Cap20_Parte2.pdf

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�OX �OY �OZ
y2
b2
� z
2
c2
= 1
z2
c2
� y
2
b2
= 1 ,
x2
a2
� z
2
c2
= 1
z2
c2
� x
2
a2
= 1 ,
x2
a2
� y
2
b2
= 1
y2
b2
� x
2
a2
= 1 ,
a b c
�OZ
Q : x
2
a2
� y
2
b2
= 1 .
XY
Q \ z = k :
8<:
x2
a2
� y
2
b2
= 1
z = k
,
(0, 0, k) �OZ
�OX
(
y = ± b
a
x
z = k .
Q XY
Q x = k Y Z
Q \ {x = k} :
8<:
y2
b2
=
k2
a2
� 1
x = k
,
•
(
y = ± b
a
p
k2 � a2
x = k
�OZ |k| > a
•
⇢
y = 0
x = a
�OZ k = a
•
⇢
y = 0
x = �a �OZ k = �a
• |k| < a k
2
a2
� 1 < 0
Q Y Z
Q \ {y = k} :
8<:
x2
a2
= 1 +
k2
b2
y = k
() Q \ {y = k} :
8<:x = ±
a
b
p
b2 + k2
y = k
�OZ k 2 R
Q XZ
�OX �OY
�OZ
y2
b2
+
z2
c2
= ax ,
x2
a2
+
z2
c2
= by ,
x2
a2
+
y2
b2
= cz ,
a b c
�OZ
Q : x
2
a2
+
y2
b2
= cz , c > 0 .
Q Y Z XZ
XY
Q \ {z = k}
Q
z = k XY
Q \ {z = k} :
8<:
x2
a2
+
y2
b2
= ck
z = k
,
• (0, 0, k)
k > 0
• (0, 0, 0) k = 0
• k < 0
XZ
Q \ {y = k} :
8<:
x2
a2
= cz � k
2
b2
y = k
() Q \ {y = k} :
8><>:x
2 = a2c
✓
z � k
2
cb2
◆
y = k
,
Vk =
✓
0, k,
k2
b2c
◆
�OZ
a2c > 0
Q \ {x = k} :
8<:
y2
b2
= cz � x
2
a2
x = k
() Q \ {y = k} :
8><>:y
2 = b2c
✓
z � k
2
ca2
◆
x = k
Vk =
✓
k, 0,
k2
a2c
◆
�OZ
Q
XZ
Q
Y Z
(x0, y0, z0) 2 Q : x
2
a2
+
y2
b2
= cz
r :
8>>><>>>:
x(t) = ↵t+ x0
y(t) = �t+ y0
z(t) = �t+ z0
, t 2 R ,
�!v = (↵, �, �) 6= (0, 0, 0)
(x0, y0, z0) (↵t+ x0, �t+ y0, �t+ z0) 2 Q
(↵t+ x0)2
a2
+
(�t+ y0)2
b2
= c(�t+ z0)
()
✓
↵2
a2
+
�2
b2
◆
t2 +
✓
2↵x0
a2
+
2�y0
b2
� c�
◆
t = 0 ,
x20
a2
+
y20
b2
= cz0
r ⇢ Q
↵2
a2
+
�2
b2
= 0
2↵x0
a2
+
2�y0
b2
� c� = 0 ,
↵ = � = � = 0
⇤
�OZ �OY
�OX
x2
a2
� y
2
b2
= cz ,
x2
a2
� z
2
c2
= by ,
y2
b2
� z
2
c2
= ax ,
a b c
�OZ
Q : x
2
a2
� y
2
b2
= cz , c < 0 .
Y Z XZ
XY
Q z = k k 2 R XY
Q \ {z = k} :
8<:
x2
a2
� y
2
b2
= ck
z = k
,
Q
z = k k > 0
Q
z = 0
Q
z = k k < 0
•
�OY
(0, 0, k)
8<:x = ±
a
b
y
z = k
k > 0 ck < 0
•
8<:y = ±
b
a
x
z = 0
k = 0
• �OX C = (0, 0, k)8<:y = ±
b
a
x
z = k
k < 0 ck > 0
XZ
Q \ {y = k} :
8><>:x
2 = a2
✓
cz +
k2
b2
◆
= a2c
✓
z +
k2
b2c
◆
y = k
,
�OZ
✓
0, k,� k
2
cb2
◆
k 2 R
a2c < 0
Q y =
k 2 R
Q \ {x = k} :
8><>:y
2 = b2
✓
k2
a2
� cz
◆
= �b2c
✓
z � k
2
a2c
◆
x = k
,
�OZ
V =
✓
k , 0 ,
k2
a2c
◆
�b2c > 0
Q x = k
S : x
2
4
� y
2
9
= z
A = (4, 3, 3) 2 S
r = { ( at + 4, bt + 3, ct + 3 ) ; t 2 R }
�!v = (a, b, c) 6= (0, 0, 0) A
r ⇢ S
(at+ 4)2
4
� (bt+ 3)
2
9
= ct+ 3
()
✓
a2
4
� b
2
9
◆
t2 +
✓
8a
4
� 6b
9
� c
◆
t+
16
4
� 9
9
� 3 = 0
()
✓
a2
4
� b
2
9
◆
t2 +
✓
2a� 2b
3
� c
◆
t = 0
() t
✓
a2
4
� b
2
9
◆
t+
✓
2a� 2b
3
� c
◆�
= 0
t 2 R
8>><>>:
a2
4
=
b2
9
2a� 2b
3
� c = 0
()
8>><>>:
b = ±3
2
a
c = 2a� 2
3
b = 2a� 2
3
⇣
±3
2
⌘
a
() �!v =
⇣
a,
3
2
a, a
⌘ �!v = ⇣a,�3
2
a, 3a
⌘
() �!v k (1, 3
2
, 1) �!v k (1,�3
2
, 3).
r = { (t+4, 3
2
t+3, t+3) ; t 2 R } s = { (t+4,�3
2
t+3, 3t+3) ; t 2 R }
S A ⇤
P
S S P
�OX �OY
�OZ
y2
b2
= cz
z2
c2
= by ,
x2
a2
= cz
z2
c2
= ax ,
x2
a2
= by
y2
b2
= ax ,
a b c
�OY
Q : x
2
a2
= cz c > 0
Q Y Z
XZ XY
Q y = k
XZ
c > 0
Q y = k
XZ
Q \ {y = k} :
8<:x2 = ca2zy = k
Vk = (0, k, 0)
�OZ
XY
Q \ {z = k} :
8<:x2 = ca2kz = k ,
•
8<:x = ±
p
ca2k
z = k
�OY k > 0
•
8<:x = 0z = 0 �OY k = 0
• k < 0
Q XY
Q z = k
XY k > 0
Q \ {x = k} :
8<:z =
k2
a2c
x = k
�OY
Q x = k Y Z
S P
S P S
•
•
R3
Ax2 +By2 + Cz2 = R
Ax2 +By2 = Sz
R � 0 S � 0
A B C R
Ax2 +By2 + Cz2 = R ,
R > 0 A B C
• =) Q
• =) Q
• =) Q
• =) Q
• =) Q
• =) Q
• =) Q
• =) Q
• =) Q
R = 0 A B C
• =) Q
• =) Q
• =) Q
• =) Q
• =) Q
A B S
Ax2 +By2 = Sz ,
S > 0 A B
• =) Q
• =) Q
• =) Q
S = 0 A B
• =) Q
• =) Q
• =) Q
Q ⇡
Q \ ⇡
Q : x
2
4
� y2 + z
2
16
= 1 ⇡ : y =
p
3
Q �OY
Q \ ⇡ :
8<:
x2
4
+
z2
16
= 1 + 3
y =
p
3
() Q \ ⇡ :
8<:
x2
16
+
z2
64
= 1
y =
p
3
C = (0,
p
3, 0) y =
p
3
` = { (0,p3, t) ; t 2 R } �OZ `0 = { (t,p3, 0) ;
t 2 R } �OX A1 = (0,
p
3,�8) A2 = (0,
p
3, 8)
B1 = (�4,
p
3, 0) B2 = (4,
p
3, 0)
F1 = (0,
p
3,�2p12) F2 = (0,
p
3, 2
p
12)
c =
p
64� 16 = p48 = 2p12 ⇤
Q : �x
2
4
+ y2 = 4z ⇡ : y = 2
Q �OZ
Q \ ⇡ :
8<: �
x2
16
+ 4 = 4z
y = 2
() Q \ ⇡ :
8<: x2 = �16⇥ 4(z � 1)y = 2
V = (0, 2, 1) y = 2
` = { (0, 2, t) ; t 2 R } �OZ 4p = 16 ⇥ 4 () p = 16
F = (0, 2, 1� 16) = (0, 2,�15) L = { (t, 2, 1+16) ; t 2 R }
�OX ⇤
Q : x2 � y
2
4
� z2 = 1 ⇡ : y = 2
Q �OX
Q \ ⇡ :
8<: x2 � z2 = 1 +
4
4
y = 2
() Q \ ⇡ :
8<: x2 � z2 = 2y = 2
y = 2 C = (0, 2, 0)
` = { (t, 2, 0) ; t 2 R } �OX
`0 = { (0, 2, t) ; t 2 R } �OZ A1 = (�
p
2, 2, 0)
A2 = (
p
2, 2, 0) B1 = (0, 2,�
p
2) B2 = (0, 2,
p
2)
F1 = (�2,�1, 0) F2 = (2, 1, 0) r+ = { (t, 2, t) ; t 2 R }
r� = { (t, 2,�t) ; t 2 R } r+ :
⇢
z = x
y = 2
r� :
⇢
z = �x
y = 2 ⇤
Q : x
2
2
� z
2
16
= 1 ⇡ : z = 4
Q �OY
Q \ ⇡ :
8<:
x2
2
= 1 +
16
16
z = 4
() Q \ ⇡ :
8<: x2 = 4z = 4 () Q \ ⇡ :
8<: x = ±2z = 4
�OY r+ = { (2, t, 4) ; t 2 R }
r� = {(�2, t, 4) ; t 2 R } ⇤
P0 = (1, 1,�1) � :
⇢
4y2 + 2z2 = 3
x = 2
.
Q : Ax2 +By2 + Cz2 = R
P0 2 Q � ⇢ Q
� :
(
By2 + Cz2 = R� 4A
x = 2
,
� 6= 0 B = 4� C = 2� R� 4A = 3�
Q : Ax2 + 4�y2 + 2�z2 = R () Q : A
�
x2 + 4y2 + 2z2 =
R
�
() Q : A0x2 + 4y2 + 2z2 = R0 ,
R0 � 4A0 = 3 .
P0 = (1, 1,�1) 2 Q
A0 + 4 + 2 = R0 () R0 = A0 + 6 .
A0 + 6� 4A0 = 3 =) A0 = 1 R0 = 7
Q : x2 + 4y2 + 2z2 = 7
a =
p
7 b =
p
7
2
c =
r
7
2eQ
P0 2 eQ � ⇢ eQeQ eQ : By2 + Cz2 = Ax ,eQ \ {x = 2}
� :
⇢
By2 + Cz2 = 2A
x = 2 ,
� 6= 0 B = 4� C = 2� 2A = 3�
eQ : 4�y2 + 2�z2 = 3�
2
x () eQ : 4y2 + 2z2 = 3
2
x .
P0 = (1, 1,�1) eQ 4 + 2 6= 32
⇤
Q
� :
8<:2x2 + 3y2 = 5z = 1 � :
8<:4z2 � 3y2 = 1x = 1 .
↵ = Q \ {z = 1}
� = Q \ {x = 1} Q
Q : Ax2 +By2 + Cz2 = R
� ⇢ Q � ⇢ Q
� :
8<:Ax2 +By2 = R� Cz = 1 � :
8<:By2 + Cz2 = R� Ax = 1 ,
� 6= 0 µ 6= 0
A = 2� B = 3� R� C = 5� B = �3µ C = 4µ R� A = µ
3� = �3µ
µ = �1 � = 1
A = 2 B = 3 C = �4 R = C + 5� = �4 + 5 = 1 = 2� 1 = A+ µ
Q : 2x2 + 3y2 � 4z2 = 1
�OZ ⇤
� 2 R
(�3 � �)x2 + �2y2 + (�+ 1)z2 = �2 + 1 .
�1 < � < �1 � = �1 �1 < � < 0 � = 0 0 < � < 1 � = 1 1 < � <1
�3 � � � 0 + 0 � 0 +
�2 + + + 0 + + +
�+ 1 � 0 + + + + +
�2 + 1 + + + + + + +
• �OY � 2 (�1,�1)
• y = ±p2 � = �1
• � 2 (�1, 0)
• z = ±1 � = 0
• �OX � 2 (0, 1)
• y2 + 2z2 = 2 �OX � = 1
• � 2 (1,+1) ⇤
� :
8<:x2 = 2
⇣
y +
1
4
⌘
z = 1
� :
8<:x2 = 2(y + 1)z = 2 .
�OY
�OY
Ax2 + Cz2 = Sy .
� :
(
Ax2 = Sy � C
z = 1
� :
(
Ax2 = Sy � 4C
z = 2
,
� 6= 0 µ 6= 0
A = � S = 2� �C = �
2
A = µ S = 2µ �4C = 2µ
� = µ 6= 0 � = µ = 1
A = � = µ = 1 C = ��
2
= �1
2
= �2µ
4
S = 2� = 2µ = 2 ,
�OY
x2 � z
2
2
= 2y .
⇤
� P0 = (1, 3, 1)
� :
⇢
x2 + 5z2 = 2
y = 1
.Q : Ax2 +By2 + Cz2 = R
� ⇢ Q P0 2 Q
� :
⇢
Ax2 + Cz2 = R� B
y = 1
� 6= 0 A = � C = 5� R� B = 2�
Q : �x2 +By2 + 5�z2 = B + 2�
P0 = (1, 3, 1) 2 Q
�+ 9B + 5� = B + 2�() 8B = �4�() B = ��
2
.
Q : �x2 � �
2
y2 + 5�z2 = ��
2
+ 2� =
3�
2
() Q : x2 � y
2
2
+ 5z2 =
3
2
�OY a =
p
3p
2
b =
p
3 c =
p
3p
10
Q0 : Ax2 + Cz2 = Sy
�OY � ⇢ Q0 P0 2 Q0
� :
⇢
Ax2 + Cz2 = S
y = 1
,
� 6= 0 A = � C = 5� S = 2�
Q0 : �x2 + 5�z2 = 2�y () Q0 : x2 + 5z2 = 2y ,
�OY
P0 = (1, 3, 1) 2 Q0 1 + 5 ⇥ 1 = 6 = 2 ⇥ 3 Q0
� ⇢ Q0 P0 2 Q0
(x, y, z) Q \Q08<:x2 + 5z2 =
3
2
+
y2
2
x2 + 5z2 = 2y
()
8<:x
2 + 5z2 = 2y
y2
2
+
3
2
= 2y
()
8<:x2 + 5z2 = 2yy2 � 4y + 3 = 0 .
y2 � 4y + 3 = 0 y = 1 y = 3
Q \Q0 = � [ �
� �
� :
⇢
x2 + 5z2 = 2
y = 1
� :
⇢
x2 + 5z2 = 6
y = 3
Q Q0 � �
Q Q0⇤
� 2 R
Q : (�3 + �2)x2 + (�2 � 1)y2 + (�+ 2)z2 = � .
�<�2 �=�2 �2<�<�1 �=�1 �1<�<0 �=0 0<�<1 �=1 �>1
�3+�2 � � � 0 + 0 + + +
�2�1 + + + 0 � � � 0 +
�+2 � 0 + + + + + + +
� � � � � 0 + + +
• �OY � 2 (�1,�2)
• �4x2 + 3y2 = �2 �OZ � = �2
• �OX � 2 (�2,�1)
• z2 = �1 � = �1
• �OY � 2 (�1, 0)
• y = ±p2 z � = 0
• �OY � 2 (0, 1)
• 2x2 + 3z2 = 1 �OY � = 1
• � 2 (1,+1) ⇤
Q
x = k k 2 R
�OY 8<:x2 + z2 = 1y = p2
Y Z
�OY
�OY
�OX
� :
8<:x2 + z2 = 1y = p2
�OY
Q : y
2
b2
� x
2
a2
� z
2
c2
= 1 .
� :
8<:
x2
a2
+
z2
c2
=
2
b2
� 1
y =
p
2
,
a2 = c2 a2
⇣
2
b2
� 1
⌘
= 1
Q \ {x = k} :
8<:
y2
b2
� z
2
c2
= 1 +
k2
a2
x = k
,
b2 = c2
a2 = b2 = c2
a2
⇣
2
a2
� 1
⌘
= 1() 2� a2 = 1() a2 = 1
Q : y2 � x2 � z2 = 1
⇤
H z = 1 C = (0, 0, 1)
�OY F = �0,p5, 1�
r : 2y � x = 0
Q H ⇢ Q (0, 0, 0) 2 Q
c = d(C, F ) =
p
5
b
a
= 2 r : x = 2y
H �OY
5 = c2 = a2 + b2 = a2 + 4a2 a = 1 b = 2
H :
8<:y2 �
x2
4
= 1
z = 1
.
Q : Ax2 +By2 + Cz2 = R
H ⇢ Q (0, 0, 0) 2 Q
R = 0
H = Q \ {z = 1} :
8<:Ax2 +By2 = �Cz = 1 .
� 6= 0 A = ��
4
B = � C = ��
Q : ��
4
x2 + �y2 � �z2 = 0 .
� = 1
Q : �1
4
x2 + y2 � z2 = 0() Q : 1
4
x2 + z2 = y2 ,
�OY
Q0 : Ax2 +By2 = Sz
�OZ
(0, 0, 0) 2 Q0
H : Q0 \ {z = 1} :
8<:Ax2 +By2 = Sz = 1 .
� 6= 0 A = ��
4
B = � S = �
� = 1
Q0 : �x
2
4
+ y2 = z
�OZ
(x, y, z) Q \Q08><>:�
x2
4
+ y2 = z2
�x
2
4
+ y2 = z
()
8<:�
x2
4
+ y2 = z2
z2 = z
()
8<:�
x2
4
+ y2 = z2
z = 0
8<:�
x2
4
+ y2 = z2
z = 1
.
Q \Q0 = � [ �
� = H :
8<:�
x2
4
+ y2 = 1
z = 1
� :
8<:x = ±2yz = 0 .
Q Q0
Q \Q0 = H [ �
⇤
Ax2 +By2 + Cz2 +Gx+Hy + Iz + J = 0
• A B C
• A 6= 0 B = C = 0 H = 0 I = 0
• B 6= 0 A = C = 0 G = 0 I = 0
• C 6= 0 A = B = 0 G = 0 H = 0
• A 6= 0 B = C = 0 H 6= 0 I 6= 0
• B 6= 0 A = C = 0 G 6= 0 I 6= 0
• C 6= 0 A = B = 0 G 6= 0 H 6= 0
S : 4x2 + y2 + 3z2 � 8x+ 2y + 6z = �7
S : x2 � y2 � z2 + 8x� 2y + 6z = �5
S : 4(x2 � 2x) + (y2 + 2y) + 3(z2 + 2z) = �7
() S : 4(x� 1)2 + (y + 1)2 + 3(z + 1)2 = �7 + 4 + 1 + 3 = 1
() S : (x� 1)
2
1/4
+ (y + 1)2 +
(z + 1)2
1/3
= 1 .
OX Y Z O = C = (1,�1,�1)
OX OY OZ
OX OY OZ
(x, y, z) = (x, y, z) + (1,�1,�1) ,
x2
1/4
+ y2 +
z2
1/3
= 1
x y z
C = (1,�1,�1) a = 1
2
b = 1
c =
p
3
3
S : (x2 + 8x)� (y2 + 2y)� (z2 � 6z) = �5
() S : (x+ 4)2 � (y + 1)2 � (z � 3)2 = �5 + 16� 1� 9 = 1 .
OX Y Z OXY Z
O = C = (�4,�1, 3)
(x, y, z) = (x, y, z) + (�4,�1, 3) ,
S : x2 � y2 � z2 = 1
x y z
S
r = {(�4,�1, 3) + t(1, 0, 0) | t 2 R }
�OX a = b = c = 1 ⇤