46
ใบความรทู ี่ 1.7 (ภาคตัดกรวย)
⌫   
⌫ ⌫  ⌦
„µ¦Á¨ºÉ°œÂ„œšµŠ…œµœ (Translation of Axes)
„µ¦Á¨ºÉ°œÂ„œšµŠ…œµœ ®¤µ¥™¹Š„µ¦Áž¨¸É¥œÂž¨ŠÂ„œ¡·„´—Á—·¤°¥nµŠœo°¥®œ¹ÉŠÂ„œ („œ X ®¦º°Â„œ Y)
ץĮo„œ¡·„´—Ä®¤n…œµœ„´Â„œ¡·„´—Á—·¤
„µ¦Á¨ºÉ°œÂ„œšµŠ…œµœœ´Áž}œ¡ºÊœ“µœš¸É­Îµ‡´š¸É‹³nª¥Äœ„µ¦«¹„¬µÁ„¸É¥ª„´£µ‡˜´—„¦ª¥Ä—o­³—ª„¥·ÉŠ…¹Êœ
Ĝ¦³Â„œ¡·„´—Œµ„ Á¦µÄo„œ X ¨³ Y ­Îµ®¦´°oµŠ°·Š¡·„´—®¦º°˜ÎµÂ®œnŠ…°Š‹»—Äœ¦³œµ
Y
‹»— P(x, y) Áž}œ‹»—š¸É°¥¼n®nµŠ‹µ„„œ Y ŞšµŠ…ªµ¤º°Áž}œ¦³¥³ x ®œnª¥
X
P(x, y)
¨³°¥¼n®nµŠ‹µ„„œ X Ž¹ÉŠ°¥¼nÁ®œº°Â„œ X Áž}œ¦³¥³ y ®œnª¥
y
X
O
Á¤ºÉ°Á¨ºÉ°œÂ„œ ‹»— P(x, y) ¥´Š‡Šš¸É ˜n¡·„´—…°Š‹»— P ‹³Áž¨¸É¥œÅžÁ¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n —´Š¦¼ž
Yc
x
P(x, y)
xc y Pc(xc, yc)
yc
Oc (h, k)
X
h k
X
O
‹µ„¦¼ž
„œ¡·„´—Ä®¤n Xc ¨³ Yc …œµœ„´Â„œ¡·„´—Á—·¤ X ¨³ Y ˜µ¤¨Îµ—´ ¡·„´—…°Š‹»—„εÁœ·—Ä®¤nÁ¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤
‡º°‹»— Oc(h, k) œ´Éœ‡º°Â„œ¡·„´—Ä®¤nÁ„·—‹µ„„µ¦Á¨ºÉ°œÂ„œ˜µ¤Âœªœ°œ h ®œnª¥ ¨³˜µ¤Âœª˜´ÊŠ k ®œnª¥
Ä®o (x, y) Áž}œ¡·„´—…°Š‹»— P Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤
(xc, yc) Áž}œ¡·„´—…°Š‹»— P Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n ¨³ h, k Áž}œ‹Îµœªœ‹¦·Š
—´Šœ´Êœ ‹³Å—o
x = xc + h
xc = x - h
®¦º°
y = yc + k
yc = y - k
˜´ª°¥nµŠš¸É 1 ™oµÁ¨ºÉ°œÂ„œÅžÃ—¥Äo‹»— (-2, 3) Áž}œ‹»—„εÁœ·—Ä®¤n Ž¹ÉŠ A(0, 2), B(-5, 4), C(4, -1) ¨³ D(-3, -5) Áž}œ¡·„´—
…°Š‹»—Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤ ‹Š®µ¡·„´—…°Š‹»—Á®¨nµœ¸ÊÁ¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n Yc Y
ª·›¸šÎµ
Ä®o (x, y) Áž}œ¡·„´—…°Š‹»—Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤
B
¨³ (xc, yc) Áž}œ¡·„´—…°Š‹»—Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n
A
(-2, 3)
Ĝš¸Éœ¸Ê (h, k) = (-2, 3) œ´Éœ‡º° h = -2 ¨³ k = 3
‹µ„ xc = x - h ¨³ yc = y - k
‹³Å—o xc = x + 2 ¨³ yc = y - 3
O
C
(1) A(0, 2)
Ž¹ÉŠ x = 0 , y = 2
‹³Å—o xc = 0 + 2 = 2 ¨³ yc = 2 - 3 = -1
D
—´Šœ´Êœ ¡·„´—…°Š‹»— A(0, 2) Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n ‡º° ‹»— (2, -1)
(2) B(-5, 4) Ž¹ÉŠ x = -5 , y = 4
‹³Å—o xc = -5 + 2 = -3 ¨³ yc = 4 - 3 = 1
—´Šœ´Êœ ¡·„´—…°Š‹»— B(-5, 4) Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n ‡º° ‹»— (-3, 1)
Xc
X
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47
Ž¹ÉŠ x = 4 ¨³ y = -1
‹³Å—o xc = 4 + 2 = 6 ¨³ yc = -1 - 3 = -4
—´Šœ´Êœ ¡·„´—…°Š‹»— C(4, -1) Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n ‡º° ‹»— (6, -4)
(4) D(-3, -5) Ž¹ÉŠ x = -3 ¨³ y = -5
‹³Å—o xc = -3 + 2 = -1 ¨³ yc = -5 - 3 = -8
—´Šœ´Êœ ¡·„´—…°Š‹»— D(-3, -5) Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n ‡º° ‹»— (-1, -8)
˜´ª°¥nµŠš¸É 2 ™oµÁ¨ºÉ°œÂ„œÅžÃ—¥Äo‹—» (3, -4) Áž}œ‹»—„εÁœ·—Ä®¤n Ž¹ÉŠ P(-4, 3), Q(-5, -2) ¨³ R(2, 7) Áž}œ¡·„—´ …°Š‹»—
Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n ‹Š®µ¡·„´—…°Š‹»—Á®¨nµœ¸ÊÁ¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤
ª·›¸šµÎ Ä®o (x, y) Áž}œ¡·„´—…°Š‹»—Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤ ¨³(xc, yc) Áž}œ¡·„´—…°Š‹»—Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Ä®¤n
Ĝš¸Éœ¸Ê (h, k) = (3, -4) œ´Éœ‡º° h = 3 ¨³ k = -4
‹µ„ x = xc+ h ¨³ y = yc + k ‹³Å—o x = xc+ 3 ¨³ y = yc - 4
Y
Yc
(1) P(-4, 3) Ž¹ÉŠ xc = -4 ¨³ yc = 3
R
‹³Å—o x = -4 + 3 = -1 ¨³ y = 3 - 4 = -1
—´Šœ´Êœ ¡·„´—…°Š‹»— P(-4, 3) Á¤ºÉ°Áš¸¥„´Â„œ¡·„—´ Á—·¤ ‡º° ‹»— (-1, -1)
O
(2) Q(-5, -2) Ž¹ÉŠ xc = -5 ¨³ yc = -2
P
‹³Å—o x = -5 + 3 = -2 ¨³ y = -2 - 4 = -6
—´Šœ´Êœ ¡·„´—…°Š‹»— Q(-5, -2) Á¤ºÉ°Áš¸¥„´Â„œ¡·„´—Á—·¤ ‡º° ‹»— (-2, -6)
(3, -4)
Q
(3) R(2, 7) Ž¹ÉŠ xc = 2 ¨³ yc = 7
‹³Å—o x = 2 + 3 = 5 ¨³ y = 7 - 4 = 3
—´Šœ´Êœ ¡·„´—…°Š‹»— R(2, 7) Á¤ºÉ°Áš¸¥„´Â„œ¡·„—´ Á—·¤ ‡º° ‹»— (5, 3)
⌦
⌦
(3) C(4, -1)
X
Xc
…o°˜„¨Š "„µ¦Á¨ºÉ°œÂ„œšµŠ…œµœÃ—¥¤¸‹—» (h, k) Áž}œ‹»—„εÁœ·—Ä®¤n" Á¦¸¥„­´ÊœÇ ªnµ "„µ¦Á¨ºÉ°œÂ„œÅžš¸É‹»— (h, k)"
˜´ª°¥nµŠš¸É 3 ™oµÁ¨ºÉ°œÂ„œÅžš¸É‹»— (-3, 4) „¦µ¢…°Š­¤„µ¦ y = | x + 3 | + 4 ‹³¤¸­¤„µ¦Áš¸¥„´Â„œÄ®¤n
Ž¹ÉŠÄo¡·„´— (xc, yc) šœ¡·„´— (x, y) Áž}œ°¥nµŠÅ¦
ª·›¸šÎµ ‹µ„Ëš¥r Á¨ºÉ°œÂ„œÅžš¸É‹»— (-3, 4) ‹³Å—o (h, k) = (-3, 4) œ´Éœ‡º° h = -3, k = 4
ÁœºÉ°Š‹µ„ x = xc + h ¨³ y = yc + k ‹³Å—o x = xc - 3 ¨³ y = yc + 4
‹µ„­¤„µ¦ y = | x + 3 | + 4 šœ‡nµ x —oª¥ xc - 3 ¨³Âšœ‡nµ y —oª¥ yc + 4
‹³Å—o yc + 4 = | xc - 3 + 3 | + 4
yc + 4 - 4 = | xc - 3 + 3 |
‹³Å—o yc = | xc| ‹³Áž}œ­¤„µ¦Áš¸¥„´Â„œÄ®¤n…°Š„¦µ¢¦¼žœ¸Ê
˜´ª°¥nµŠš¸É 4 ™oµÁ¨ºÉ°œÂ„œÅžš¸É‹»— (3, -7) „¦µ¢…°Š­¤„µ¦ x2 - 6x + y2 + 14y - 2 = 0 ‹³¤¸­¤„µ¦Áš¸¥„´Â„œÄ®¤n
Ž¹ÉŠÄo¡·„´— (xc, yc) šœ¡·„´— (x, y) Áž}œ°¥nµŠÅ¦
ª·›¸šÎµ ‹µ„Ëš¥r Á¨ºÉ°œÂ„œÅžš¸É‹»— (3, -7) ‹³Å—o (h, k) = (3, -7) œ´Éœ‡º° h = 3, k = -7
ÁœºÉ°Š‹µ„ x = xc + h ¨³ y = yc + k ‹³Å—o x = xc+ 3 ¨³ y = yc - 7
‹µ„­¤„µ¦ x2 - 6x + y2 + 14y - 2 = 0 šœ‡nµ x —oª¥ xc+ 3 ¨³Âšœ‡nµ y —oª¥ yc - 7
‹³Å—o
(xc+ 3)2 - 6(xc+ 3) + (yc - 7)2 + 14(yc - 7) - 2 = 0
(xc)2 + 6xc + 9 - 6xc- 18 + (yc)2 - 14 yc + 49 + 14yc - 98 - 2 = 0
(xc)2 + (yc)2 = 60
‹³Å—o (xc)2 + (yc)2 = 60 ‹³Áž}œ­¤„µ¦Áš¸¥„´Â„œÄ®¤n…°Š„¦µ¢¦¼žœ¸Ê
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48
⌫   
⌫ ⌫  ⌦
˜´ª°¥nµŠš¸É 5 ‹µ„­¤„µ¦ÄœÂ˜n¨³…o°˜n°Åžœ¸Ê ™oµ˜o°Š„µ¦Á¨ºÉ°œÂ„œ°oµŠ°·ŠÄ®oŗo­¤„µ¦Äœ¦¼žš¸É„ε®œ—Ä®o¨oª
‹³Á¨º°„‹»—Ä—Áž}œ‹»—„εÁœ·—
(1) 2x - 3y + 12 = 0 ˜o°Š„µ¦Ä®o°¥¼nĜ¦¼ž 2xc = 3yc
ª·›¸šÎµ ‹µ„­¤„µ¦ 2x - 3y + 12 = 0
‹³Å—o
2x = 3y - 12
2x = 3(y - 4) ..............................(1)
Ä®o xc = x ¨³ yc = y - 4
šœ x —oª¥ xc ¨³ šœ y - 4 —oª¥ yc ¨ŠÄœ­¤„µ¦ (1)
‹³Å—o ­¤„µ¦ 2xc = 3yc °¥¼nĜ¦¼žš¸É˜o°Š„µ¦ ¨³‹»—„εÁœ·—Ä®¤n‡º°‹»— (0, 4)
®¦º° ‹µ„­¤„µ¦ 2x - 3y + 12 = 0 ‹³Å—o 2x + 6 = 3y - 6
2(x + 3) = 3(y - 2) ...............................(2)
Ä®o xc = x + 3 ¨³ yc = y - 2
šœ x + 3 —oª¥ xc ¨³ šœ y - 2 —oª¥ yc ¨ŠÄœ­¤„µ¦ (2)
‹³Å—o ­¤„µ¦ 2xc = 3yc °¥¼nĜ¦¼žš¸É˜o°Š„µ¦ ¨³‹»—„εÁœ·—Ä®¤n‡º°‹»— (-3, 2)
…o°­´ŠÁ„˜ ‹³Á®Èœªnµ 2x - 3y + 12 = 0 Áž}œ­¤„µ¦Á­oœ˜¦Š ‹³Á¨º°„‹»—„εÁœ·—Ä®¤nÄ—Ç „Èŗoš¸ÉÁž}œ‹»—°¥¼nœÁ­oœ˜¦Šœ¸Ê
(2) y(x - 5) = 3 ˜o°Š„µ¦Ä®o°¥¼Än œ¦¼ž ycxc = 3
ª·›¸šÎµ ‹µ„­¤„µ¦ y(x - 5) = 3 ...........................................(1)
Ä®o xc = x - 5 ¨³ yc = y
šœ x - 5 —oª¥ xc ¨³ šœ y —oª¥ yc ¨ŠÄœ­¤„µ¦ (1)
‹³Å—o ­¤„µ¦ ycxc = 3 °¥¼nĜ¦¼žš¸É˜o°Š„µ¦ ¨³‹»—„εÁœ·—Ä®¤n‡º°‹»— (5, 0)
(3) x2 + y2 - 8x + 6y + 24 = 0 ˜o°Š„µ¦Ä®o°¥¼nĜ¦¼ž (xc)2 + (yc)2 = 1
ª·›¸šÎµ ‹µ„­¤„µ¦
x2 + y2 - 8x + 6y + 24 = 0
‹³Å—o
(x2 - 8x + 16) + (y2 + 6y + 9) = -24 + 16 + 9
(x - 4)2 + (y + 3)2 = 1 ...............................(1)
Ä®o xc = x - 4 ¨³ yc = y + 3
šœ x - 4 —oª¥ xc ¨³ šœ y + 3 —oª¥ yc ¨ŠÄœ­¤„µ¦ (1)
‹³Å—o ­¤„µ¦ (xc)2 + (yc)2 = 1 °¥¼nĜ¦¼žš¸É˜o°Š„µ¦ ¨³‹»—„εÁœ·—Ä®¤n‡º°‹»— (4, -3)
c2
c2
(4) 25x2 - 9y2 + 50x + 36y = 236 ˜o°Š„µ¦Ä®o°¥¼Än œ¦¼ž ( x ) ( y ) 1
9
25
ª·›¸šÎµ ‹µ„­¤„µ¦ 25x2 - 9y2 + 50x + 36y = 236
‹³Å—o
(25x2 + 50x) - (9y2 - 36y) = 236
25(x2 + 2x) - 9(y2 - 4y ) = 236
25(x2 + 2x + 1) - 9(y2 - 4y + 4) = 236 + 25 -36
25(x + 1)2 - 9(y - 2)2 = 225
( x 1) 2 ( y 2 ) 2
1 .............................................(1)
9 25
Ä®o xc = x + 1 ¨³ yc = y - 2
šœ x + 1 —oª¥ xc ¨³ šœ y - 2 —oª¥ yc ¨ŠÄœ­¤„µ¦ (1)
( xc) 2 ( yc) 2
‹³Å—o ­¤„µ¦ 9 25 1 °¥¼nĜ¦¼žš¸É˜o°Š„µ¦ ¨³‹»—„εÁœ·—Ä®¤n‡º°‹»— (-1, 2)
sm.tm
„µ¦Á¨ºÉ°œÂ„œšµŠ…œµœ„´„µ¦Á…¸¥œ„¦µ¢
„µ¦Á…¸¥œ„¦µ¢Ã—¥„µ¦Á¨ºÉ°œÂ„œšµŠ…œµœÅžš¸É‹»— (h, k) š¸ÉÁ®¤µ³­¤ ‹³Á…¸¥œŠnµ¥„ªnµ„µ¦Á…¸¥œ„¦µ¢Äœ¦³
¡·„´—Œµ„š¸É¤‹¸ —» „εÁœ·—š¸É‹»— (0, 0) ×¥Áž¨¸É¥œ¡·„´—‹»— P(x, y) Ä—Ç Äœ¦³Á—·¤ Áž}œ P(xc, yc) Ĝ¦³Ä®¤n ×¥š¸É xc = x - h
¨³ yc = y - k ‹³šÎµÄ®o­¤„µ¦Áš¸¥„´Â„œÄ®¤n¤¸¦¼žŽ¹ÉŠ­³—ª„˜n°„µ¦Á…¸¥œ„¦µ¢ —´Š˜´ª°¥nµŠ˜n°Åžœ¸Ê
Yc Y
˜´ª°¥nµŠš¸É 6 ‹ŠÁ…¸¥œ„¦µ¢…°Š­¤„µ¦˜n°Åžœ¸Ê
(1) y = | x + 3 | + 4
ª·›¸šÎµ ‹µ„­¤„µ¦ y = | x + 3 | + 4
Xc
(-3, 4)
‹´—Å—oÁž}œ y - 4 = | x + 3 |
¨³Á¨ºÉ°œÂ„œÅžš¸É‹—» (-3, 4)
X
O
‹³Å—o ­¤„µ¦Áš¸¥„´Â„œÄ®¤n ‡º° yc = | xc|
(2)
ª·›¸šÎµ
y = (x - 5)2
‹µ„­¤„µ¦ y = (x - 5)2
Á¨ºÉ°œÂ„œÅžš¸É‹»— (5, 0)
‹³Å—o ­¤„µ¦Áš¸¥„´Â„œÄ®¤n ‡º° yc = (xc)2
Yc
Y
O
(5, 0)
(3)
ª·›¸šÎµ
(4)
ª·›¸šÎµ
(5)
ª·›¸šÎµ
y = (x - 5)2 + 3
‹µ„­¤„µ¦ y = (x - 5)2 + 3
‹´—Å—oÁž}œ y - 3 = (x - 5)2
¨³Á¨ºÉ°œÂ„œÅžš¸É‹—» (5, 3)
‹³Å—o ­¤„µ¦Áš¸¥„´Â„œÄ®¤n ‡º° yc = (xc)2
y = x3 + 3x2 + 3x + 3
‹µ„­¤„µ¦ y = x3 + 3x2 + 3x + 3
‹´—Å—oÁž}œ y = (x3 + 3x2 + 3x + 1) + 2
y - 2 = x3 + 3x2 + 3x + 1
y - 2 = (x + 1)3
¨³Á¨ºÉ°œÂ„œÅžš¸É‹—» (-1, 2)
‹³Å—o ­¤„µ¦Áš¸¥„´Â„œÄ®¤n ‡º° yc = (xc)3
y = x3 + 3x2 + 3x
‹µ„­¤„µ¦ y = x3 + 3x2 + 3x
‹´—Å—oÁž}œ y + 1 = x3 + 3x2 + 3x + 1
y + 1 = (x + 1)3
¨³Á¨ºÉ°œÂ„œÅžš¸É‹—» (-1, -1)
‹³Å—o ­¤„µ¦Áš¸¥„´Â„œÄ®¤n ‡º° yc = (xc)3
Xc
X
Yc
Y
Xc
(5, 3)
X
O
Yc Y
(-1, 2)
Xc
X
O
Yc Y
O
X
Xc
(-1, -1)
sm.tm
⌦
⌦
49

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