Cap20_Parte1.pdf

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x y
x y z
Ax2 +By2 + Cz2 +Dxy + Exz + Fyz +Gx+Hy + Iz + J = 0 ,
A B C D E F G H I J
A B C D E F
• •
• •
• •
Q\ ⇡ ⇡
Q
• XY (x, y, z) 2 Q() (x, y,�z) 2 Q
• XZ (x, y, z) 2 Q() (x,�y, z) 2 Q
• Y Z (x, y, z) 2 Q() (�x, y, z) 2 Q
• (x, y, z) 2 Q() (�x,�y,�z) 2 Q
Q
XY XZ Y Z
Q : x
2
a2
+
y2
b2
+
z2
c2
= 1 ,
a b c
Q
x2 + y2 + z2 = R2
a = b = c = R
Q z = k k 2 R
XY
Q \ {z = k} :
8<:
x2
a2
+
y2
b2
= 1� k
2
c2
z = k
,
� (0, 0, k) k 2 (�c, c)
� (0, 0, c) k = c
� (0, 0,�c) k = �c
� |k| > c
{z = k} Q
Q
XZ
Q \ {y = k} :
8<:
x2
a2
+
z2
c2
= 1� k
2
b2
y = k
,
� (0, k, 0) k 2 (�b, b)
� (0, b, 0) k = b
� (0,�b, 0) k = �b
� |k| > b
{y = k} Q
Q
Y Z
{x = k} Q
Q \ {x = k} :
8<:
y2
b2
+
z2
c2
= 1� k
2
a2
x = k
,
� (k, 0, 0)
k 2 (�a, a)
� (a, 0, 0) k = a
� (�a, 0, 0) k = �a
� |k| > a
r :
8>>><>>>:
x(t) = ↵t+ x0
y(t) = �t+ y0
z(t) = �t+ z0
; t 2 R ,
(x0, y0, z0) Q
(↵, �, �) 6= (0, 0, 0)
(↵ t+ x0, � t+ y0, � t+ z0) 2 Q
() (↵ t+ x0)
2
a2
+
(� t+ y0)2
b2
+
(� t+ z0)2
c2
= 1
()
✓
↵2
a2
+
�2
b2
+
�2
c2
◆
t2 + 2
✓
↵x0
a2
+
�y0
b2
+
�z0
c2
◆
t+
x20
a2
+
y20
b2
+
z20
c2
= 1
()
✓✓
↵2
a2
+
�2
b2
+
�2
c2
◆
t+ 2
✓
↵x0
a2
+
�y0
b2
+
�z0
c2
◆◆
t = 0 ,
x20
a2
+
y20
b2
+
z20
c2
= 1 (x0, y0, z0) 2 Q
↵2
a2
+
�2
b2
+
�2
c2
> 0
t = 0 t =
�2
✓
↵x0
a2
+
�y0
b2
+
�z0
c2
◆
↵2
a2
+
�2
b2
+
�2
c2
⇤
�OX �OY
�OZ
Q \ {z = k} (0, 0, k)
z = k
�x
2
a2
+
y2
b2
+
z2
c2
= 1 ,
x2
a2
� y
2
b2
+
z2
c2
= 1 ,
x2
a2
+
y2
b2
� z
2
c2
= 1 ,
a b c
�OZ
Q : x
2
a2
+
y2
b2
� z
2
c2
= 1 .
Q z = k XY
Q \ {z = k} :
8<:
x2
a2
+
y2
b2
=
k2
c2
+ 1
z = k
,
(0, 0, k) k 2 R
Q \ {x = k} :
8<:
y2
b2
� z
2
c2
= 1� k
2
a2
=
a2 � k2
a2
x = k
,
• k 2 (�a, a) 8>><>>:
y2
b2
✓
a2 � k2
a2
◆ � z2
c2
✓
a2 � k2
a2
◆ = 1
x = k
(k, 0, 0) �OY(
z = ±c
b
y
x = k
a2 � k2
a2
> 0
(k, 0, 0) x = k Q \ {x = k}
• k = a
(
y = ±b
c
z
x = a
(a, 0, 0)
(a, 0, 0) Q \ {x = a}
• k = �a
8<:y = ±
b
c
z
x = �a
(�a, 0, 0)
(�a, 0, 0) Q \ {x = �a}
• |k| > a 8>>><>>>:
z2
c2
✓
k2 � a2
a2
◆ � y2
b2
✓
k2 � a2
a2
◆ = 1
x = k
(k, 0, 0) �OZ
8<:y = ±
b
c
z
x = k
,
k2 � a2
a2
> 0
Q \ {x = k} |k| > a
Q y = k
XZ
Q \ {y = k} :
8<:
x2
a2
� z
2
c2
= 1� k
2
b2
=
b2 � k2
b2
y = k
,
• k 2 (�b, b) (0, k, 0) �OX8<:z = ±
c
a
x
y = k
b2 � k2
b2
> 0
Q \ {y = k} �b < k < b
• k = b
8<:z = ±
c
a
x
y = b
(0, b, 0)
Q \ {y = b}
• k = �b
8<:z = ±
c
a
x
y = �b
(0,�b, 0)
• |k| > b
Q \ {y = k} :
8>>><>>>:
z2
c2
✓
k2 � b2
b2
◆ � x2
a2
✓
k2 � b2
b2
◆ = 1
y = k
(0, k, 0) �OZ
8<:x = ±
a
c
z
y = k
k2 � b2
b2
> 0 .
Q \ {y = k} k = �b Q \ {y = k} k > b
�OY
S : 4x2 � y
2
4
+ z2 = 4 .
S P = (1, 2, 1) 2 S .
r = {(at+1, bt+2, ct+1) ; t 2 R} �!v = (a, b, c) 6=
(0, 0, 0) P = (1, 2, 1)
r 2 S
4(at+ 1)2 � (bt+ 2)
2
4
+ (ct+ 1)2 = 4
()
✓
4a2 � b
2
4
+ c2
◆
t2 + (8a� b+ 2c)t+ 4� 4
4
+ 1� 4 = 0
() t
✓
4a2 � b
2
4
+ c2
◆
t+ 8a� b+ 2c
�
= 0,
t 2 R
4a2 � b24 + c2 = 0 8a� b+ 2c = 0
() 4a2 � 14(8a+ 2c)2 + c2 = 0 b = 8a+ 2c
() 4a2 � (4a+ c)2 + c2 = 0 b = 8a+ 2c
() 4a2 � 16a2 � 8ac = 0 b = 8a+ 2c
() �8a2 � 8ac = 0 b = 8a+ 2c
() ac = �a2 b = 8a+ 2c
() a 6= 0 , c = �a b = 6a
a = 0 b = 2c
() �!v k (1, 6,�1) �!v k (0, 2, 1).
r = {(t + 1, 6t + 2,�t + 1) ; t 2 R} l = {(1, 2t + 2, t + 1) ; t 2 R}
S P ⇤
P
Q Q
P
⇡ : 4x� 5y � 10z = 20
S : x
2
25
+
y2
16
� z
2
4
= 1
x2
25
+
y2
16
� z
2
4
= 1 () 16x2 � 4⇥ 25z2 = 25⇥ 16� 25y2
() (4x� 10z)(4x+ 10z) = 25(4� y)(4 + y) .
(x, y, z) 2 S \ ⇡ (x, y, z)8><>: 4x� 5y � 10z = 20(4x� 10z)(4x+ 10z) = 25(4� y)(4 + y)
()
8><>: 4x� 10z = 20 + 5y(20 + 5y)(4x+ 10z) = 25(4� y)(4 + y)
()
8><>: 4x� 10z = 20 + 5y(4 + y)(4x+ 10z) = 5(4� y)(4 + y)
y 6= �4
4x+ 10z = 20� 5y ,
(x, y, z) ⇡0 : 4x+ 5y + 10z = 20
(x, y, z)
` :
8<:4x� 5y � 10z = 204x+ 5y + 10z = 20 ,
�����4 �5 �104 5 10
����� = (0,�80, 40) k (0,�2, 1)
(5, 0, 0)
` = {(5,�2t, t) | t 2 R } ⇢ S \ ⇡ .
(x,�4, z) 2 ⇡
S ⇡ ` `0
4x+20� 10z = 20
x =
5
2
z ⇡ \ {y = �4}
`0 = {(5t,�4, 2t) | t 2 R }
`0
S t 2 R
25t2
25
+
16
16
� 4t
2
4
= 1 .
`0 ⇢ S \ ⇡
S \ ⇡ = ` [ `0 ⇤
�OX �OY
�OZ
x2
a2
� y
2
b2
� z
2
c2
= 1 ,
�x
2
a2
+
y2
b2
� z
2
c2
= 1 ,
�x
2
a2
� y
2
b2
+
z2
c2
= 1 ,
a b c
�OZ
�x
2
a2
� y
2
b2
+
z2
c2
= 1 .
Q z = k k 2 R XY
Q \ {z = k} :
8<:
x2
a2
+
y2
b2
=
k2
c2
� 1 = k
2 � c2
c2
z = k ,
� k 2 (�c, c)
� (0, 0, c) k = c
� (0, 0,�c) k = �c
� 8>>><>>>:
x2
a2
✓
k2 � c2
c2
◆ + y2
b2
✓
k2 � c2
c2
◆ = 1
z = k ,
(0, 0, k) k 2 (�1, c) [ (c,+1)
Q z =
XZ
Q \ {y = k} :
8<:�
x2
a2
+
z2
c2
= 1 +
k2
b2
y = k
() Q \ {y = k} :
8>>><>>>:
z2
c2
✓
1 +
k2
b2
◆ � x2
a2
✓
1 +
k2
b2
◆ = 1
y = k
,
(0, k, 0) �OZ
8<:x = ±
a
c
z
y = k
1 +
k2
b2
> 0 k 2 R
Q y = Q x =
Q x = k k 2 R
Y Z
Q \ {x = k} :
8<:
z2
c2
� y
2
b2
= 1 +
k2
a2
x = k
() Q \ {x = k} :
8>>><>>>:
z2
c2
✓
1 +
k2
a2
◆ � y2
b2
✓
1 +
k2
a2
◆ = 1
x = k
,
(k, 0, 0) �OZ8<:y = ±
b
a
z
x = k
k 2 R
(x0, y0, z0) 2 Q
r :
8><>:
x(t) = ↵t+ x0
y(t) = �t+ y0
z(t) = �t+ z0
, t 2 R ,
(↵, �, �) 6= (0, 0, 0) (x0, y0, z0)
(↵t+ x0, �t+ y0, �t+ z0) 2 Q
�(↵t+ x0)
2
a2
� (�t+ y0)
2
b2
+
(�t+ z0)2
c2
= 1
() t2
✓
�↵
2
a2
� �
2
b2
+
�2
c2
◆
+ 2t
✓
�↵x0
a2
� �y0
b2
+
�z0
c2
◆
= 0 ,
�x
2
0
a2
� y
2
0
b2
+
z20
c2
= 1 r ⇢ Q
�↵
2
a2
� �
2
b2
+
�2
c2
= 0
�↵x0
a2
� �y0
b2
+
�z0
c2
= 0 .
� 6= 0 ↵ = � = 0
(↵, �, �) 6= (0, 0, 0)
� 6= 0 r XY
r ⇢ Q Q \ XY = ?
⇤
�OX �OY �OZ
�x
2
a2
+
y2
b2
+
z2
c2
= 0 ,
x2
a2
� y
2
b2
+
z2
c2
= 0 ,
x2
a2
+
y2
b2
� z
2
c2
= 0 ,
a b c
Q
z =
�OZ
Q : x
2
a2
+
y2
b2
=
z2
c2
.
Q
XY
Q \ {z = k} :
(
x2
a2
+
y2
b2
=
k2
c2
z = k
,
(0, 0, k) k 6= 0
(0, 0, 0) k = 0
Q \ {y = k} k > 0
Q y = k
k 2 R XZ
Q \ {y = k} :
8<: �
x2
a2
+
z2
c2
=
k2
b2
y = k
,
(0, k, 0)
�OZ(
x = ± c
a
z
y = k ,
k 6= 0
(
x = ± c
a
z
y = 0 ,
k = 0
Q \ {y = 0} Q \ {y = k} k < 0
Q Y Z
C \ {x = k} :
8<:
z2
c2
� y
2
b2
=
k2
a2
x = k
,
Q \ {x = k} k > 0
(k, 0, 0)
�OZ(
y = ±c
b
z
x = k
,
k 6= 0
(
y = ±c
b
z
x = 0
,
k = 0
x y z
y2
b2
+
z2
c2
= 1 ,
x2
a2
+
z2
c2
= 1 ,
x2
a2
+
y2
b2
= 1 ,
a b c
�OZ
Q : x
2
a2
+
y2
b2
= 1 .
XY
Q \ {z = k} :
8<:
x2
a2
+
y2
b2
= 1
z = k
,
(0, 0, k) �OZ
Q XY
Q y = k k 2 R XZ
Q \ {y = k} :
8<:
x2
a2
= 1� k
2
b2
y = k
,
• �OZ
8<:x = ±
a
b
p
b2 � k2
y = k
k 2 (�b, b)
• �OZ
⇢
x = 0
y = b
k = b
• �OZ
⇢
x = 0
y = �b k = �b
• |k| > b
Q XZ
Q \ {x= k} :
8<:
y2
b2
= 1� k
2
a2
x = k
• �OZ
(
y = ± b
a
p
a2 � k2
x = k
k 2 (�a, a)
• �OZ
⇢
y = 0
x = a
k = a
• �OZ
⇢
y = 0
x = �a k = �a
• k 2 (�1,�a) [ (a,1)
Q Y Z
Q : x
2
a2
+
y2
b2
= 1 �OZ P = (x0, y0, z0)
Q r P
�OZ Q
x02
a2
+
y02
b2
= 1 P 2 Q r = {(x0, y0, z0 + t) ; t 2 R}
(x0, y0, z0 + t) 2 Q
t 2 R ⇤