P2 EA Noturno Gab Algebra linear

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A = (2,�2) B = (1, 5) C = (�2, 0) 30�
M =
" p
3
2 � 12
1
2
p
3
2
#
N =
 �1 0
0 �1
�
A0 =
��1�p3,�1 +p3� B0 = � 12 (5�p3),� 12 (1 + 5p3�
C 0 =
�p
3, 1
�
P = (x, y) F1 = (1, 0) F2 = (3, 0)
5
G = {P = (x, y)/d(P, F1) + d(P, F2) = 5}
G :
p
(x� 1)2 + y2 + p(x� 3)2 + y2 = 5 G : px2 + y2 � 2x+ 1 + px2 + y2 � 6x+ 9 = 5
w = x2 + y2 G :
p
w � 2x+ 1 +pw � 6x+ 9 = 5
G : 2w � 8x+ 10 + 2p(w � 2x+ 1) (w � 6x+ 9) = 25
2
p
w2 � 8xw + 10w + 12x2 � 24x+ 9 = �2w + 8x+ 15
4w2 � 32xw + 40w + 48x2 � 96x+ 36 = 4w2 + 64x2 + 225� 32xw � 60w + 240x.
100w � 16x2 � 336x = 189
w 84x2 � 336x+ 100y2 = 189
x 84(x� 2)2 + 100y2 = 525
(x� 2)2�
25
4
� + y2� 21
4
� = 1.
x2 � 6x + 5y � 11 = 0 R = (�1, 1)
P : (x� 3)2 = �5(y � 4)
x = 3
(�1, 1) L : y � 1 = m(x + 1)
x2 + (5m� 6)x+ 5m� 6 = 0
(5m� 6)2� 4(5m� 6) = 0
R m = 65 m = 2
m = 65 L1 : y � 1 = 65 (x+ 1) L2 : y � 1 = 2(x+ 1).
x = 3 Q1 =
�
3, 295
�
Q2 = (3, 9)
A =
24 1 0 00 3 �2
0 �2 �3
35 .
det
0@24 1� � 0 00 3� � �2
0 �2 �3� �
351A = 0
�1 = �
p
13 �2 =
p
13 �3 = 1
(A� �I)X = 0
V1 =
24 0p13� 3
2
35 V2 =
24 0�p13� 3
2
35 V3 =
24 10
0
35
kV1k =
p
26� 6p13 kV1k =
p
26 + 6
p
13
A =
2664
0 0 1
�3+p13p
26�6p13
�3�p13p
26+6
p
13
0
2p
26�6p13
2p
26+6
p
13
0
3775
24 �p13 0 00 p13 0
0 0 1
35
2664
0 �3+
p
13p
26�6p13
2p
26�6p13
0 �3�
p
13p
26+6
p
13
2p
26+6
p
13
1 0 0
3775 .
x2 + y2 + 2z2 � 2x+ 4z + 4y = 18�
x2 � 2x�+ �y2 + 4y�+ 2 �z2 + 2z� = 18�
x2 � 2x+ 1�+ �y2 + 4y + 4�+ 2 �z2 + 2z + 1� = 18 + 1 + 4 + 2
(x� 1)2 + (y + 2)2 + 2 (z + 1)2 = 25 C = (1,�2,�1)
a = b = 5 c = 52
p
2
z