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Tentukan persamaan bayangan dari elips≡x2+3y2=27 yang didilatasi oleh [(−1,2), D(−2)].

Pertanyaan

Tentukan persamaan bayangan dari begin mathsize 14px style elips identical to x squared plus 3 y squared equals 27 end style yang didilatasi oleh open square brackets left parenthesis negative 1 comma 2 right parenthesis comma space straight D left parenthesis negative 2 right parenthesis close square brackets.

O. Rahmawati

Master Teacher

Mahasiswa/Alumni UIN Sunan Gunung Djati Bandung

Jawaban terverifikasi

Jawaban

persamaan bayangan dari elips adalah x squared plus 3 y squared minus 6 x minus 36 y plus 9 equals 0.

Pembahasan

Sebuah titik apabila di dilatasi oleh open square brackets P left parenthesis a comma b right parenthesis comma space D left parenthesis k right parenthesis close square brackets maka bayangannya adalah

open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row k 0 row 0 k 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 plus open parentheses table row a row b end table close parentheses

Menentukan persamaan bayangan dari elips identical to x squared plus 3 y squared equals 27 yang didilatasi oleh open square brackets left parenthesis negative 1 comma 2 right parenthesis comma space straight D left parenthesis negative 2 right parenthesis close square brackets:

open parentheses table row cell x apostrophe end cell row cell y apostrophe end cell end table close parentheses equals open parentheses table row k 0 row 0 k 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 plus open parentheses table row a row b end table 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 negative 2 end cell 0 row 0 cell negative 2 end cell end table close parentheses open parentheses table row cell x plus 1 end cell row cell y minus 2 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row 2 end table 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 negative 2 x minus 2 end cell row cell negative 2 y plus 4 end cell end table close parentheses plus open parentheses table row cell negative 1 end cell row 2 end table 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 negative 2 x minus 3 end cell row cell negative 2 y plus 6 end cell end table close parentheses

diperoleh: 

x apostrophe equals negative 2 x minus 3 rightwards arrow x equals fraction numerator negative x apostrophe minus 3 over denominator 2 end fraction... left parenthesis 1 right parenthesis y apostrophe equals negative 2 y plus 6 rightwards arrow y equals fraction numerator negative y apostrophe plus 6 over denominator 2 end fraction... left parenthesis 2 right parenthesis

Substitusi persamaan (1) dan (2) ke persamaan elips:

table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus 3 y squared end cell equals 27 row cell open parentheses fraction numerator negative x apostrophe minus 3 over denominator 2 end fraction close parentheses squared plus 3 open parentheses fraction numerator negative y apostrophe plus 6 over denominator 2 end fraction close parentheses squared end cell equals 27 row cell open parentheses fraction numerator open parentheses x apostrophe close parentheses squared plus 6 x apostrophe plus 9 over denominator 4 end fraction close parentheses plus 3 open parentheses fraction numerator open parentheses y apostrophe close parentheses squared minus 12 y apostrophe plus 36 over denominator 4 end fraction close parentheses end cell equals 27 row cell open parentheses x apostrophe close parentheses squared plus 6 x apostrophe plus 9 plus 3 open parentheses y apostrophe close parentheses squared minus 36 y apostrophe plus 108 end cell equals 108 row cell open parentheses x apostrophe close parentheses squared plus 3 open parentheses y apostrophe close parentheses squared minus 6 x apostrophe minus 36 y apostrophe plus 9 end cell equals 0 end table

Jadi, persamaan bayangan dari elips adalah x squared plus 3 y squared minus 6 x minus 36 y plus 9 equals 0.

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