ellips (irisan kerucut)
TRANSCRIPT
![Page 1: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/1.jpg)
Assalamualaikum Wr. Wb.
Kita akan mempresentasikan tentang....ELIPS
![Page 2: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/2.jpg)
Nama Kelompok :1.Dewi Ulul Azmi (01)
2.Ernita Khusnalia (03)3.Fransiska Anggraini (06)4.M. Febrian Bachtiar (13)5.Mawaddatul Hikmawati
(16)6.Moh. Prima T. (18)7.Rizky Ari S.P. (30)8.Upik Nurhalizah (33)9.Windri Ayu Atika Suri (36)
![Page 3: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/3.jpg)
APA ITU “ELIPS”?
Elips adalah tempat kedudukan titik-titik pada bidang datar yang jumlah jaraknya terhadap dua titik tertentu yang diketahui adalah tetap (konstan).
Dua titik tertentu itu disebut fokus atau titik api (F1 dan
F2), jarak F1 dan F2 adalah
2c, dan jumlah jarak tetap
adalah 2a (a > 0).
![Page 4: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/4.jpg)
• Perhatikan gambar berikut!
A1 (-a, 0)
PA2 (a, 0)
(- c, 0) (c, 0)
(0, b)
(0, - b)
F2 F1
E L
KT
D
B2
B1
![Page 5: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/5.jpg)
Keterangan :1. (F1 dan F2) disebut fokus. Jika T adalah
sembarang titik pada elips maka TF1 + TF2 = 2a. F1F2 = 2c, dengan 2a > 2c
2. A1A2 merupakan sumbu panjang (sumbu mayor) yang panjangnya sama dengan jarak tetap yaitu 2a. B1B2 merupakan sumbu pendek (sumbu minor) yang panjangnya sama dengan 2b. Karena itu a > b.
3. Lactus rectum yaitu segmen garis yang dibatasi elips, tegak lurus sumbu mayor, dan melalui fokus (DE dan KL) panjang lactus rectum DE = KL =
4. Titik pusat (P) yaitu titik potong sumbu mayor dengan sumbu minor
5. Titik puncak elips yaitu A1 ,A2 ,B1 , dan B2
![Page 6: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/6.jpg)
F1 F2A B
D
C
F1 dan F2 disebut titik-titik api atau fokusTitik- titik A , B, disebut puncak-puncak ellips
Menggambar Elips
![Page 7: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/7.jpg)
F1(-c,0)A1(-a,0) F2(c,0)O
b
c
a
A2(a, 0)
B1(0, b)
B2(0, -b)
P(x, y)
JARAK TITIK
![Page 8: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/8.jpg)
F1(-c,0)A1(-a,0) F2(c,0)O
b
c
a
A2(a, 0)
B1(0, b)
B2(0, -b)
P(x, y)
Jika titiknya A2, maka :
A2F1 + A2F2 = 2a
(a + c) + (a – c) = 2a
2a = 2a
![Page 9: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/9.jpg)
F1(-c,0)A1(-a,0) F2(c,0)O
b
c
a
A2(a, 0)
B1(0, b)
B2(0, -b)
P(x, y)
Jika titiknya B1, maka :
222
22
22
2222
2111
22
2
2
acb
acb
acb
acbcb
aFBFB
a
a2 = b2 + c2
b2 = a2 - c2
c2 = a2 - b2
Phytagoras dalam Elips
![Page 10: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/10.jpg)
F1(-c,0)A1(-a,0) F2(c,0)O A2(a, 0)
B1(0, b)
B2(0, -b)
P(x, y)
(x, 0)
(y, 0)
Persamaan Elips dengan pusat O (0,0)
![Page 11: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/11.jpg)
![Page 12: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/12.jpg)
![Page 13: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/13.jpg)
![Page 14: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/14.jpg)
![Page 15: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/15.jpg)
Karena
![Page 16: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/16.jpg)
Persamaan Elips (horizontal)
Pusat O (0,0)
12
2
2
2
by
ax
![Page 17: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/17.jpg)
O
F1(0,-c)
F2(0,c)P(x, y)
x
y
x2
b2y2
a21
a
b
(0,-a)
(0,a)
Elips Vertikal dengan pusat O (0,0)
![Page 18: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/18.jpg)
elips horizontal elips vertikal
Persamaan Elips
Fokus (-c,0) , (c,0) (0,-c) , (0, c)
Puncak (-a,0) , (a,0) (0 ,-a) , (0,a)
Sumbu mayor Sumbu x Sumbu y
Sumbu minor Sumbu y Sumbu x
12
2
2
2
by
ax 12
2
2
2
bx
ay
Perbedaan Persamaan Ellips (0,0)
![Page 19: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/19.jpg)
Selidiki dan buat sketsa grafik dari persamaan
225259 22 yx
Contoh soal
![Page 20: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/20.jpg)
225259 22 yx
1225
252259 22
yx
1925
22
yx
252 a5a
92 b3b
222 bac
9252 c4c
x
y
(0,3)
(0,-3)
(0,5)(0,-5)
(0,4)(0,-4)
pembahasan
![Page 21: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/21.jpg)
Diketahui elips dengan persamaan
.Tentukan fokus, titik puncak, panjang sumbu mayor, panjang sumbu minor dan panjang lactus rectumnya !
Contoh soal
![Page 22: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/22.jpg)
Diketahui persamaan elips = 81 ⇔ a = 9 = 25 ⇔ b = 5
= = ⇔ c =
Fokus (0, - ) dan (0, )Titik puncak (0, -9) dan (0, 9)Panjang sumbu mayor ⇔ 2a = 18Panjang sumbu minor ⇔ 2b = 10Panjang lactus rectum (LR) ⇔
pembahasan
![Page 23: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/23.jpg)
O’ = S(g,h)
y
xO (0,0)
y’(x=g)
x’(y=h)
P 1''2
2
2
2
by
ax
1)()(2
2
2
2
bhy
agx
a
b
Persamaan Ellips dengan pusat S (g,h)
![Page 24: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/24.jpg)
O’ = S (g,h)
y
xO (0,0)
y’(x=g)
x’ ( y=h)
P
1''2
2
2
2
ay
bx
1)()(2
2
2
2
ahy
bgxa
b
Ellips Vertikal dengan pusat S (g,h)
![Page 25: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/25.jpg)
elips horizontal elips vertikal
Persamaan Elips
Fokus ((g-c),h) , ((g+c),h) (g,(h-c)) , (g,(h+c))
Puncak ((g-a),h) , ((g+a),h) (g,(h-a)) , (g,(h+a))
Sumbu mayor x’ atau y=h y’ atau x=g
Sumbu minor y’ atau x=g x’ atau y=h
1)()(2
2
2
2
bhy
agx
1)()(2
2
2
2
bgx
ahy
Perbandingan Persamaan Ellips (g,h)
![Page 26: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/26.jpg)
Gambarlah ellips yang mempunyai persamaan
(x – 2)236
(y + 5)216+ =
1
Contoh soal
![Page 27: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/27.jpg)
x
y
a2 = 36a = ±6
b2 = 16b = ±4
(x – 2)2
36(y + 5)2
16+ = 1
pusat = (2,-5)
(8,-5) (-4,-5)
(2,-1) (2,-9)
pembahasan
![Page 28: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/28.jpg)
Gambarlah ellips yang mempunyai persamaan
(x – 2)2
36(y + 5)2
16+ = 1
Contoh soal
![Page 29: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/29.jpg)
x
y
a2 = 25a = ±5
b2 = 81b = ±9
(x + 3)2
25
(y + 1)2
81+ = 1
Titik pusat = (-3,-1)
Titik puncak : (-3,8) (-3,-10)
(-8,-1) (2,-1)
pembahasan
![Page 30: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/30.jpg)
Bentuk Umum Persamaan ElipsPersamaan elips memiliki bentuk umum:
![Page 31: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/31.jpg)
i. Persamaan Garis Singgung Melalui Titik (x1, y1) pada Elips
Persamaan Garis singgung ellips
![Page 32: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/32.jpg)
ii. Persamaan Garis Singgung dengan gradien p
![Page 33: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/33.jpg)
![Page 34: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/34.jpg)
Persamaan elipsPersamaan garis singgung
Melalui titik ( Dengan gradien p
Persamaan garis singgung elips dititik P(
![Page 35: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/35.jpg)
Persamaan garis singgung elips dititik P(m, n)
Persamaan elipsPersamaan garis singgung
Melalui titik ( Dengan gradien p
![Page 36: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/36.jpg)
Tentukan persamaan garis singgung elips
, pada titik (5, -3)
Contoh soal
![Page 37: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/37.jpg)
Diketahui : Pusat (m, n) ⇔ (1, -2) (5, -3) ⇔ =5 dan = -3Persamaan garis singgung :⇔ ⇔⇔⇔ ⇔ 2(x – 1) - (y + 2) = 9⇔ 2x – y = 13
pembahasan
![Page 38: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/38.jpg)
Tentukan persamaan garis singgung elips
, dengan gradien 2
Contoh soal
![Page 39: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/39.jpg)
Diketahui :⇔ m = -3, n = 4, = 15, = 4, dan p = 2Persamaan garis singgung :
⇔⇔⇔y =2x + 6 8 +4⇔ y = 2x + 18 dan y = 2x + 2
pembahasan
![Page 40: Ellips (Irisan Kerucut)](https://reader033.vdocuments.net/reader033/viewer/2022061606/58ed1d791a28abd8098b4599/html5/thumbnails/40.jpg)