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KELOMPOK 2 KELOMPOK IRISAN KERUCUT HIPERBOLA Alfra Putriana D / 03 Arjun Duan P / 0! "#natan Mar$%llin Mar$%llin# #M/ &' Ri()a Nurlaill* / 32
P%r(a)aan Hi,%r-#la .i ,u(at P , 1
P%n+%rtia n Hi,%r-#la ari( Sin++un+ Hi,%r-#la P%r(a)aan Hi,%r-#la .i ,u(at O 0 01
B%ntu4 U)u) P%r(a)aa n Hi,%r-#la
P%n+%rtian Hi,%r-#la •
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Hiperbola )%ru,a4an 4ur5a l%n+4un+ t%r-u4a *an+ t%r.iri ata( .ua -a+ian atau $a-an+ *an+ (alin+ (i)%tri( t%ra.a, .ua (u)-u (i)%tri6 Hiperbola .i,%r#l% .%n+an )%n+iri( .ua -an+un 4%ru$ut *an+ (alin+ -%rt#la4 -%la4an+ (%(uai -i.an+ *an+ t%+a4 luru( ala( 4%ru$ut t%ta,i ti.a4 )%)#t#n+ ,un$a4 4%ru$ut6 Dala) -i.an+ 4##r.inat hiperbola .i.%fni(i4an (%-a+ai t%),at 4%.u.u4an titi47titi4 *an+ (%li(i jara4n*a t%ra.a, .ua titi4 t%rt%ntu (%lalu t%ta,6 K%.ua titi4 t%rt%ntu itu .ina)a4an titik fokus. B%ntu4 u)u) ,%r(a)aan i,%r-#la a.ala .%n+an A B C D E -ilan+an r%al .an A B 8 06
Un(ur Hi,%r-#la
Keterangan: Titi4 O .i(%-ut ,u(at i,%r-#la6 Titi4 .an .i(%-ut ;#4u( i,%r-#la6 Titi4 .an .i(%-ut titi4 ,un$a4 i,%r-#la6 Rua( +ari( .i(%-ut (u)-u )a*#r/(u)-u ,anjan+ .an )%)ili4i ,anjan+ *an+ .in*ata4an .%n+an 2a. Rua( +ari( .i(%-ut (u)-u i)ajin%r/ (u)-u )in#r .an )%)ili4i ,anjan+ *an+ .in*ata4an .%n+an 2b6 Su)-u (i)%tri *an+ )%lalui 4%.ua ;#4u( .i(%-ut (u)-u uta)a atau (u)-u tran(5%r(al6 Su)-u (i)%tri *an+ )%lalui titi4 ,u(at .an t%+a4 luru( (u)-u uta)a .i(%-ut (u)-u (%4a
P%r(a)aan i,%r-#la ? Pu(at i,%r-#la O 0 01 @#4u( i,%r-#la @&7$ 01 .an @2$ 01 .%n+an $2=a2-2 Pun$a4 i,%r-#la A&7a 01 .an A2a 01 Su)-u n*ata a.ala * = 0 (u)-u i)ajin%r a.ala = 0 Panjan+ (u)-u )a*#r = 2a ,anjan+ (u)-u )in#r = 2P%r(a)aan a(i)t#t? * =
P%)-u4tian ,a.a ,u(,%r(a)aan i,%r-#la at O 0 01 (u)-u uta)a (u)-u,a.a i,%r-#la (%in++a -%rla4u ,%r(a)aan Titi4(%jajar P *1 t%rl%ta4 P@: P@ = 2a666666&1 D%n+an )%n++una4an ru)u( untu4 )%n$ari jara4 antara 2 titi4 .i,%r#l% P@:= .an P@= A4i-atn*a ,%r(a)aan &1 )%nja.i
D%n+an )%n+4ua.rat4an 4%.ua rua( .i,%r#l%
$ 7 a2 = D%n+an )%)-a+i 4%.ua rua( .%n+an -ilan+an .i,%r#l% $ 7 a2 = a S%lanjutn*a 4%.ua rua( .i4ua.rat4an .an .i,%r#l% a $22 7 2a2$ = a 2 7 $1 2 *2F a $22 7 2a2$ = a $22 7 2a2$ = a a2$2 = a22 7 $22 a2*2
K%.ua rua( .i-a+i .%n+an -ilan+an 7& (%in++a .i,%r#l% a2$2 7 a21 = $2 7 a212 7 a2*2 K%.ua rua( .i-a+i .%n+an a 2$2 7 a21 .i,%r#l% & = T%la .ij%la(4an (%-%lu)n*a -a
P%r(a)aan i,%r-#la ? Pu(at i,%r-#la .i , 1 Titi4 ;#4u( , 7 $ 1 .an , $ 1 .%n+an = Titi4 ,un$a4 , 7 a 1 .an , a 1 Su)-u n*ata a.ala * = .an (u)-u i)ajin%r a.ala = , P%r(a)aan a(i)t#t ? * = ,1
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Pu(at P , 1 Su)-u Uta)a S%jajar Su)-u G P%r(a)aan i,%r-#la ? Pu(at i,%r-#la .i , 1 Titi4 ;#4u( , $1 .an , $1 .%n+an = Titi4 ,un$a4 , a1 .an , a1 Su)-u n*ata a.ala = , .an (u)-u i)ajin%r a.ala * = P%r(a)aan a(i)t#t ? * = ,1
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C#nt# S#al .an P%)-aa(an &6 S%-ua i,%r-#la )%),un*ai ,%r(a)aan ' 2 *2 3! * ! = 06 T%ntu4an titi4 ,u(at titi4 ,un$a4 .an titi4 ;#4u( i,%r-#la t%r(%-ut
Pembahasan '2 *2 3! * ! = 0 '2 3! *2 * = ! '2 1 *2 2*1 = ! '2 1 *2 2* &1 = ! 3! ? & ' 212 * &12 = 3! * &12 ' 212 = 3! 7 =& P%r(a)aan i,%r-#la ini )%)ili4i (u)-u uta)a *an+ (%jajar .%n+an (u)-u *6 Ma4a •
S%in++a ? Titi4 ,u(at ,1 = 2 7&1 Titi4 ,un$a4 , a1 = 2 7&1 31 = 2 21 Titi4 ,un$a4 , 7 a1 = 2 7&1 31 = 2 71 Titi4 ;#4u( , $1 = 2 7&1 1 Titi4 ;#4u( , 7 $1 = 2 7&1 7 1
26 T%ntu4an ,%r(a)aan i,%r-#la *an+ ;#4u(n*a .i J21 .an 4##r.inat 4%.ua ,un$a4n*a a.ala 72 21 .an ! 21 Pembahasan •
)i(al ,u(atn*a , 1 ,un$a4n*a 72 21 .an ! 21 )a4a = 2 .an (u)-u n*ata (%jajar (u)-u > *aitu * = 2 ,a=! , a = 72 2, = , = 2 titi4 ,u(at 2 21 P=2 ,a ,$=J =! 2$=J 2a $= = != 7 = 7 = 2 &! = ' Ma4a a = S%in++a ,%r(a)aan i,%r-#lan*a a.ala ?
B%ntu4 u)u) ,%r(a)aan i,%r-#la • •
B%ntu4 u)u) ,%r(a)aan i,%r-#la a.ala .%n+an A B C D E -ilan+an r%al .an A B 8 06
C#nt# S#al .an P%)-aa(an &6 T%ntu4an 4##r.inat titi4 ,un$a4titi4 ;#4u(titi4 ,u(at.an ,%r(a)aan a(i)t#t .ari i,%r-#la .%n+an ,%r(a)aan -%ri4ut6 '2 7 &!*2 7 & 7 !* 7 &'' = 0 Pembahasan :
'2 7 &!*2 7 & 7 !* &'' = 0 '2 7 &!*2 7 & 7 !* = &'' '2 7 21 7 &!*2 *1 = &'' '2 7 2 &1 7 &!*2 * 1 = &'' ' 7 &12 7 &!* 212 = &'' ' 7 !
26 D%n+an ,%r(a)aan i,%r-#la 27*2707 2*'t%ntu4an A6 K##r.inat titi4 ,u(at B6 K##r.inat titi4 ;#4u( C6 K##r.inat titi4 ,un$a4 D6 P%r(a)aan A(i)t#t
Pembahasan :
2 7 *2 7 0 7 2* ' = 0 2 7 *2 7 0 7 2* = 7' 2 7 &0 21 7 *2 2* &1 = 7' &00 7 & ? 2 2 7 1 7 * &1 = 7 12 7 = &
A2 = & B2 = A=& B=2 C= S%in++a ? A6 K##r.inat Titi4 ,u(at 7&1 B6 K##r.inat Titi4 @#4u( 77&1 .an 7&1 C6 K .inat Titi4 P 4 7&1 .an
ari( Sin++un+ Hi,%r-#la a6 ari( Sin++un+ .i Suatu Titi4 ,a.a Hi,%r-#la •
,%r(a)aan +ari( (in++un+ .i P 1 ,a.a i,%r-#la ?
&1 a.ala 21 a.ala 31 a.ala 1 a.ala
-6 ari( Sin++un+ Hi,%r-#la .ari Suatu Titi4 .i Luar Hi,%r-#la •
Seperti pada parabola untuk menentukan persamaan garis singgung di titik P () di luar hiperbola dapat dilakukan dengan menentukan lebih dahulu garis polar (garis kutub). G
P
Mi(al P 1 .i luar i,%r-#la 6 P .an PR a.ala +ari( (in++un+ .%n+an titi4 (in++un+ .an R6 ari( R a.ala +ari( ,#lar +ari( 4utu-1 titi4 P6 Ma4a ,%r(a)ann*a ?
$6 ari( Sin++un+ Hi,%r-#la .%n+an ra.i%n T%rt%ntu P%r(a)aan +ari( (in++un+ .%n+an +ra.i%n ) ,a.a i,%r-#la ?
&1 21 31 1
= & a.ala * = ) = & a.ala * = ) a.ala * = ) ,1 = & a.ala * = ) ,1
C#nt# S#al .an P%)-aa(an &6 T%ntu4an ,%r(a)aan +ari( (in++un+ i,%r-#la .i titi4 ! 21 •
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Pembahasan Titi4 ! 21 t%rl%ta4 ,a.a i,%r-#la6 P%r(a)aan +ari( (in++un+ .i titi4 ! 21 a.ala ?
&*&=2 *=
26 T%ntu4an ,%r(a)aan +ari( (in++un+ i,%r-#la = & *an+ (%jajar +ari( * = 2 & •
Pembahasan = & )a4a * = 2 & ) = 2 )a4a +ra.i%n +ari( (in++un+ = 2 P%r(a)aan +ari( (in++un+ a.ala? * = 2 * = 2 * = 2 •
T%ri)a 4a(i
