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Tentukan persamaan bayangan dari setiap rotasi berikut: c. lingkaran L≡x2+y2+2x−2y−5=0 terhadap (4,5) sejauh 3π​ berlawanan arah jarum jam.

Pertanyaan

Tentukan persamaan bayangan dari setiap rotasi berikut:

c. lingkaran L identical to x squared plus y squared plus 2 x minus 2 y minus 5 equals 0 terhadap begin mathsize 14px style open parentheses 4 comma 5 close parentheses end style sejauh begin mathsize 14px style straight pi over 3 end style berlawanan arah jarum jam.

Pembahasan Soal:

Rotasi terhadap  left parenthesis 4 comma 5 right parenthesis sejauh begin mathsize 14px style straight pi end style berlawanan jarum jam maka,

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell cos space 60 end cell cell negative sin space 60 end cell row cell sin space 60 end cell cell cos space 60 end cell end table close parentheses open parentheses table row cell x minus 4 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half end cell cell negative 1 half square root of 3 end cell row cell 1 half square root of 3 end cell cell 1 half end cell end table close parentheses open parentheses table row cell x minus 4 end cell row cell y minus 5 end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half open parentheses x minus 4 close parentheses minus 1 half square root of 3 left parenthesis y minus 5 right parenthesis end cell row cell 1 half square root of 3 left parenthesis x minus 4 right parenthesis plus 1 half left parenthesis y minus 5 right parenthesis end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half x minus 2 minus 1 half square root of 3 y minus 5 over 2 end cell row cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y minus 5 over 2 end cell end table close parentheses plus open parentheses table row 4 row 5 end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half x minus 1 half square root of 3 y minus 2 plus 4 minus 5 over 2 end cell row cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y minus 5 over 2 plus 5 end cell end table close parentheses end cell row cell open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses end cell equals cell open parentheses table row cell 1 half x minus 1 half square root of 3 y minus 1 half end cell row cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y plus 5 over 2 end cell end table close parentheses end cell row cell x apostrophe end cell equals cell 1 half x minus 1 half square root of 3 y minus 1 half end cell row cell 1 half x end cell equals cell x apostrophe plus 1 half square root of 3 y plus 1 half end cell row x equals cell 2 x apostrophe plus square root of 3 y plus 1 end cell row blank blank blank row cell y apostrophe end cell equals cell 1 half square root of 3 x minus 2 square root of 3 plus 1 half y plus 5 over 2 end cell row cell 1 half y end cell equals cell y apostrophe minus 1 half square root of 3 x plus 2 square root of 3 minus 5 over 2 end cell row y equals cell 2 y apostrophe minus square root of 3 x plus 4 square root of 3 minus 5 end cell row blank blank blank end table

Maka di dapat table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 5 end table. Cari dulu pusat lingkaran L identical to x squared plus y squared plus 2 x minus 2 y minus 5 equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus y squared plus 2 x minus 2 y minus 5 end cell equals 0 row cell x squared plus y squared plus A x plus B y plus C end cell equals 0 row P equals cell open parentheses negative A over 2 comma negative B over 2 close parentheses end cell row P equals cell open parentheses negative 2 over 2 comma negative fraction numerator left parenthesis negative 2 right parenthesis over denominator 2 end fraction close parentheses end cell row P equals cell open parentheses negative 1 comma 1 close parentheses end cell end table

Di dapatkan x equals negative 1 dan y equals 1 kemudian masukkan ke persamaan,

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell 2 x apostrophe plus square root of 3 y plus 1 end cell row cell 2 x apostrophe end cell equals cell x minus square root of 3 y minus 1 end cell row cell x apostrophe end cell equals cell fraction numerator x minus square root of 3 y minus 1 over denominator 2 end fraction end cell row cell x apostrophe end cell equals cell fraction numerator negative 1 minus square root of 3 space left parenthesis 1 right parenthesis minus 1 over denominator 2 end fraction end cell row cell x apostrophe end cell equals cell fraction numerator negative 2 over denominator 2 end fraction minus fraction numerator square root of 3 over denominator 2 end fraction end cell row cell x apostrophe end cell equals cell negative 1 minus 1 half square root of 3 end cell row blank blank blank row y equals cell 2 y apostrophe minus square root of 3 x plus 4 square root of 3 minus 5 end cell row cell 2 y apostrophe end cell equals cell y plus square root of 3 x minus 4 square root of 3 plus 5 end cell row cell y apostrophe end cell equals cell fraction numerator y plus square root of 3 x minus 4 square root of 3 plus 5 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 1 plus square root of 3 left parenthesis negative 1 right parenthesis minus 4 square root of 3 plus 5 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 6 minus square root of 3 minus 4 square root of 3 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell fraction numerator 6 minus 5 square root of 3 over denominator 2 end fraction end cell row cell y apostrophe end cell equals cell 3 minus 5 over 2 square root of 3 end cell end table

Kemudian cari jari-jari,

table attributes columnalign right center left columnspacing 0px end attributes row cell square root of open parentheses fraction numerator negative A over denominator 4 end fraction close parentheses squared plus open parentheses fraction numerator negative B over denominator 4 end fraction close parentheses squared minus C end root end cell equals cell square root of open parentheses fraction numerator negative 2 over denominator 4 end fraction close parentheses squared plus open parentheses fraction numerator negative left parenthesis negative 2 right parenthesis over denominator 4 end fraction close parentheses squared minus left parenthesis negative 5 right parenthesis end root end cell row blank equals cell square root of open parentheses fraction numerator negative 1 over denominator 2 end fraction close parentheses squared plus open parentheses 1 half close parentheses squared plus 5 end root end cell row blank equals cell square root of 1 fourth plus 1 fourth plus 5 end root end cell row blank equals cell square root of 2 over 4 plus 20 over 4 end root end cell row blank equals cell square root of 22 over 4 end root end cell row blank equals cell 1 half square root of 22 end cell end table

Maka di dapatkan, table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table dan table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank apostrophe end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 5 over 2 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 3 end cell end table. Masukkan ke persamaan lingkaran,

table attributes columnalign right center left columnspacing 0px end attributes row cell left parenthesis x minus a right parenthesis squared plus left parenthesis x minus b right parenthesis squared end cell equals cell r squared space end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell open parentheses 1 half square root of 22 close parentheses squared end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell 1 fourth times 22 end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell 22 over 4 end cell row cell open parentheses x minus open square brackets negative 1 minus 1 half square root of 3 close square brackets close parentheses squared minus open parentheses y minus open square brackets 3 minus 5 over 2 square root of 3 close square brackets close parentheses squared end cell equals cell 5 comma 5 end cell end table
 

Jadi, bayangan persamaan garisnya adalah Error converting from MathML to accessible text..

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

G. Albiah

Mahasiswa/Alumni Universitas Galuh Ciamis

Terakhir diupdate 06 Oktober 2021

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