เรขาคณิตวิเคราะห์ 25-12-53
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1
1.1
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2
1.2
P (x1, y1) Q Q(x2, y2)
P (x1, y1) Q(x2, y2)
P Q
1.1.
1. P (3, 2) Q(1, −1)
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1.2 3
2. P (3, 0) Q(6, 0)
3. P (1, −2) Q(1, 2)
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4
1.3
P (x1, y1) Q(x2, y2) x1 = x2
m P Q
m = y2 − y1
x2 − x1
= tan θ
θ X 0◦ ≤ θ < 180◦
1. X (θ = 0◦
)
2. X (0◦ < θ < 90◦)
3. Y (θ = 90◦)
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1.3 5
4. X
(90◦ < θ < 180◦
)
1.
2. ( Y)
3. ( Y) −1
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6
1.4 1. X Y (0, b)
y = b
2. Y X (a, 0)
x = a
3. X Y
• (x1, y1) m
y − y1 = m(x − x1)
• m Y (0, c)
y = mx + c
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1.4 7
4.
Ax + By + C = 0
A, B C A B
Ax + By + C = 0
By = −Ax − C
y = −A
Bx − C
B
Ax + By + C = 0 − A
B Y −C
B
X (a, 0) a X
(x-intercept) Y (0, b) b Y
(y-intercept)
1.2. Y
3x − 2y − 6 = 0
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8
1.5
l Ax + By + C = 0 P (x1, y1)
l d P
l
d = |Ax1 + By1 + C |
√ A2
+ B2
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1.5 9
1.3. ( A-NET )
l1 (5, 2) (1, −6) l2 (3, −1)
−1 (a, b) a+b
1. − 2 2. − 1 3. 1 4. 2
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10
1.6
1.4. (circle)
(center) (radius)
1.6.1
• (0, 0) r
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1.6 11
x2 + y2 = r2
• (h, k) r
(x
−h)2 + (y
−k)2 = r2
(3, −2) 5
(x − 3)2 + (y + 2)2 = 52
x2 + y2 − 6x + 4y − 12 = 0
•
x2 + y2 + Ax + By + C = 0
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12
x2 + y2 + Ax + By + C = 0
(x2 + Ax) + (y2 + By) = −C
(x2 + 2(A
2 )x +
A2
4 ) + (y2 + 2(
B
2 )y +
B2
4 ) = −C +
A2
4 +
B2
4
(x +
A
2 )2
+ (y +
B
2 )2
= −C +
A2
4 +
B2
4
x2 + y2 + Ax + By + C = 0
(−A
2, −B
2 )
– −C + A2
4 +
B2
4 > 0
r =
−C +
A2
4 +
B2
4
– −C + A2
4 +
B2
4 = 0
r = 0 ()
– −C +A2
4 +
B2
4 < 0
(imaginary circle)
1.5.
x2 + y2 − 12x + 10y + 45 = 0
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1.6 13
1.6.2
1. X
(h, k) X |k| (h, 0)
2. Y
(h, k) Y |h| (0, k)
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14
3. X Y
(h, k) Ax + By + C = 0 r = |Ah + Bk + C |√
A2 + B2
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1.6 15
1.6. ( A-NET )
a ax + 12y + 15 = 0
x2 + y2 − 14x + 4y + 49 = 0 a 1. (0, 4] 2. (4, 8] 3. (8, 12] 4. (12, 16]
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16
1.7. ( A-NET )
l x2 − 2x + y2 + 10y − 39 = 0 (2, 3) l
1. (4, −43
8 ) 2. (5, −9
2)
3. (2, −13) 4. (3, −11)
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1.7 17
1.7 1.8. (parabola)
(focus)
(directrix)
(parabola axis)
(axis of symmetry) (latus rectum)
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18
(0, 0)
(x)2 = 4c(y) (y)2 = 4c(x)
(0, 0)
c > 0 c < 0 c > 0 c < 0
(0, c) (c, 0)
y = −c x = −c
x = 0 y = 0
|4c|
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1.7 19
(h, k)
(x − h)2 = 4c(y − k) (y − k)2 = 4c(x − h)
(h, k)
c > 0 c < 0 c > 0 c < 0
(h, k + c) (h + c, k)
y = k − c x = h − c
x = h y = k
|4c|
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20
x2 + Ax + By + C = 0
y2 + Ax + By + C = 0
x2 + Ax + By + C = 0
x2 + 2(A
2 ) +
A2
4 = −By − C +
A2
4
(x + A
2 )2 =
−B(y
−C − A2
4
−B )
(x + A
2 )2 = 4(
−B
4 )(y +
C − A2
4B
)
x2+Ax+By+C = 0 (−A2
, −C − A2
4B
)
B
4 < 0
B
4 > 0
y2 + Ax + By + C = 0
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1.7 21
1.9. (3,
−8)
y = −2
1.10.
y2 + 2y + 8x + 25 = 0
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22
1.11. ( 2548 )
y2 − 4y − 16x − 12 = 0
l
3x − 2y + 5 = 0
y l
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1.7 23
1.12. ( A-NET )
y = 5 x2 + y2 − 6x + 2y − 6 = 0 1. x2 − 6x + 12y − 15 = 0 2. x2 − 6x − 12y + 33 = 0
3. x2 − 6x + 12y + 21 = 0 4. x2 − 6x − 12y − 3 = 0
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24
1.8
1.13. (Ellipse) 2
2 F F
P F F
(vertex) A
A
A A
(major axis) C
(center)
B B
(minor axis) a
b c
a2 = b2 + c2
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1.8 25
P (x, y) |P F | + |P F | = |2a| (0, 0)
x2
a2 +
y2
b2 = 1 a > b
x2
b2 +
y2
a2 = 1 a > b
(0, 0) (0, 0)
(±a, 0) (0, ±a)
(±c, 0) (0, ±c)
(0, ±b) (±b, 0)
|2a| |2a|
|2b| |2b|
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26
(h, k)
(x
−h)2
a2 +
(y
−k)2
b2 = 1
(x
−h)2
b2 +
(y
−k)2
a2 = 1
(h, k) (h, k)
(h ± a, k) (h, k ± a)
(h ± c, k) (h, k ± c)
(h, k ± b) (h ± b, k)
|2a| |2a|
|2b| |2b|
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1.8 27
Ax2 + By2 + Dx + Ey + F = 0
A B
. A > B > 0
. B > A > 0
1.14.
16x2 + 25y2 + 128x − 50y − 119 = 0
1.15. (5, 2) (5, −2) 12
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28
1.16. ( A-NET )
x = y (x − 1)2
9 + (y − 1)2
4 = 1 A
B F 1 F 2 AF 1 + AF 2 + BF 1 + BF 2
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1.8 29
1.17. ( A-NET )
C x2 + 2y2 + 4x − 4y + 2 = 0 C y = −x A B AB
1.√
3
2.
√ 5
3.
√ 6 4.
√ 8
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30
1.18. ( A-NET )
X 4x2 + 3y2 − 16x + 4 = 0
1. y2 = x 2. y2 = 4x
3. 2y2 = x 4. 8y2 = x
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1.9 31
1.9 1.19. (Hyperbola) 2
P P F P
F A A
A
A
C C
AA AA
BB
BB
(conjugate axis) |2c| |2a| |2b|
c2 = b2 + a2
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32
(0, 0)
(x)
2
a2 − (y)
2
b2 = 1 y
2
a2 − x
2
b2 = 1
(0, 0) (0, 0)
(±a, 0) (0, ±a)
(±c, 0) (0, ±c)
(0, ±b) (±b, 0)
|2a
| |2a
||2b| |2b|
asymtotex
a ± y
b = 0
y
a ± x
b = 0
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1.9 33
(h, k)
(x − h)2
a2 − (y − k)
2
b2 = 1 (y − k)
2
a2 − (x − h)
2
b2 = 1
(h, k) (h, k)
(h ± a, k) (h, k ± a)
(h ± c, k) (h, k ± c)
(h, k ± b) (h ± b, k)
|2a| |2a|
|2b| |2b| asymtote
x − h
a ± y − k
b = 0
y − k
a ± x − h
b = 0
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34
Ax2 + By2 + Cx + Dy + E = 0
A B
. A > 0 B < 0
. A < 0 B > 0
1.20.
3x2 − y2 + 18x + 4y + 20 = 0
1.21. (2, −1) (2, 3) 8
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1.9 35
1.22. ( A-NET )
9x2 − 16y2 − 18x − 64y − 199 = 0
3x + 4y − 8 = 0
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36
1.23. F 1 F 2 16y2−9x2+36x−32y−164 = 0 V x2+y2+2x−4y+1 = 0 F 1F 2V
1. 13 + 2√
5
2. 13 + 3√
5
3. 15 + 2√ 54. 15 + 3
√ 5
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1.9 37
1.24. ( A-NET )
F 6x2−10y2−12x−40y−94 = 0
F
1. y2 + 4y−
4x−
9 = 0
2. y2 + 4y − 4x + 24 = 0
3. y2 + 4y − 16x − 44 = 0
4. y2 + 4y − 16x + 84 = 0
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38
1.25. ( A-NET )
(−4, 0) (−5, 0) (1, 0)
1. (x + 2)2
4 +
y2
5 = 1
2. (x + 2)2
5 +
y2
9 = 1
3. (x − 2)2
4 +
y2
9 = 1
4. (x
−2)2
4 +
y2
5 = 1
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1.9 39
( A-NET)
1. (5,-1) y =
x
2 X
1. y = 7 2. y = 7
2 3. y = 5
2 4. y = 6
2. y = 5 x2 + y2 − 6x + 2y − 6 = 0
1. x
2
−6x+12y−15 = 0 2. x
2
−6x−12y+33 = 03. x2−6x+12y +21 = 0 4. x2−6x−12y−3 = 0
3. 4x2 + 9y2 − 48x + 72y + 144 = 0 3x + 4y = 5
1. 4x − 3y + 12 = 0 2. 4x − 3y − 12 = 03. 4x − 3y − 36 = 0 4. 4x − 3y + 36 = 0
4. E 6x2 + 5y2 + 12x − 20y − 4 = 0 H E E
E H
1. x2 − 5y2 − 2x − 20y + 14 = 0
2. x2 − 5y2 + 2x + 20y − 14 = 0
3. x2 − 5y2 + 2x + 20y − 18 = 0
4. 5x2
−y2
−2x + 20y + 18 = 0
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40
5. (h, k) 2x+3y = 6 h, k 2x − y = 1 2x + 7 = −3 1. [2, 4] 2. [4, 5] 3. [5, 6] 4. [6, 7]
6. x2 + y2 + 2ax + 2by + c = 0 y
2 x 1. a2 > c b2 > c 2. a2 > c b2 < c
3. a2 < c b2 > c 4. a2 < c b2 < c
7. (10, 7)
5x2 + 5y2 − 20x − 10y − 100 = 0
8. A 5x − 4y − 15 = 0 x +
y − 12 = 0 B C 4y2 − 9x2 − 16y + 18x − 29 = 0 ABC
9. P x2 + 8y + 2x + a = 0 a < 0 y = 4 P x A A P
10.
186 162 ( )
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[1] , A-NET ,www.neutron.rmutphysics.com
[2] , . , www.atom.rmutphysics.com
[3] , .4-5-6 , ,.