- 1 -

conic ections

ParabolaElipseHypebola

0,0

0,0

퐹(0, 푃)퐹(푃, 0)

푦 = −푝푥 = −푝

푥 = 0푦 = 0

- 2 -

푦 = 4푝푥

0,0

0,0

푦 = 12푥푦 = 4푝푥

4푝 = 12푝 = 3F(3,0)

F(P, 0)

푥 = −푝

푥 = −3

- 3 -

푦 = −6푥푦 = 4푝푥

4푝 = −6

ℎ, 퐾

- 4 -

(푦 − 퐾) = 4푝(푥 − ℎ)(푥 − ℎ) = 4푝(푦 − 퐾)푦 = 4푝푥푥 = 4푝푦

−ℎ푥 −푘푦

퐹(ℎ, 푝 + 퐾)퐹(푝, 0)

퐹(ℎ, 퐾 + 푝)퐹(0, 푝)

푥 = ℎ − 푃푥 = −푃

푦 = 퐾푦 = 0

푦 = 퐾 − 푃푦 = −푃

푥 = ℎ푥 = 0

(푥 + 3) + (푦 − 2) = 0(푦 − 2) = −(푥 − 3)

(푦 − 퐾) = 4P(푥 − ℎ)4푝 = −1

푝 =−14h = −3,K = 2

- 5 -

푦 − 퐾 = 0

푦 = 퐾

퐹(ℎ + 푝,퐾)

퐹(−3 −14, 2) = 퐹(

−134

, 2)

푥 = ℎ − 푃

푥 = −3 +14

푥 =−114

4푝 = −8푝 = −2ℎ = −2K = −22,-2

h, k

푦 + 4y + 8푥 − 12 = 0

푦 + 4y + 0 = −8푥 + 12 + 0

푦 + 4y + 4 = −8푥 + 12 + 4

(푦 − 2) = −8(푥 − 2)

(푦 − 퐾) = 4P(푥 − ℎ)

퐹(ℎ + 푝,퐾)

퐹(, )퐹(0, −2)

푥 = ℎ − 푃푥 = 2 + 2

푥 = 4

푦 − 퐾 = 0푦 = 퐾

푦 = −2

- 6 -

4푝 = −2

푝 =−24

푝 =−12

ℎ = 1K = −1

2푥 + 푦 + 2y − 1 = 0

푦 + 2y + 1 = −2푥 + 1 + 1

(푦 + 1) = −2(푥 + 1)

(푦 − 퐾) = −2(푥 − ℎ)

푦 − 퐾 = 0

푦 = −1

퐹(ℎ + 푝,퐾)

퐹(1 −12,−1) = 퐹(

12, −1)

푥 = ℎ − 푃

푥 = 1 +12

푥 =32

- 7 -

(ℎ, 퐾)

(3,2)

(1,2)

푦 = 2

ℎ = 3K = 2

퐹(ℎ + 푝,퐾) = (1,2)ℎ + 푝 = 1, 퐾 = 2

3 + 푝 = 1 ⇒ 푃 = −2

(푦 − 퐾) = 4P(푥 − ℎ)

(푦 − 2) = 4(−2)(푥 − 3)

(푦 − 2) = −8(푥 − 3)

푦 = −2

ℎ = 0K = 4

(0,4)

y

(푥 − ℎ) = 4푝(푦 − 퐾)

−2 = 4 − 푃 ⇒ 푃 = 6

푦 = 퐾 − 푃 ⇒ 푦 = −2

- 8 -

푥 = −2(2,2)

푥

푥 = ℎ − 푃

−2 = ℎ − 푃. . . . . .1

퐹(ℎ + 푝,퐾) = (2,2)ℎ + 푝 = 2. . . .2, 퐾 = 2

−2 = ℎ − 푃

2 = ℎ + 푃0 = 2ℎ

ℎ = 0

h

−2 = 0 − 푃

푃 = 2

- 9 -

(4,0)(0,0)(2,4)

0,0

푝 =416⇒ 푝 =

−14

1-6197

y푥 = 2푦

- 10 -

x

푦 = 8푥

x

푦 = −4푥

y

푥 = −6푦

0,0

푭ퟐ, 푭ퟏ풙푭ퟐ, 푭ퟏ풚

- 11 -

푎

푎푥퐹 , 퐹푥

- 12 -

푎

푏

푐

푐, 푏, 푎

푐 = 푎 − 푏

푎 > 푐, 푎 > 푏

휋푎푏

푎푐푏

: 2푎

: 2푏

5푥 + 7푦 = 70 ÷ 70

5푥70

+7푦70

=7070

- 13 -

푥14

+푦10

= 1

푏 = 10

푏 = √10

푎 = 14

푎 = √14

푐 = 푎 − 푏

푐 = 14 − 10 = 4

푐 = 2

*0,0

푐 = 2

퐹 (퐶, 0), 퐹 (−퐶, 0)퐹 (2,0), 퐹 (−2,0)

(푎, 0), (−푎, 0)√14, 0 , −√14, 0

2푎 =

2푎 = 2√142푏 =

2푏 = 2√10

- 14 -

(0,0)

푐 = 2(2,0)

푎 = 3(3,0)

푥

푐 = 푎 − 푏

4 = 9 − 푏

푏 = 5

푥9+푦5= 1

(0,±5)

- 15 -

14=

푐 = 푎 − 푏

25 = 49 − 푏

푏 = 24

ℎ, 푘0,0

풂ퟐ풙ퟐ

풂ퟐ풚ퟐ

- 16 -

퐹 (ℎ ± 퐶, 퐾)퐹(ℎ, 퐾 ± 퐶)

푎푐푏

9푥 + 25푦 − 36푥 + 150푦 + 36 = 0

9푥 + 25푦 − 36푥 + 150푦 + 36 = −36

9(푥 + 4푥 + 4) + 25(푦 + 6푦 + 9) = −36 + 36 + 225

9(푥 − 2) + 25(푦 + 3) = 225 ÷ 225

- 17 -

(푥 − 2)25

+(푦 + 3)

9= 1

푎 = 25 푏 = 9 푐 = 푎 − 푏

푐 = 25 − 9 = 16 푎 = 5 푏 = 3

*2,-3

푐 = 4

퐹 (ℎ + 퐶, 퐾), 퐹 (ℎ − 퐶, 퐾)

퐹 (6,−3), 퐹 (−2,−3)

(ℎ + 푎, 퐾), (ℎ − 푎, 퐾)

(7,−3), (−3,−3)

(3,1)(3,9)

- 18 -

(ℎ, 퐾 + 푎), (ℎ, 퐾 − 푎)

(3,9), (3,1)

퐾

(1,2)(3,2)(1,6)

(1,2)ℎ = 1퐾 = 2

- 19 -

(푥 − 1)푏

+(푦 − 2)

푎= 1

1,6

(1 − 1)푏

+(6 − 2)

푎= 1 ⇒ 0 +

16푎

= 1

푎 = 16

3,2

(3 − 1)푏

+(2 − 2)16

= 1 ⇒4푏

+ 0 = 1

푏 = 16

(푥 − 1)4

+(푦 − 2)16

= 1

- 20 -

0,0

풙ퟐ푭ퟏ, 푭ퟐ풙

풚ퟐ푭ퟏ, 푭ퟐ풚

푎

퐹(±퐶, 0)퐹(0,±퐶)

- 21 -

ℎ, 푘

풙ퟐ

풚ퟐ

푏 =

푐 =

푎 =

퐹(ℎ ± 퐶, 푘)퐹(0,±퐶)

- 22 -

9푥 − 푦 − 36푥 − 6푦 + 18 = 0

9푥 − 36푥 − 푦 − 6푦 = −18

9 × 4 = 36 − 9

9(푥 − 4푥 + 4) − (푦 + 6푦 + 9) = −18 + 36 − 9

9(푥 − 2) − (푦 + 3) = 9 ÷ 9

(푥 − 2)1

+(푦 + 3)

9= 1

푎 = 1 푏 = 9 ℎ = 2

푘 = −3 푎 = 5 푏 = 3

푐 = 푎 + 푏

2,-3

푐 = √10

퐹 (ℎ + 퐶, 퐾), 퐹 (ℎ − 퐶, 퐾)

퐹 (2 + √10,−3), 퐹 (2 − √10,−3)

(ℎ + 푎, 퐾), (ℎ − 푎, 퐾)

(3, −3), (1,−3)

- 23 -

(±1,0)

3ℎ, 푘, 푎푦 = ± 푥

푎 = 1ℎ = 0푘 = 0

(0,0)

- 24 -

퐹 , 퐹 푥

푥푎

+푦푏

= 1

푦 = ±31푥

푦 = ±푏푎푥푏 = 3

+ = 1

(2,±3)

3,5푦 = ± 푥

푎 = 3ℎ = 2푘 = 0

(2,0)

3,5

- 25 -

2푎

2푎

푦 = 8푥

- 26 -

(푦 + 2) = 8(푥 − 1)(+1,−2)

푦 = 8푥

(0,0)

퐹(푃, 0)

푦 = 8푥

푦 = 4푝푥

4푝 = 8

푝 = 2

푝 = 2(푦 + 2) = 8(푥 − 1)

(1,−2)

퐹(ℎ + 푃, 푘)

ℎ = 1

푘 = −2

- 27 -

6-2

푒 =2퐶2푎

푒 =퐶푎

2,0

푭ퟏ, 푭ퟐ풙푭ퟏ, 푭ퟐ풚

푥 = ±

- 28 -

ℎ, 푘

푥 − ℎ = ±

퐹푒dp

푃퐹 = 푒푃퐷

푒 = 1

푒 < 1

푒 > 1

- 29 -

(0,±6.5)

0.75 = (푒)

퐶

푒

0

푎

(0,±8.7)

16푥 − 9푦 = 144

푥9+푦16

= 1

푏 = 16 푎 = 9

푏 = 4 푎 = 3

푐 = 푎 + 푏

푒 =퐶푎

- 30 -

푒 =53

푃퐷 = |푥 − 4|

(0,0)푥 = 4퐹(3,0)

푃(푥, 푦)퐹(3,0)

푥 = 4

푥 =푎푒

4 =푎푒

푎 = 4푒

푒 =c푎

푒 =34푒

푒 =34

푒 =√32

(푥 − 3) + 푦 =34(푥 − 8푥 + 16)

푥 − 6푥 + 9 + 푦 =34(푥 − 8푥 + 16)

4푥 − 24푥 + 36 + 4푦 = 3푥 − 24푥 + 48

푥 + 4푦 = 48 − 36

푥 + 4푦 = 12 ÷ 12

+ =1

푦 = 2퐹(1, −3)푒 =

- 31 -

푒 =96

푒 = C = 9

푎 = 6

푒 =퐹(1, −3)

푦 = 2푃퐷 = |푥 − 4|

푃(푥, 푦)

(푥 − 1) + 푦 + 6푦 + 9 =94(푦 − 4푦 + 4)

4(푥 − 1) + 4푦 + 24푦 + 36 = 9푦 − 36푦 + 36

4(푥 − 1) − 5푦 + 60푦 = 0

4(푥 − 1) − 5(푦 + 12푦 + 36) = −180

4(푥 − 1) − 5(푦 − 6) = −180 ÷ −180

(푦 − 6)36

−(푥 − 1)45

= 1

(0,0)

푒 =45

퐶 = 4

- 32 -

푎 = 5

푐 = 푎 + 푏

퐴퐵 = |푦 − 푦 |

푦

퐴퐵 = |푥 − 푥 |

푥

(1,4)ℎ = 1푘 = 4

2푎 = |7 − 1|2푎 = 6푎 = 32푏 = | − 1 − 3|2푏 = 4푏 = 2

(푥 − ℎ)b

+(푦 − K)

푎= 1

( ) + ( ) = 1

푐 = 푎 + 푏

푐 = 9 − 4 = 5

푐 = √5

푒 =퐶푎=√53

- 33 -

퐹 (ℎ, 퐾 + 퐶), 퐹 (ℎ, 퐾 − 퐶)

퐹 1,4 + √5 , 퐹 1,4 − √5

(1,4)(0,±1)푒 = 3

푦푎

−푥푏

= 1

푒

3

퐶 = 3

푐 = 푎 + 푏

9 = 1 + 푏

푏 = 8

푦1−푥8= 1

4푥 + 푦 + 푎푥 + 푏푦 + 푐 = 0

푎, 푏, 푐푎, 푏, 푐

−1,2

4(−1) + (2) + 푎(−1) + 푏(2) + 푐 = 0

4 + 4 − 푎 + 2푏 + 푐 = 0

2푏 − 푎 + 푐 = −8 …………1

- 34 -

0,0

4(0) + (0) + 푎(0) + 푏(0) + 푐 = 0

푐 = 0

푥0,0푦̀ = 0

8푥 + 2푦푦̀ + 푎 + 푏푦̀ + 0 = 0

8(0) + 2(0)(0) + 푎 + 푏(0) + 0 = 0

푎 = 0

C, a

4푥 + 푦 + 푎푥 + 4푦 = 0

4푥 + 푦 − 4푦 + 4 = 0 + 4

4푥 + (푦 − 2) = 4 ÷ 4

푥1+(푦 − 2)

4= 1

푏 = 1

푏 = 1

푐 = 푎 + 푏

푐 = 4 − 1 = 3

푐 = √3

푎 = 4

푎 = 2

푒 =퐶푎

푒 =√32

- 35 -

6=

(푥 − ℎ)푎

+(푦 − K)

푏= 1

퐹 (ℎ + 퐶, 퐾), 퐹 (ℎ − 퐶, 퐾)

퐹 (10,3),퐹 (2,2)

ℎ

ℎ + 푐 = 10. . . . . . . .1

ℎ − 푐 = 2. . . . . . . . .2

2ℎ = 12

ℎ = 6

푐 = 푎 + 푏

16 = 9 + 푏

푏 = 7

- 36 -

6= 퐹 (10,2), 퐹 (2,2)

(푥 − 10) + (푦 − 2) − (푥 − 2) + (푦 − 2) = ±6(푥 − 10) + (푦 − 2) = ±6 + (푥 − 2) + (푦 − 2)

푎 =14푏 = 1

4푥 − 9푦 = 36 ÷ 36

푥9−푦4= 1