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Pertanyaan

Jika lingkaran x squared plus y squared minus a x plus b y minus c equals 0 yang berpusat di titik left parenthesis negative 2 comma space 1 right parenthesis menyinggung garis y equals 2 x minus 5, nilai left parenthesis a minus b plus c right parenthesis adalah ....

  1. 11

  2. 13

  3. 15

  4. 17

  5. 21

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah B.

Pembahasan

Ingat!

  • Lingkaran yang berpusat di open parentheses a comma space b close parentheses dan menyinggung garis A x plus B y plus C equals 0 mempunyai jari-jari:

r equals open vertical bar fraction numerator A a plus B b plus C over denominator square root of A squared plus B squared end root end fraction close vertical bar

  • Persamaan lingkaran straight L identical to x squared plus y squared plus A x plus B y plus C equals 0 berpusat di titik open parentheses negative A over 2 comma space minus B over 2 close parentheses dan berjari-jari straight r equals square root of 1 fourth straight A squared plus 1 fourth straight B squared minus straight C end root.

Berdasarkan konsep tersebut maka lingkaran x squared plus y squared minus a x plus b y minus c equals 0 berpusat di titik

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses negative A over 2 comma space minus B over 2 close parentheses end cell equals cell left parenthesis negative 2 comma space 1 right parenthesis end cell row cell open parentheses negative fraction numerator negative a over denominator 2 end fraction comma space minus b over 2 close parentheses end cell equals cell left parenthesis negative 2 comma space 1 right parenthesis end cell row cell open parentheses a over 2 comma space minus b over 2 close parentheses end cell equals cell left parenthesis negative 2 comma space 1 right parenthesis end cell row cell a over 2 end cell equals cell negative 2 left right double arrow a equals negative 2 times 2 equals negative 4 end cell row cell negative b over 2 end cell equals cell 1 left right double arrow b equals 1 times open parentheses negative 2 close parentheses equals negative 2 end cell end table 

dan berjari-jari

table attributes columnalign right center left columnspacing 0px end attributes row straight r equals cell square root of 1 fourth straight A squared plus 1 fourth straight B squared minus straight C end root end cell row blank equals cell square root of 1 fourth open parentheses negative a close parentheses squared plus 1 fourth open parentheses b close parentheses squared minus open parentheses negative c close parentheses end root end cell row blank equals cell square root of a squared over 4 plus b squared over 4 plus c end root end cell row blank equals cell square root of open parentheses negative 4 close parentheses squared over 4 plus open parentheses negative 2 close parentheses squared over 4 plus c end root end cell row blank equals cell square root of 16 over 4 plus 4 over 4 plus c end root end cell row blank equals cell square root of 4 plus 1 plus c end root end cell row blank equals cell square root of 5 plus c end root end cell end table ...(*)

Lingkaran yang berpusat di left parenthesis negative 2 comma space 1 right parenthesis dan menyinggung garis y equals 2 x minus 5 left right double arrow 2 x minus y minus 5 equals 0 mempunyai jari-jari:

table attributes columnalign right center left columnspacing 0px end attributes row r equals cell open vertical bar fraction numerator A a plus B b plus C over denominator square root of A squared plus B squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator 2 times open parentheses negative 2 close parentheses plus open parentheses negative 1 close parentheses times 1 minus 5 over denominator square root of open parentheses negative 2 close parentheses squared plus open parentheses negative 1 close parentheses squared end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 4 minus 1 minus 5 over denominator square root of 4 plus 1 end root end fraction close vertical bar end cell row blank equals cell open vertical bar fraction numerator negative 10 over denominator square root of 5 end fraction close vertical bar end cell row blank equals cell fraction numerator 10 over denominator square root of 5 end fraction end cell row blank equals cell fraction numerator 10 over denominator square root of 5 end fraction cross times fraction numerator square root of 5 over denominator square root of 5 end fraction end cell row blank equals cell 10 over 5 square root of 5 end cell row blank equals cell 2 square root of 5 end cell end table ...(**)

Substitusikan (**) ke (*) menjadi

table attributes columnalign right center left columnspacing 0px end attributes row straight r equals cell square root of 5 plus c end root end cell row cell 2 square root of 5 end cell equals cell square root of 5 plus c end root end cell row blank blank cell open parentheses Kuadratkan space kedua space ruas close parentheses end cell row cell open parentheses 2 square root of 5 close parentheses squared end cell equals cell open parentheses square root of 5 plus c end root close parentheses squared end cell row cell 4 times 5 end cell equals cell 5 plus c end cell row 20 equals cell 5 plus c end cell row c equals cell 20 minus 5 end cell row c equals 15 end table

Dengan demikian diperoleh nilai a equals negative 4 comma space b equals negative 2 space dan space c equals 15 sehingga nilai

 table attributes columnalign right center left columnspacing 0px end attributes row cell a minus b plus c end cell equals cell negative 4 minus open parentheses negative 2 close parentheses plus 15 end cell row blank equals cell negative 4 plus 2 plus 15 end cell row blank equals 13 end table

Jadi, jika lingkaran x squared plus y squared minus a x plus b y minus c equals 0 yang berpusat di titik left parenthesis negative 2 comma space 1 right parenthesis menyinggung garis y equals 2 x minus 5, nilai left parenthesis a minus b plus c right parenthesis adalah 13.

Oleh karena itu, jawaban yang benar adalah B.

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