 2y =1 y = x√3 2√3 y = x√3 2√3+2 y = 1 x 2+ √3 2 2 + y + 1 4 2 = 9 16 x 7 2 2 + y + 3 2 2 = 125 2 ( 1, 2) (1, 0) (5, 4) (3, 6) y = 3x ± 2√10 3y = x ± 2√10 y = 5
=1, 2,..., 19 (1, 5, 25,...) (1, 6, 36,...) a 1 q n q 1 q 2n 1 q
=1 b = 1 c =3 1/2 (4+√35)/3 4/3 (√35 4)/3 3/2, 2, 5/2 2/3 (5+√13)/3 (5 √13)/3 1/2 6 3/2 2, 4
(2, 1) (2, 1) ( 2, 1) ( 2, 1) (9, 7) (9, 7) ( 9, 7) ( 9, 7)
(9, 4) (9, 4) ( 9, 4) ( 9, 4) (4, 2) ( 4, 2) √15, 1 3 √15 √15, 1 3 √15
(6, 4) ( 6, 4) (8/√3, 2√3) ( 8/√3, 2/√3) 10|a| √39 , |a| √39 , 10|a| √39 , −|a| √39 (2|a|, −|a|) ( 2|a|, |a|) (5, 1) ( 5, 1) (√13, √13) ( √13, √13)
(2, 1) ( 2, 1) 8 3 , 1 3 , 8 3 , 1 3 (12, 3) ( 12, 3) ( a,b) (a, b) ( b,a) (b, a) (3, 1) ( 3, 1) (√2, 2√2) ( √2, 2√2) (2, 8) ( 2, 8) 17 2 , 5 , 17 5 , 5
(√2, 3√2) ( √2, 3√2) √14, 3 7 √14 , √14, 3 7 √14
( 2, 10/3) (13/2, 1/2) (3√2, √2) ( 3√2, √2) ( 6, 6) (6/5, 3/5) (9, 9) ( 9/5, 9/10) (5, 4), ( 5, 4) (11/5, 3/5), (7/5, 9/5) ( 11/5, 3/5), ( 7/5, 9/5) (5, 2), (25/2, 11/2) (4, 3), (2/3, 1/3) 14+ √109 3 , 8+ √109 3 14 √109 3 , 8 √109 3
( 4, 2) ( 4, 3) (3, 2) (3, 3) (1, 3), ( 1/5, 3/5) (5, 4), ( 5, 4) ab + ab(ab 4) 2b , ab + ab(ab 4) 2a ab ab(ab 4) 2b , ab ab(ab 4) 2a ab2 a2 + b2 , a2 b a2 + b2
√1111 29 , 6+2√1111 29
ПРОГРАМ МЗ
(ПРИРОДНО-МАТЕМАТИЧКИ СМЕР ГИМНАЗИЈЕ)
ПРОГРАМ M14
(ХИДРОМЕТЕОРОЛОШКА СТРУКА)
5 часова недељно; 180 часова годишње
Полиедри (25 часова)
Рогаљ, триедар. Полиедар, Ојлерова тео-
рема; правилан полиедар.
Призма и пирамида; равни пресеци при-
зме и пирамиде.
Површина полиедра; површина призме,
пирамиде и зарубљене пирамиде.
Запремина полиедра; запремина квадра,
Кавалијеријев принцип. Запремина при-
зме, пирамиде и зарубљене пирамиде.
Обртна тела (20 часова)
Цилиндрична и конусна површ, обртна
површ.
Прав ваљак, права купа и зарубљена пра-
ва купа. Површина и запремина правог
кружног ваљка, праве кружне купе и за-
рубљене кружне купе.
Сфера и лопта; равни пресеци сфере и
лопте. Површина лопте, сферне калоте и
појаса. Запремина лопте.
Уписана и описана сфера полиедра, пра-
вог ваљка и купе.
Вектори (15 часова)
Правоугли координатни систем у простору, пројекције вектора; координате
вектора.
Скаларни, векторски и мешовити прои-
звод вектора, детерминанте другог и трећег реда. Неке примене вектора.
Аналитичка геометрија у равни (50 часова)
Растојање две тачке. Подела дужи у датој размери. Површина троугла.
Права, разни облици једначине праве; угао између две праве; растојање тачке
од праве.
Системи линеарних једначина, Гаусов
поступак. Систем линеарних неједначина са две непознате и његова графичка
интерпретација; појам линеарног програмирања.
Криве линије другог реда: кружница, елипса, хипербола, парабола (једначине; међусобни односи праве и кривих другог реда, услов додира, тангента; заједничка својства).
Математичка индукција. Низови (38 часова)
Математичка индукција и њене примене.
Елементарна теорија бројева (дељивост, прости броjеви, конгруенције).
Основни појмови о низовима (дефиниција, задавање, операције).
Аритметички низ; геометријски низ; примене.
Једноставније диференцне једначине.
Гранична вредност низа; број е.
Комплексни бројеви и полиноми (20 часова)
Појам и примери алгебарских структура (група, прстен, поље).
Поље комплексних бројева. Тригонометријски облик комплексног броја. Моаврова формула. Неке примене комплексних бројева.
Полиноми над пољем комплексних бројева. Основна теорема алгебре и неке њене последице. Вијетове формуле.
Системи алгебарских једначина вишег реда.
