首页 高等代数与解析几何 习题解答12

高等代数与解析几何 习题解答12

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高等代数与解析几何 习题解答12 1›�Ù õ�õ‘ª § 1 õ�õ‘ª 1. � f(x1, · · · , xn)´ê K þ� n�àgõ‘ª. y²: XJ3ê K þ� n�õ‘ª g(x1, · · · , xn) † h(x1, · · · , xn), ¦ f(x1, · · · , xn) = g(x1, · · · , xn)h(x1, · · · , xn), K g(x1, · · · , xn) † h(x1, · · · , xn)Ñ´àgõ‘ª. y²: � deg f = m, deg g = k...

高等代数与解析几何 习题解答12
1›�Ù õ�õ‘ª § 1 õ�õ‘ª 1. � f(x1, · · · , xn)´ê K þ� n�àgõ‘ª. y²: XJ3ê K þ� n�õ‘ª g(x1, · · · , xn) † h(x1, · · · , xn), ¦ f(x1, · · · , xn) = g(x1, · · · , xn)h(x1, · · · , xn), K g(x1, · · · , xn) † h(x1, · · · , xn)Ñ´àgõ‘ª. y²: � deg f = m, deg g = k, deg h = l. - g = gp + gp+1 + · · · + gk, h = hq + hq+1 · · ·+ hl, Ù¥ gi, hj ©O i, j gàgõ‘ª, … gp, hq ´©)¥gê$�àgõ‘ ª, k + l = m, K f = gphq + m∑ t=p+q+1 (∑ i+j=t gihj ) . Ïd� p + q < mž f Ø´àgõ‘ª. p+ q = k + l = m ŒíÑ p = k, q = l, Ïd g = gk, h = hl Ñ´àgõ‘ª. 2. � f(x, y) ∈ K[x, y]. y²: XJ f(x, x) = 0, K x− y | f(x, y). y²: � f(x, y) = n∑ k=0 ak(x)y k, K f(x, y) = f(x, y)− f(x, x) = n∑ k=0 ak(x)(y k − xk) = (y − x) n∑ k=1 ak(x)(y k−1 + yk−2x+ · · · + yxk−2 + xk−1). Ïd x− y | f(x, y). · 160 · 1›�Ù õ�õ‘ª ∗3. OŽe�1�ª: ∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 x1 − a1 1 x1 − a2 · · · 1 x1 − an 1 x2 − a1 1 x2 − a2 · · · 1 x2 − an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 xn − a1 1 xn − a2 · · · 1 xn − an ∣∣∣∣∣∣∣∣∣∣∣∣∣ . ): r�1�ªP Dn(x1, · · · , xn, a1, · · · , an). K Dn(x1, · · · , xn, a1, · · · , an) = G(x1, · · · , xn, a1, · · · , an) F (x1, · · · , xn, a1, · · · , an) , Ù¥ G † F Ñ´ x1, · · · , xn, a1, · · · , an �õ‘ª. ´ F (x1, · · · , xn, a1, · · · , an) = ∏ 1≤i,j≤n (xi − aj). du Dn(x1, · · · , xi, · · · , xi, · · · , xn, a1, · · · , an) = 0, Dn(x1, · · · , xn, a1, · · · , ai, · · · , ai, · · · , an) = 0, Œ� G(x1, · · · , xn, a1, · · · , an) = ∏ 1≤i xi † aj �gê (Ñ´ n− 1 g), Œ G1 = cn ´˜‡~ê. Ïd Dn(x1, · · · , xn, a1, · · · , an) = ∏ 1≤i n. y²: � g(x) = n∑ i=1 xk+1i f(x) x− xi , K g(x) = 0½ deg g(x) < n. xk+1f ′(x)− g(x) = n∑ i=1 xk+1f(x) x− xi − n∑ i=1 xk+1i f(x) x− xi = ( xk+1 − xk+1i x− xi ) f(x) = n∑ i=1 k∑ j=0 (xk−jxjif(x) = k∑ j=0 ( n∑ i=1 (xk−jxji ) f(x) = ( k∑ j=0 xk−jsj ) f(x) = (s0x k + s1x k−1 + · · ·+ sk−1x+ sk)f(x). =�¤y. (2) '��ª xk+1f ′(x) = (s0x k + s1x k−1 + · · ·+ sk−1x+ sk)f(x) + g(x) ü> n g‘Xê, du g(x)�gê < n½ g(x) = 0, ¤± xk+1f ′(x)�ng‘Xê=(s0x k+ s1x k−1 + · · ·+ sk−1x+ sk)f(x)�ng‘Xê, ¤±� k ≤ nž, (n− k)(−1)kσk = sk − σ1sk−1 + σ2sk−2 + · · ·+ (−1)kσks0, = sk − σ1sk−1 + σ2sk−2 + · · ·+ (−1)k−1σk−1s1 + (−1)kkσk = 0. ∗§ 3 (ª · 165 · � k > n, 0 = sk − σ1sk−1 + σ2sk−2 + · · ·+ (−1)nσnsk−n, =�¤y. ∗8. ^Ð�é¡õ‘ªL« s2, s3, s4, s5. ): s2 = σ 2 1 − 2σ2, s3 = σ 3 1 − 3σ1σ2 + 3σ3, s4 = σ 4 1 − 4σ21σ2 + 2σ22 + 4σ1σ3 − 4σ4, s5 = σ 5 1 − 5σ31σ2 + 5σ1σ22 + 5σ21σ3 − 5σ2σ3 − 5σ1σ4 + 5σ5. ∗§ 3 (ª 1. OŽe�õ‘ª�(ª: (1) f(x) = x3 − 3x2 + 2x+ 1, g(x) = 2x2 − x− 1; (2) f(x) = 2x3 − 3x2 − x+ 2, g(x) = x4 − 2x3 − 3x+ 4; ): (1) Res(f, g) = (−1)2·3 Res(2x2−x−1, f) = (−1)6 ·23 ·f ( −1 2 ) f(1) = −7. (2) f(x), g(x) kú�Š 1, ¤±(ª Res(f, g) = 0. 2. � λ�ۊž, e�õ‘ªkú�Š: (1) f(x) = x3 − λx+ 2, g(x) = x2 + λx+ 2; (2) f(x) = x3 + λx2 − 9, g(x) = x3 + λx− 3. ): (1) Res(f, g) = −4(λ+ 1)2(λ− 3), �� λ = −1½ 3žkú�Š. (2) Res(f, g) = 9(λ2 + 12)(λ2 + 2), �� λ = ±2√3i½ ±√2ižkú� Š. 3. ¦e�­‚�†�‹I§: (1) x = t2 + t− 1, y = 2t2 + t− 1; (2) x = t− 1 t2 + 1 , y = t2 + t− 1 t2 + 1 . ): (1) 4x2 − 4xy + y2 + 5x− 3y + 1 = 0. (2) 5x2 − 6xy + 2y2 + 5x− 3y + 1 = 0. 4. � λÛŠž, e�õ‘ªk­Š? (1) f(x) = x3 − 3x+ λ; (2) f(x) = x4 − 4x3 + (2− λ)x2 + 2x− 2. ): (1) 2,−2; (2) −1,−3 2 , 7 2 + 9 2 √ 3i, 7 2 − 9 2 √ 3i. · 166 · 1›�Ù õ�õ‘ª 5. ¦e�§|�): (1)  5x 2 − 6xy + 5y2 = 16, 2x2 − xy + y2 − x− y = 4; (2)  x 2 + y2 + 4x− 2y = −3, x2 + 4xy − y2 + 10y = 9. ): (1) Resy(f, g) = 32(y 4 − y3 − 3y2 + y + 2),  x = 1y = −1  x = −1y = 1  x = 2y = 2 (2) Resx(f, g) = 4(5x 4 + 40x3 + 106x2 + 104x + 33),  x = −1y = 2  x = −3y = 0   x = −2 + 3 5 √ 5 y = 1 + 1 5 √ 5   x = −2− 3 5 √ 5 y = 1− 1 5 √ 5 6. ¦e��I­‚��:‹I: (1) � x2 + y2 − 3x− y = 0 †V­‚ x2 + 2xy − y2 − 4y − 2 = 0; (2) V­‚ 4x2 − 7xy + y2 + 13x − 2y − 3 = 0 †V­‚ 9x2 − 14xy + y2 + 28x − 4y − 5 = 0. ): (1) (1,−1), ( 3 2 + 1 2 √ 2, 1 2 + √ 2 ) , ( 3 2 − 1 2 √ 2, 1 2 −√2 ) ; (2) (0,−1), (1, 2), (2, 3), (−2, 1). 7. y²(ª�e�5Ÿ: � f(x), g(x)©O´ n g† m gõ‘ª. K (1) Res(f, g) = (−1)mnRes(g, f); (2) Res(af, bg) = ambnRes(f, g); ∗(3) Res((x− a)f, g) = g(a)Res(f, g). y²: (1), (2) w,. 8y (3). � f(x) = a0x n + a1x n−1 + · · ·+ an, g(x) = b0xm + b1xm−1 + · · ·+ bm, K (x− a)f(x) = a0xn+1 + (a1 − a0a)xn + · · ·+ (an − an−1a)x− ana. ∗§ 3 (ª · 167 · Res((x− a)f, g) =∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ a0 a1−a0a a2−a1a · · · an−an−1a −ana a0 a1−a0a a2−a1a · · · an−an−1a −ana . . . . . . . . . · · · . . . . . .  n a0 · · · · · · · · · an−an−1a −ana b0 b1 b2 · · · bm−1 bm b0 b1 b2 · · · bm−1 bm . . . . . . . . . · · · . . . . . .  m+1 b0 · · · · · · · · · bm−1 bm ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ g1˜�å, ˆ�¦ a \��˜�, †–�˜�, Œ� Res((x− a)f, g) =∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ a0 a1 a2 · · · an 0 a0 a1 a2 · · · an 0 . . . . . . . . . · · · . . . . . .  n a0 · · · · · · · · · an 0 b0 b1+b0a b2+b1a+b0a 2 · · · · · · g(a) · · · g(a)am b0 b1+b0a · · · · · · · · · · · · g(a)am−1 . . . . . . . . . · · · . . . . . . ...  m+1 b0 · · · · · · · · · · · · g(a) ∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣ l�˜1å, ˆ1¦ (−a) \�c˜1, †�1 n+ 11, 2U�˜�Ðm, Œ� Res((x− a)f, g) = g(a)Res(f, g). ∗8. � f(x) = a(x− x1) · · · (x− xn), g(x) = b(x− y1) · · · (x− ym). y²: Res(f, g) = am n∏ i=1 g(xi) = (−1)mnbn m∏ j=1 f(yj) = a mbn n∏ i=1 m∏ j=1 (xi − yj). y²: Res(f, g)= amRes((x− x1) · · · (x− xn), g(x)) = amg(x1)Res((x− x2) · · · (x− xn), g(x)) = amg(x1)g(x2) · · · g(xn) = ambn n∏ i=1 m∏ j=1 (xi − yj) = (−1)mnbn m∏ j=1 f(yj). ∗9. y²: Res(f(x), g1(x)g2(x)) = Res(f(x), g1(x))Res(f(x), g2(x)). · 168 · 1›�Ù õ�õ‘ª y²: � f(x) = a(x− x1) · · · (x− xn), K Res(f, g1g2) = a deg g1g2 n∏ i=1 g1(xi)g2(xi) = adeg g1 n∏ i=1 g1(xi)a deg g2 n∏ i=1 g2(xi) = Res(f, g1)Res(f, g2). ∗10. � f Ä˜õ‘ª, y²: é?¿õ‘ª h, Res(f, g) = Res(f, g + hf). y²: � f(x) = (x− x1)(x− x2) · · · (x− xn), K Res(f, g + hf) = n∏ i=1 (g(xi) + h(xi)f(xi)) = n∏ i=1 g(xi) = Res(f, g). ∗11. |^SK 7– 10y²�(ª5ŸOŽe�õ‘ª�(ª: (1) f(x) = xn + x+ 1, g(x) = x2 − 3x+ 2; (2) f(x) = xn + 1, g(x) = (x− 1)n; (3) f(x) = a0x n + a1x n−1 + · · ·+ an−1x+ an, g(x) = a0x n−1 + a1x n−2 + · · ·+ an−2x+ an−1. (4) f(x) = xn − 1 x− 1 , g(x) = xm − 1 x− 1 ; ): (1) Res(f, g) = (−1)2n(1 + 1 + 1)(2n + 2 + 1) = 3(2n + 3). (2) Res(f, g) = (−1)n · 2n. (3) du f(x) = xg(x) + an, ¤± Res(f, g) = (−1)n(n−1) Res(g, f) = (−1)n(n−1) Res(g, an) = an−1n . (4) (a) X (m,n) = d > 1, K xn − 1 x− 1 † xm − 1 x− 1 kú�Š, Ïd Res(f, g) = 0. (b) X (m,n) = 1, ؔ� n > m, K n = mq + r, 0 ≤ r < m. w, (m, r) = 1. K xn − 1 x− 1 = xmqxr − 1 x− 1 = (xmq − 1)xr + xr − 1 x− 1 , ∗§ 4 Ǟ�{ · 169 · l Res ( xn − 1 x− 1 , xm − 1 x− 1 ) = (−1)m−1)(n+r) Res ( xr − 1 x− 1 , xm − 1 x− 1 ) . ·‚y² (m− 1)(n + r)˜½´óê. X m − 1´óê, K(ؤá. y� m− 1´Ûê, K móê, l n ´Ûê, r´Ûê, u´ n+ r´óê. l Res ( xn − 1 x− 1 , xm − 1 x− 1 ) = Res ( xr − 1 x− 1 , xm − 1 x− 1 ) . 2^ rØm, ŠâÎ=ƒØ{��n, d (m, r) = 1 Œ� Res ( xr − 1 x− 1 , xm − 1 x− 1 ) = · · · = Res ( xr ′ − 1 x− 1 , 1 ) = 1. =� (m,n) = 1ž Res ( xn − 1 x− 1 , xm − 1 x− 1 ) = 1. ∗12. � f(x) = a0x n + a1x n−1 + · · ·+ an−1x+ an ∈ K[x], y²: f(x)��Oª D(f) = (−1)n(n−1)2 a−10 Res(f, f ′). y²: D(f) = a2n−20 ∏ 1≤iread ‘d:/mapleuser/wsolve2‘: >P1:=-12*x2^2+7*x1*x2-2; >P2:=-2*x3+x1^2; >P3:=-x3^2+x1*x2+2; >PS1:={P1,P2,P3}; >ord:=[x3,x2,x1]; >B1:=basset(PS1,ord); B1 := [−2x3 + x21,−12x22 + 7x1x2 − 2] >R1:=remseta(PS1,B1,ord); R1 := {8 + 4x1x2 − x41} >PS2:={op(B1)} union R1; >B2:=basset(PS2,ord); B2 := [−2x3 + x21, 8 + 4x1x2 − x41] >R2:=remseta(PS2,B2,ord); R2 := {64x21 + 192− 48x41 − 7x61 + 3x81} >PS3:={op(B2)} union R2; >B3:=basset(PS3,ord); B3 := [−2x3 + x21, 8 + 4x1x2 − x41, 64x21 + 192 − 48x41 − 7x61 + 3x81] >R3:=remseta(PS3,B3,ord); R3 := {} >J:=Initial(B3[1],ord)*Initial(B3[2],ord)*Initial(B3[3], ord); J := x1 >solveas(B3,ord,{x1}); ù�)� 8 |): ∗§ 4 Ǟ�{ · 171 ·   x1 = 2, x2 = 1, x3 = 2;   x1 = −2, x2 = −1, x3 = 2;   x1 = √ 3i, x2 = − √ 3 12 i, x3 = −3 2 ;   x1 = − √ 3i, x2 = √ 3 12 i, x3 = −3 2 ;   x1 = √ −6 + 6√13 3 i, x2 = 2(2 + √ 13) 3 √ −6 + 6√13 i, x3 = 1−√13 3 ;   x1 = − √ −6 + 6√13 3 i, x2 = − 2(2 + √ 13) 3 √ −6 + 6√13 i, x3 = 1−√13 3 ;   x1 = √ 6 + 6 √ 13 3 i, x2 = 2(−2 +√13) 3 √ 6 + 6 √ 13 i, x3 = 1 + √ 13 3 ;   x1 = − √ 6 + 6 √ 13 3 i, x2 = −2(−2 + √ 13) 3 √ 6 + 6 √ 13 i, x3 = 1 + √ 13 3 . 2. )õ‘ª§|:  2x23 − x21 − x22 = 0, x1x3 − 2x3 + x1x2 = 0, x21 − x22 = 0. ): ùp�Ñ©Ú{�L§, ¿�ÑÜ©$Ž(J. >read ‘d:/mapleuser/wsolve2‘: >P1:=2*x3^2-x1^2-x2^2; >P2:=x1*x3-2*x3+x1*x2; >P3:=x1^2-x2^2; >PS1:={P1,P2,P3}; >ord:=[x3,x2,x1]; >B1:=basset(PS1,ord); B1 := [x1x3 − 2x3 + x1x2, x21 − x22] >R1:=remseta(PS1,B1,ord); R1 := {x21(x1 − 1)} · 172 · 1›�Ù õ�õ‘ª >PS2:={op(B1)} union R1; >B2:=basset(PS2,ord); B2 := [x1x3 − 2x3 + x1x2, x21 − x22, x21(x1 − 1)] >R2:=remseta(PS2,B2,ord); R2 := {} >J:=Initial(B2[1],ord)*Initial(B2[2],ord)*Initial(B2[3], ord); J := x1 − 2 >solveas(B2,ord,{x1-2}); ù�)� 4 |):  x1 = 0, x2 = 0, x3 = 0,   x1 = 1, x2 = −1, x3 = −1,   x1 = 1, x2 = 1, x3 = 1. ∗§ 5 AÛ½n�Åìy² 1. y²: !/�é�‚pƒR†. A(0, 0) B(u1, 0) C(x2, x3)D(u2, x1) 1 1K y²: Šâb�^‡Œ±��e�õ‘ª§: P1 def = u22 + x 2 1 − u21 = 0, (|AD| = |AB|) P2 def = (x2 − u2)2 + (x3 − x1)2 − u21 = 0, (|DC| = |AB|) P3 def = (x2 − u1)2 + x23 − u21 = 0. (|BC| = |AB|) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2, P3}. ½n(Ø´ AC ⊥ BD, Œ±8(õ‘ª§ G def = (u2 − u1)x2 + x1x3 = 0. ∗§ 5 AÛ½n�Åìy² · 173 · �Cþ�S x1, x2, x3, ¦�ü‡A�� Ci = {Ci1, Ci2, Ci3}©O: C11 = x 2 1 − u21 + u22, C12 = x2, C13 = x3. ±9 C21 = x 2 1 − u21 + u22, C22 = −x2 + u1 + u2, C33 = x1x3 − u21 + u22. l C1 �Ñ x2 = x3 = 0, w,´OŠ. OŽ Rem(G,C2) = 0, Œ½n¤á. šòz^‡ J2 = x1, lAÛ¿Âw, ù´ØŒ±�. 2. y²: ��F/.�ƒ�. A(0, 0) B(u1, 0) C(x1, x2)D(u2, u3) 1 2K y²: Šâb�^‡Œ±��e�õ‘ª§: P1 def = x2 − u3 = 0, (AB//CD) P2 def = u1 − x1 − u2 = 0. (−−→AD † −−→CB 3 AB þ�ÝKƒ�) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2}. ½n(Ø´ ∠BAD = ∠CBA, Œ±8(õ‘ª§ G def = −u21u3(x1 − u1)− u21u2x2 = 0. ¦�A�� C = {C1, C2}Ò´P gC, … J = 1, vkšòz^‡. O Ž Rem(G,C ) = 0, Œ½n¤á. XJr½n�1�‡^‡U¤ P2 def = u22 + u 2 3 − (x1 − u1)2 − x22 = 0, (|AD| = |BC|) OŽ�¬��ü‡A��, ˜‡A��Óc, ,˜‡A��´ C1 = −x1 + u1 + u2, · 174 · 1›�Ù õ�õ‘ª C2 = x2 − u3. G'uù‡A���{ª�u u21u2u3, Ò´`(ØØé. lAÛ¿Â5w, C1 = 0 ƒ�u u1 = x1 − u2, = ABCD´²1o>/ („eã). ù´ØÎÜ K¿�OŠ, Ïd ∠BAD 6= ∠CBA. A(0, 0) B(u1, 0) C(x1, x2)D(u2, u3) 1 2K�,˜«œ/ 3. y²: n�/�ü^¥‚��:©º:†é>¥:¤ 2 : 1. A(0, 0) B(2u1, 0) C(2u2, 2u3) D(u1+u2, u3)E(u2, u3) G(x1,x2) 1 3K y²: Šâb�^‡Œ±��e�õ‘ª§: P1 def = u3x1 − (u1 + u2)x2 = 0, (AGD �‚) P2 def = (u2 − 2u1)x2 − (x1 − 2u1)u3 = 0, (BGE �‚) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2}. ½n(Ø´ −→ AG = 2 3 −−→ AD, Œ±8(õ‘ª§ G1 def = 3x1 − 2(u1 + u2) = 0, G2 def = 3x2 − 2u3 = 0. �Cþ�S x1, x2, ¦�A�� C = {C1, C2}: C1 = 3x1 − 2u1 − 2u2, C2 = −3x2 + 2u3. … J = 1, vkšòz^‡. w, G1, G2 ÑU� C ئ. 4. y²: †�n�/�>þ�p´�>þü‚ã�'~¥‘. ∗§ 5 AÛ½n�Åìy² · 175 · A(0, 0) B(u1, 0) C(0, u2) D(x1, x2) 1 4K y²: Šâb�^‡Œ±��e�õ‘ª§: P1 def = u1x1 − u2x2 = 0, (AD ⊥ BC) P2 def = (x2 − u2)u1 + u2x1 = 0. (BDC �‚) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2}. ½n(Ø´ |AD|2 = |CD||DB|, Œ±8(õ‘ª§ G def = (x21 + x 2 2) 2 − (x21 + (x2 − u2)2)((u1 − x1)2 + x22) = 0. �Cþ�S x1, x2, ¦�A�� C = {C1, C2}: C1 = (u 2 1 + u 2 2)x1 − u1u22, C2 = −(u21 − u22)x2 + u21u2. OŽ Rem(G,C ) = 0, Œ½n¤á. šòz^‡´ u2±9 J = (u 2 1 +u 2 2) 2, lAÛ¿Âw, ù œ/Ñ´Ø#N�. 5. Xã, � △ABC ¥ ∠A´†�, M1,M2,M3 ©O´ AB,AC,BC > �¥:. AH ⊥ BC ¿… H ´Rv. y² M1,M2,M3,H o:��. b A(0, 0) B(u1, 0) C(0, u2) M1(x1, 0) M2(0, x2) M3(x3, x4) H(x5, x6) O(x7, x8) 1 5K · 176 · 1›�Ù õ�õ‘ª y²: Šâb�^‡Œ±��e�õ‘ª§: P1 def = u1 − 2x1 = 0, (M1 ´ AB �¥:) P2 def = u2 − 2x2 = 0, (M2 ´ AC �¥:) P3 def = u1 − 2x3 = 0, (M3 ´ BC �¥:) P4 def = u2 − 2x4 = 0, (M3 ´ BC �¥:) P5 def = u1x5 − u2x6 = 0, (AH ⊥ BC) P6 def = u1(x6 − u2) + u2x5 = 0. (BHC �‚) P7 def = (x1 − x7)2 − x27 = 0, (|OM1| = |OA|) P8 def = (x2 − x8)2 − x28 = 0. (|OM2| = |OA|) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2, · · · , P8}. ½n(Ø´ |OH| = |OM3| = |OA|, Œ±8(õ‘ª§ G1 def = (x5 − x7)2 + (x6 − x8)2 − x27 − x28 = 0, G2 def = (x3 − x7)2 + (x4 − x8)2 − x27 − x28 = 0. �Cþ�S x1, x2, · · · , x8, ¦�A�� C = {C1, C2, · · · , C8}: C1 = −2x1 + u1, C2 = −2x2 + u2, C3 = −2x3 + u1, C4 = −2x4 + u2, C5 = (u 2 1 + u 2 2)x5 − u1u22, C6 = −(u21 + u22)x6 + u21u2, C7 = −4x7 + u1, C8 = −4x8 + u2. OŽ Rem(G1,C ) = 0, Rem(G2,C ) = 0, Œ½n¤á. šòz^‡´ u2 ±9 J = (u21 + u 2 2) 2, lAÛ¿Âw, ù œ/Ñ´Ø#N�. 6. Xã, A,B,C n:3˜^†‚þ, A′, B′, C ′ n:3,˜^†‚þ. P,Q,R´§‚ë‚��:. y²: P,Q,Rn:�‚. ∗§ 5 AÛ½n�Åìy² · 177 · b b b A(u1, 0) B(u2, 0) C(u3, 0) A′(0, u4) B′(0, u5) C′(0, u6) P (x1, x2) R(x5, x6) Q (x3, x4) 1 6K y²: Ϗù´‡•�¯K, ÏdŒïá•�‹IXXþ㤫. Šâb�^‡Œ±��e�õ‘ª§: P1 def = x1(−u4)− u2(x2 − u4) = 0, (A′PB �‚) P2 def = x1(−u5)− u1(x2 − u5) = 0, (B′PA�‚) P3 def = x3(−u4)− u3(x4 − u4) = 0, (A′QC �‚) P4 def = x3(−u6)− u1(x4 − u6) = 0, (C ′QA�‚) P5 def = x5(−u5)− u3(x6 − u5) = 0, (B′RC �‚) P6 def = x5(−u6)− u2(x6 − u6) = 0. (C ′RB �‚) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2, · · · , P6}. ½n(Ø´ PQR�‚, Œ±8(õ‘ª§ G def = (x3 − x1)(x6 − x2)− (x5 − x1)(x4 − x2) = 0, �Cþ�S x1, x2, · · · , x6, ¦�A�� C = {C1, C2, · · · , C6}: C1 = (u1u4 − u2u5)x1 − u1u2(u4 − u5), C2 = (u1u4 − u2u5)x2 − (u1 − u2)u4u5, C3 = (u1u4 − u3u6)x3 − u1u3(u4 − u6), C4 = (u1u4 − u3u6)x4 − (u1 − u3)u4u6, C5 = −(u2u5 − u3u6)x5 + u2u3(u5 − u6), C6 = −(u2u5 − u3u6)x6 + (u2 − u3)u5u6. OŽ Rem(G,C ) = 0, Œ½n¤á. šòz^‡´ u2, u3 ±9 J = (u1u4 − u2u5)2(u1u4 − u3u6)2(u2u5 − u3u6)2. éšòz^‡�AۿŠ©Û: (1) e u2 = 0½ u3 = 0, §�Aۿ´ B ½ C † A ′B′C ′ �‚, l P,Q,RØ(½, ¯KÿÂ. · 178 · 1›�Ù õ�õ‘ª (2) u1u4 − u2u5 = 0 =⇒ A′B//B′A, u1u4 − u3u6 = 0 =⇒ A′C//C ′A, u2u5− u3u6 = 0 =⇒ B′C//C ′B, ?ۘ«œ/ÑyѬ¦ P,Q,R¥�˜‡ :Ã{(½, ¯KÿÂ. 7. y²: �%��uƒA�±��ü�. A(u1, x1) B(u2, x2) C(u3, x3) O(0, 0) r 1 7K y²: Šâb�^‡Œ±��e�õ‘ª§: P1 def = u21 + x 2 1 − r2 = 0, (|OA| = r) P2 def = u22 + x 2 2 − r2 = 0, (|OB| = r) P3 def = u23 + x 2 3 − r2 = 0, (|OC| = r) ù�½nb�Œ±8(¤˜‡õ‘ª| P = {P1, P2, P3}. ½n(Ø´ ∠AOB = 2∠ACB, = tan∠AOB = tan(2∠ACB) = 2 tan∠ACB 1− tan2 ∠ACB . (∗) du ∠AOB = kOB − kOA 1 + kOBkOA = u1x2 − u2x1 u1u2 + x1x2 , ∠ACB = kCB − kCA 1 + kCBkCA = (u1 − u3)(x2 − x3)− (u2 − u3)(x1 − x3) (u1 − u3)(u2 − u3) + (x1 − x3)(x2 − x3) = α β . “\ (∗)ª� u1x2 − u2x1 u1u2 + x1x2 = 2α β 1− α2 β2 = 2αβ β2 − α2 . Ïd·K�(،±8(õ‘ª§ G def = (u1x2 − u2x1)(((u1 − u3)(u2 − u3) + (x1 − x3)(x2 − x3))2 − ((u1 − u3)(x2 − x3)− (u2 − u3)(x1 − x3))2)− 2(u1u2 + x1x2) × ((u1 − u3)(x2 − x3)− (u2 − u3)(x1 − x3))((u1 − u3)(u2 − u3) + (x1 − x3)(x2 − x3)) = 0. �Cþ�S x1, x2, x3, ¦�A�� C Ò´�5�õ‘ª| P. OŽ Rem(G,C ) = 0, Œ½n¤á. …vkšòz^‡.
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