PEMBAHASAN SOAL GEOMETRI ANALITIK R OTASI S UMBU 1. Letty Andrias M Eva Putri Karunia Kinanthi Mustika Ayu Iffatun luthfiah

T RANSFORM EACH EQUATION TO AN X ’ Y ’ EQUATION UNDER THE GIVEN ROTATION N O. 6 HALAMAN 92

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 xy + 9y 2 – 60x + 80y = 0 N O. 17 HALAMAN 93

Penyelesaian: x 7 24 r ingat kembali pada teorema 3.8 yaitu Tan 2a = Melalui persamaan, didapat A = 9, B = -24, C = 16 Maka didapat tan 2a = r = = = 25 cos 2a = 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 2a – (A – C) sin 2a 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 25y’ x’ = 0

T RANSFORM EACH EQUATION BY A ROTATION TO ELIMINATE THE X ’ Y ’ TERM 16 X XY + 9 Y 2 – 60 X + 80 Y = 0 ingat kembali pada teorema 3.8 yaitu Tan 2a = Melalui persamaan, didapat A = 16, B = 24, C = 9 Maka didapat tan 2a = x 7 24 r r = = = 25 cos 2a = sin a = = cos a = = N O. 18 HALAMAN 93

Ingat pada T. 3.7 A’ = A cos 2 a + B sin a cos a + C sin 2 a B’ = B cos 2a – (A – C) sin 2a 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’ y’ = 0

U SE THEOREM 3.9 TO NAME THE GRAPH OF EACH AQUATION. I N EACH CASE ASSUME THE GRAPH EXIST AND IS OT DEGENERATE. N O. 19 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 maka grafik dari persamaan tersebut adalah hiperbola N O. 20 HALAMAN 93

N O. 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 3x²+3xy+3y²-x=0 Solution: A=3, B=3, C=3 ; didapat B²- 4AC = 3²-4(3)(3) = 9 – 36 = < 0 Karena nilai B² - 4AC < 0 maka grafik dari persamaan tersebut adalah elips.

N O. 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

S HOW THAT AFTER THE GENERAL QUADRATIC EQUATION IS TRASNFORMED BY USING THE ROTATION FORMULAS OF THEOREM 3.7. N O. 31 HALAMAN 94

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

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