PEMBAHASAN SOAL GEOMETRI ANALITIK ROTASI SUMBU 1 Letty

PEMBAHASAN SOAL GEOMETRI ANALITIK ROTASI SUMBU 1. Letty Andrias M. 4101412003 2. Eva Putri Karunia 4101412047 3. Kinanthi Mustika Ayu 4101412118 4. Iffatun luthfiah 4101412158

NO. 6 HALAMAN 92 TRANSFORM EACH EQUATION TO ANX’Y’ EQUATION UNDER THE GIVEN ROTATION

NO. 17 HALAMAN 93 Transform each equation by a rotation to eliminate the x’y’ term. Sketch the graph showing the x, y and x’, y’ axes 16 x 2 - 24 xy + 9 y 2 – 60 x + 80 y = 0

Penyelesaian: ingat kembali pada teorema 3. 8 yaitu Tan 2 a = Melalui persamaan, didapat A = 9, B = -24 , C = 16 Maka didapat tan 2 a = r= r cos 2 a = 24 x 7 = 25 = sin a = = cos a = =

Ingat pada T. 3. 7 A’ = A cos 2 a + B sin a cos a + C sin 2 a B’ = B cos 2 a – (A – C) sin 2 a C’ = A sin 2 a - B sin a cos a + C cos 2 a D’ = D cos a + E sin a E’ = E cos a – D sin a F’ = F

Diperoleh : A’ = 0 B’ = 0 C’ = 25 D’ = -150 E’ = 0 F’ = 0 Substitusikan ke A’x’ 2 +B’x’y’+ C’y’ 2 +D’x’ + E’y’+ F’= 0 Persamaannya menjadi 25 y’ 2 - 150 x’ = 0

Y Y’ 5 5 5 -3 -3 X

NO. 18 HALAMAN 93 TRANSFORM EACH EQUATION BY A ROTATION TO ELIMINATE THEX’Y’ TERM 16 X 2 + 24 XY + 9 Y 2 – 60 X + 80 Y = 0 ingat kembali pada teorema 3. 8 yaitu Tan 2 a = Melalui persamaan, didapat A = 16, B = 24 , C = 9 r= Maka didapat tan 2 a = r = 25 = cos 2 a = 24 x sin a = = 7 cos a = =

Ingat pada T. 3. 7 A’ = A cos 2 a + B sin a cos a + C sin 2 a B’ = B cos 2 a – (A – C) sin 2 a C’ = A sin 2 a - B sin a cos a + C cos 2 a D’ = D cos a + E sin a E’ = E cos a – D sin a F’ = F

Diperoleh : A’ = 25 B’ = 0 C’ = 0 D’ = 0 E’ = 84 F’ = 0 Substitusikan ke A’x’ 2 +B’x’y’+ C’y’ 2 +D’x’ + E’y’+ F’= 0 Persamaannya menjadi 25 x’ 2 + 84 y’ = 0

NO. 19 HALAMAN 93 USE THEOREM 3. 9 TO NAME THE GRAPH OF EACH AQUATIONI. N EACH CASE ASSUME THE GRAPH EXIST AND IS OT DEGENERATE.

NO. 20 HALAMAN 93 Use theorem 3. 9 to name the graph of each equation. In each assume the graph exists and is not degenerate. answer : A=3 , B=-5 , C=-1 ; didapat = = Karena nilai tersebut adalah hiperbola maka grafik dari persamaan

NO. 22 HALAMAN 93 Use theorem 3. 9 to name the graph of each equation. In each case assume the graph exists and is not degenerate 3 x²+3 xy+3 y²-x=0 Solution: A=3 , B=3, C=3 ; didapat B²- 4 AC = 3²-4(3)(3) = 9 – 36 = -27 < 0 Karena nilai B² - 4 AC < 0 maka grafik dari persamaan tersebut adalah elips.

NO. 24 HALAMAN 93 Use theorem 3. 9 to name the graph of each equation. In each assume the graph exists and is not degenerate. answer : A=4 , B=-4 , C=1 ; maka = = Karena nilai maka grafik dari persamaan tersebut adalah parabola

NO. 31 HALAMAN 94 SHOW THAT AFTER THE GENERAL QUADRATIC EQUATION IS TRASNFORMED BY USING THE ROTATION FORMULAS OF THEOREM 3. 7.

PROVE THAT : B’ 2 – 4 A’C’ = B 2 – 4 AC Bukti: Jelas A’ = ½ [(A+C) + B sin 2 a +(A–C) cos 2 a] B’ = B cos 2 a – (A-C) sin 2 a C’ = ½ [(A+C) – B sin 2 a – (A-C) cos 2 a] Persamaan : B’ 2 – 4 A’C’ = [B 2 cos 2 2 a – 2 B (A-C) cos 2 a sin 2 a + (A-C)2 sin 2 2 a] – 4 [ ½ ((A+C)+B sin 2 a + (A-C) cos 2 a)] [ ½ ((A+C) – B sin 2 a – (A-C) cos 2 a ]

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