SOAL UTBK REFERENSI 2021 Elips dengan persamaan 4x2+9y2=36 digeser (−12​)  kemudian diputar 90∘ dengan pusat (−1, 2). Persamaan bayangan elips tersebut adalah ....

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SOAL UTBK REFERENSI 2021

Elips dengan persamaan 4 x squared plus 9 y squared equals 36 digeser open parentheses table row cell negative 1 end cell row 2 end table close parentheses  kemudian diputar 90 degree dengan pusat left parenthesis negative 1 comma space 2 right parenthesis. Persamaan bayangan elips tersebut adalah ....

  1. 4 left parenthesis x minus 3 right parenthesis squared plus 9 left parenthesis y minus 3 right parenthesis squared equals 36 space

  2. 9 left parenthesis x minus 1 right parenthesis squared plus 4 left parenthesis y plus 2 right parenthesis squared equals 36 space

  3. 4 left parenthesis x minus 1 right parenthesis squared plus 9 left parenthesis y plus 2 right parenthesis squared equals 36 space

  4. 9 left parenthesis x plus 1 right parenthesis squared plus 4 left parenthesis y minus 2 right parenthesis squared equals 36 space

  5. 4 left parenthesis x plus 1 right parenthesis squared plus 9 left parenthesis y minus 2 right parenthesis squared equals 36 space

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Pembahasan

Ingat kembali bahwa:

  • Rumus traslasi:

A apostrophe open parentheses x apostrophe comma space y apostrophe close parentheses rightwards arrow with T open square brackets table row a row b end table close square brackets on top A open parentheses x plus a comma space y plus b close parentheses

Perputaran sejauh alpha degree dangan pusat open parentheses a comma space b close parentheses 

open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row cell cos space alpha end cell cell negative sin space alpha end cell row cell sin space alpha end cell cell cos space alpha end cell end table close parentheses open parentheses table row cell x minus a end cell row cell y minus b end cell end table close parentheses

Sehingga diperoleh

  • open parentheses x comma y close parentheses digeser open parentheses table row cell negative 1 end cell row 2 end table close parentheses

open parentheses x apostrophe comma space y apostrophe close parentheses rightwards arrow with T open square brackets table row cell negative 1 end cell row 2 end table close square brackets on top open parentheses x minus 1 comma space y plus 2 close parentheses 

  • Bayangan open parentheses x apostrophe comma space y apostrophe close parentheses diputar 90 degree dengan pusat left parenthesis negative 1 comma space 2 right parenthesis, sehingga:

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 alpha end cell cell negative sin space alpha end cell row cell sin space alpha end cell cell cos space alpha end cell end table close parentheses open parentheses table row cell x minus a end cell row cell y minus b end cell end table close parentheses end cell row cell open parentheses table row cell x " plus 1 end cell row cell y " minus 2 end cell end table close parentheses end cell equals cell open parentheses table row cell cos space 90 end cell cell negative sin space 90 end cell row cell sin space 90 end cell cell cos space 90 end cell end table close parentheses open parentheses table row cell x apostrophe plus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses end cell row cell open parentheses table row cell x " plus 1 end cell row cell y " minus 2 end cell end table close parentheses end cell equals cell open parentheses table row 0 cell negative 1 end cell row 1 0 end table close parentheses open parentheses table row cell x apostrophe plus 1 end cell row cell y apostrophe minus 2 end cell end table close parentheses end cell row cell open parentheses table row cell x " plus 1 end cell row cell y " minus 2 end cell end table close parentheses end cell equals cell open parentheses table row cell 2 minus y apostrophe end cell row cell x apostrophe plus 1 end cell end table close parentheses end cell row cell open parentheses table row cell x " end cell row cell y " end cell end table close parentheses end cell equals cell open parentheses table row cell 1 minus y apostrophe end cell row cell x apostrophe plus 3 end cell end table close parentheses end cell end table 

Berdasarakan persmaaan di atas, maka diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row cell x " end cell row cell y " end cell end table close parentheses end cell equals cell open parentheses table row cell 1 minus y apostrophe end cell row cell x apostrophe plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell x " end cell row cell y " end cell end table close parentheses end cell equals cell open parentheses table row cell 1 minus open parentheses y plus 2 close parentheses end cell row cell open parentheses x minus 1 close parentheses plus 3 end cell end table close parentheses end cell row cell open parentheses table row cell x " end cell row cell y " end cell end table close parentheses end cell equals cell open parentheses table row cell negative y minus 1 end cell row cell x plus 2 end cell end table close parentheses end cell end table

Diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell x " end cell equals cell negative y minus 1 end cell row y equals cell negative x " minus 1 end cell row blank blank blank row cell y " end cell equals cell x plus 2 end cell row x equals cell y " minus 2 end cell end table

Sehingga bayangan elips tersebut:

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x squared plus 9 y squared end cell equals 36 row cell 4 open parentheses y " minus 2 close parentheses squared plus 9 open parentheses negative x " minus 1 close parentheses squared end cell equals 16 row cell 4 open parentheses y minus 2 close parentheses squared plus 9 open parentheses x plus 1 close parentheses squared end cell equals 16 end table 

Dengan demikian, persamaan bayangan elips tersebut adalah  adalah table attributes columnalign right center left columnspacing 0px end attributes row cell 4 open parentheses y minus 2 close parentheses squared plus 9 open parentheses x plus 1 close parentheses squared end cell equals 16 end table

Jadi, jawaban yang tepat adalah D.

 

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