... A F F l F l F(p, 0) = p p > 0 l p 0 P(, ) H P(, ) P l PH F PF = PH PF = PH p O p ( p) + = { ( p)} = 4p l = 4p (p 0) F(p, 0) = p O 3 5
5. F(, 0) = = 4 = 4 O = 4 =. ( = = 4 ) = 4 ( 4 ), 0 = 4 4 O 4 = 4. () = 8 () = 4
.. 53 (3) = B p 0 F(0, p) p > 0 = p = 4p 5 = 4p p F = a = 4 ( 4a ) 0, 4a = 4a O p H = p. () = 4 () =
54.. A F F F F F(c, 0) F ( c, 0) a a > c > 0 P(, ) PF + PF = a P(, ) ( c) + + ( + c) + = a ( c) + = a ( + c) + F c O F c a ( + c) + = a + c (a c ) + a = a (a c ) a > c a c = b a > b > 0 b + a = a b a + = b a b P(, ) PF + PF = a a c = b c = a b F F F( a b, 0) F ( a b, 0)
A(a, 0) A ( a, 0) B(0, b) B (0, b) a c O AA BB AA BB A F a.. 55 B b b B b c a F A c a O AA BB O a + = (a > b > 0) b F( a b, 0) F 0 ( a b, 0) a 3 a b 4 O.3 6 + 9 = 3 F( 7, 0) F ( 7, 0) 4 = 8 3 = 6 4 F F O 7 7 4 3.3 () 5 + 6 =
56 () 9 + =. (3, 0) ( 3, 0) 0 a + b = a = 0 a = 5 a b = 3 b = a 3 = 5 3 = 6 5 + 6 = a =5, b =6.4 (, 0) (, 0) 6
.. 57 B a + b = b > a > 0 F F F(0, b a ), F (0, b a ) b b b a a b F O F a.5 4 + 9 =
58 C. + = 4 3 4 ( ) 3 (s, t) s, 4 t Q(s, t) P(, ) = s, = 3 4 t s = t = 4 3 s + t = 4 + ( ) 4 3 = 4 4 + 3 = + = 4 6 + 9 = 4 4 3 O 3 4 Q(s, t) P(, ) 4 a + b = + = a b a
.. 59.6 + = 3 () 3 4 () 3
60 D. 5 AB A B AB : 3 P A(s, 0) B(0, t) P(, ) s t s t A (s, 0) B (0, t) AB = 5 s + t = 5 P (, ) P AB : 3 = 3 5 s = 5 t s = 5 3 t = 5 ( ) ( ) 5 5 3 + = 5 3 + = 3 t O B 3 P(, ) A s P 9 + 4 = 3
.. 6.7 7 AB A B AB 3 : 4 P
6..3 A F F F F F(c, 0) F ( c, 0) a c > a > 0 P(, ) PF PF = ±a P(, ) ( c) + ( + c) + = ±a ( c) + = ±a + ( + c) + F c O F c ± a ( + c) + = a + c (c a ) a = a (c a ) c > a c a = b b > 0 b a = a b a b = a b P(, ) PF PF = ±a c a = b c = a + b F F F( a + b, 0) F ( a + b, 0)
.. 63 = = 0 0 = ( ) O = + > 0 P 0 P O = = = = = = 0 = = a b = = a b b a ( a) a O b = b a, = b a = b a a = b a
64 F F FF FF a b = (a, 0) ( a, 0) O O F b a O b = b a F a = b a a = (a > 0, b > 0) b F( a + b, 0), F 0 ( a + b, 0) a 3 = b a, = b a 4 O a + b = 0 a b = 0.4 9 4 = F( 3, 0), F ( 3, 0) A(3, 0) A ( 3, 0) = 3 = 3 F A 3 O A 3 F.8 () 5 6 =
.. 65 () 4 = B 5 F 6 = 9 3 9 6 = 4 O 4 3 F(0, 5) F (0, 5) (0, 3) (0, 3) = 3 4 = 3 4 a > 0 b > 0 a = b F(0, a + b ), F 0 (0, a + b ) (0, b), (0, b) = b a, = b a b 5 F
66.9 () 9 4 = () 5 6 =
.. 67..4 A F (, ) = 0 p q C Q(s, t) P(, ) C F (s, t) = 0 = s + p, = t + q s = p t = q F ( p, q) = 0 q O p P(, ) Q(s, t) F (, ) = 0 C F (, ) = 0 p q F ( p, q) = 0.5 9 + 4 = O 3 3 ( ) + 9 ( ) 4 = ( 5, 0) ( 5, 0) ( 5 +, ) ( 5 +, ) 3
68.0 4 + = 3. = 4 B a + b + c + d + e = 0.5 ( ) 9 + ( ) 4 = 4 + 9 6 8 = 0
.. 69. 4 + + 6 9 = 0 ( + + ) 4( 4 + 4) = 4 ( + ) ( ) = 4 4 =. () 4 9 6 36 56 = 0 () + 8 6 = 0 (3) + 4 + 6 + 3 = 0
70..5 A.3 k + 4 = 0, = + k + 4 = 0 = + k + 4( + k) = 0 5 + 8k + 4(k 5) = 0 D D = (8k) 4 5 4(k 5) = 6(k + 5)(k 5) O 5 k = + k k < 5, 5 < k 0 k = ±5 5 < k < 5 5.3 k = ±5
.. 7.3 k = 4 = + k
7 B.3 C(0, 3) + = C(0, 3) = m + 3 D D = 0 C = m + 3 + (m + 3) = C 3 (m + ) + m + 6 = 0 D D = (m) 4(m + ) 6 = 6(m 4) D = 0 m = ± = + 3 = + 3 O = 6m m + = m (m + )
.. 73.4 C(4, 0) = 4
74 = 4p F l H F (p, 0) l P(s, t) = p M P(s, t) l PH FH M PF = PH PHF l PM FPH p O F(p, 0) 3 PM FH FH t p PM t = p ( s) t PM P O
.. 75..6 () (, 0) () A( 3, ) B(3, ) 0 (3) (5, 0) ( 5, 0) 8
76 + = 6 3 = = + 3 () = 4( ) () ( ) + 5 6 64 + 6 = 3 (, ) = (3) 6 9 =
.. 77... A = t + (t, t ) t t= C C P(, ) = t, = t t t= O t=0 t= t= = C C C P(, ) t = f(t), = g(t) C t t F (, ) = 0 C C
78.5 t () = t + = t + 4t () = t = t t.4 = + t t t = ( + t) t t P(, ) = t, = t t t O = + P = +
.. 79.6 = + 4t + t t B θ O 3 + = 3 P(, ) OP θ = 3 cos θ, = 3 sin θ 3 3 3 sin θ O P(, ) θ 3 cos θ 3 3 + = a = a cos θ, = a sin θ
80.7 θ () + = 4 () + = a + b = + = a b a θ a O a a b P(, ) = a cos θ, = b sin θ b a.8 θ () 9 + 4 = () 6 + 5 =
.. 8 ( ).5 θ P cos θ, tan θ = = cos θ, = tan θ = cos θ tan θ = sin θ cos θ = ( ) P cos θ, tan θ = = ( ) 3.9 θ P cos θ, tan θ 9 4 = O θ P a b = = a cos θ, = b tan θ.0 5 6 = θ
8 C.4 = cos θ +, = sin θ + 3 sin θ cos θ sin θ + cos θ = sin θ = 3, cos θ = sin θ + cos θ = ( 3) ( ) + = 4 4 ( ) + ( 3) = 4 (, 3) = f(t) + p = g(t) + q = f(t) = g(t) p q. () = 3 cos θ + = 3 sin θ () = 3 cos θ + = sin θ + 3
.. 83 D P a P a θ C Q O aθ T πa πa a P O θ P (, ) C T OT = TP = aθ = OT PQ = aθ a sin θ = CT CQ = a a cos θ sin θ < 0 cos θ < 0 = a(θ sin θ), = a( cos θ). 3 P P O a θ aθ
84 A(, 0) t + = = t( + ) A P(, ) P + t ( + ) = ( + ){( + t ) + t } = 0 A t O P(, ) = t + t = t + t ( ) t P + t, t + t t P A(, 0) t (, 0) + = = t + t, = t + t
.. A.. 85 O OX P(r, θ) P OP r OX OP θ r θ θ (r, θ) P O X (r, θ) P P(r, θ) O θ OX P θ 0 θ < π θ O (0, θ) θ (, ) P (, ) (r, θ) P(r, θ) P(, ) = r cos θ, = r sin θ r = + r 0 cos θ = r, sin θ = r O r θ X
86 ( ) π.6, 3 P (, ) r = θ = π 3 = r cos θ = cos π 3 = = = r sin θ = sin π 3 = 3 = 3 P (, 3) 3 O π 3 P.3 ( π ) (, π ) (), () (3) (3, π) 6 4.7 ( 3, ) P (r, θ) = 3 = r = + = ( 3) + = 4 = cos θ = r = 3 P 3 O 5 π 6 sin θ = r = 0 θ < π θ = 5 6 π ( ) 5 P, 6 π
.. 87.4 θ 0 θ < π () (, 3) () (, ) (3) ( 3, ) B (r, θ).8 O P (r, θ) OP = r r =, θ θ O P(, θ) X r = P(r, θ).9 (, 0) P (r, θ) OP cos θ = r cos θ = r = cos θ r P(r, θ) r = cos θ O θ X P(r, θ).9 r = cos θ θ cos θ 0 θ r < 0 (r, θ) ( r, θ +π).9 (, π) (, π)
88 (.5, π ) (r, θ) (r, θ) F (r, θ) = 0 r = f(θ).0 () O OX π 4 θ = π r 4 O π 4 ( ) r, π 4 X () A (a, 0) (r, θ) a r θ A(a, 0) O a r = a cos θ X.6 () θ = π 6 () r = cos θ
.. 89 C.6 = 0 P(, ) (r, θ) = r cos θ = r sin θ = r cos θ r sin θ = r (cos θ sin θ) = r cos θ = cos θ sin θ=cos θ.7 + = 4 r θ.7 r = (cos θ + sin θ) P(r, θ) (, ) r 0 cos θ = r sin θ = r r = (cos θ + sin θ) ( r = r + ) r r = + r = 0 r = + + = + + = 0
90.7 ( ) + ( ) = A r = (cos θ + sin θ) (, π ) π 4 4 O O X.8 () r = sin θ + cos θ () r = sin θ
.. 9 D.8 r = + cos θ r + r cos θ = r cos θ = r = 4r = + r = + 4( + ) = + 3 + 4 + = 0 ( 9 + + 3 4 3) =.9 r = + cos θ
9.9 OX A(, 0) l O l P (r, θ) P l PH OP = PH OP = r, PH = r cos θ P(r, θ) r θ O l r = r cos θ r = + cos θ.30 OX A(, 0) l P(r, θ) OP l PH PH = P H A X
.. 93 OX A(a, 0) l P(r, θ) l PH e = OP PH P PH PH = r e, r e = a r cos θ ea r = + e cos θ PH = a r cos θ O r θ P(r, θ) r e a l H A X P e e 0 < e < O e = 3 e > O l O
94..3 = f() 3 A = f() 4 9 + 4 = - - = 9, = -3-4 9 3-4 -3 - - 3 4.3 () 9 4 = () = 4 () = 9 = 4 3 9 () = = 4 3-6 -4-4 6 - - -3-4 -6-4 - 4 6 - - -3-4
.. 95 B = f(t), = g(t).3 () = t +, = t () = ( t + ), = ( t ) t t () () 3-3 4 - - -3 - - 3 - -3 - -4-3 t { = f(t) = g(t) a b { = sin at = sin bt. a b () a =, b = () a =, b = 3 (3) a = 4, b = 5.5 0.5 - -.5 - -0.5 0.5.5-0.5 - -.5 -.5 0.5 - -.5 - -0.5 0.5.5-0.5 - -.5 -.5 0.5 - -.5 - -0.5 0.5.5-0.5 - -.5 -.
96.33 { { = cos t = 3 cos t () () = sin t = sin t (3) (5) { { = cos t = sin t = sin t cos t = sin t + cos t (4) (6) { { = sin t = sin t = t sin t = cos t () 3 () -3 - - 3 - - -3 (3) (4).5 0.5 - -.5 - -0.5 0.5.5-0.5 - -.5 - (5) (6) -4-3 - - 3 4.5 0.5-3 - -3 - -.5 - -0.5 0.5.5-0.5 - -.5 -.5 0.5 - -.5 - -0.5 0.5.5-0.5 - -.5 -.5.5 0.5-0.5-3 4 6
.. 97 C r = f(θ) θ 3. r = θ (θ 0) 0 5-0 -5 5 0-5 -0 a > 0 r = aθ (θ 0).34 θ 0 () r = θ () r = sin 4θ (3) r = ( + cos θ) () ().5 (3) 0 0 0.5 4 3-0 -0 0 0 -.5 - -0.5 0.5.5 - - 3 4 5 6-0 -0.5 - - -0 - -.5-3 -4 3 r = f(θ) = f(θ) cos θ = f(θ) sin θ
98 a r = sin aθ () a = 6 () a = 7 (3) a =.5.5.5.5 0.5 0.5 0.5 -.5 - -0.5 0.5.5 -.5 - -0.5 0.5.5 -.5 - -0.5 0.5.5-0.5-0.5-0.5 - - - -.5 -.5 -.5 a b r = a + b cos θ a = b r = a( + cos θ) () a =, b = 0.8 () a =, b = (3) a =, b =.5.5.5.5 0.5 0.5 0.5 - -0.5 0.5.5.5 3 - -0.5 0.5.5.5 3 - -0.5 0.5.5.5 3-0.5-0.5-0.5 - - - -.5 -.5 -.5 - - -
.. 99..4 4 θ () = cos θ, = 3 sin θ () = +, = tan θ + cos θ ( π ) 5 O A 3, A OA ( 4 r cos θ π ) = 3 4
00 6 () r = (cos θ sin θ) () r sin θ cos θ = 7 = ( )
.. 0 ( + ) ( + ) 4 () + = () ( ) ( ) = 4 9 4 5 A OA l l P (r, θ) OP cos AOP = OA OP = r P(r, θ) AOP = θ π r 4 θ 3 π OA = 3 4 O ( r cos θ π ) = 3 4 6 () ( ) + ( + ) = 5 () = ( A 3, π ) 4 l X 7 0.75 0.5 0.5 -.5 - -0.5 0.5.5-0.5-0.5-0.75 -
0.3.3. A () + 8 = 0 () 4 + 6 = (3) 8 4 = 3
.3. 03 () (, ) () 4 (, ) (3) (, 0) (, 0) = =
04 3 () + 4 3 = 0 () 4 4 = 0 (3) 9 4 8 8 3 = 0
.3. 05 4 () { = t + 4 = t + 3 () { = t = t t ( π ) ( π ) 5 A B 3, 4, O A B 6 3 OAB 6 C (r, θ ) C a r + r rr cos(θ θ ) = a
06.3. B 7 = A(, 0) P(, ) P 8 6 3 = F(3, 0) P(, ) = PH PF : PH 9 4 + + 8 4 8 = 0 3
.3. 07 0 = + k 4 + = 4 Q R k QR P t P + + t 4t + 5t t = 0 a r = a cos θ
08 8 PF : PH 0 k t () (, 0) () ( ) ( 3 4, 0 ) 3 4, 0 () = 4 () 4 + = (3) 4 = (3) ( 3, 0) ( 3, 0) 3 () ( 3 +, 0) ( 3 +, 0) () (0, ) (3) ( 3 +, ) ( 3 +, ) [ ( ) () + = () ( ) = 4( + ) 4 (3) ( ) 4 ( + ) 9 ] = 4 () ( 3) = ( 4) () = + [ 5 3 S = ( π 3 4 sin 3 π ) ] 6 ( 0) 6 O P (r, θ) OCP CP = OP + OC OP OC cos(θ θ ) 7 = 4 [ + = ] ( ) + 8 PF : PH = 3 : [ 6 3 = = 3 PF = 3 ( ) ] PH = ( ) 9 ( + ) + = (0, 3 ) (0, 3 ) 4 6 [ ] ( + ) ( ) + = 4 6
( 0 = 4 [ 5 5 < < ) 5 5.3. 09 5 k+(k 4) = 0 D > 0 5 < k < 5 + = k 5 + = ( + k) + ( + k) = 8k ( ) + + P, ( ) ] 5 k 5, 4 5 k = ( < 0) ( + t) + ( t) = t t > 0 ( + ) = a ( ) [ ] r = + cos θ = cos θ = r a 0.75 0.5 0.5 - -.5 - -0.5-0.5 0.5.5-0.5-0.75 - a