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Pertanyaan

Tentukan persamaan garis singgung lingkaran begin mathsize 14px style open parentheses x minus 3 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 9 end style tegak lurus garis begin mathsize 14px style 5 x plus 12 y minus 4 equals 0 end style!

R. Hajrianti

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

 persamaan garis singgung lingkaran begin mathsize 14px style open parentheses x minus 3 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 9 end style tegak lurus garis begin mathsize 14px style 5 x plus 12 y minus 4 equals 0 end style yaitu begin mathsize 14px style 5 y minus 12 x equals negative 7 end style atau begin mathsize 14px style 5 y minus 12 x equals negative 85 end style.

Pembahasan

Persamaan garis singgung yang bergradien begin mathsize 14px style m end style pada lingkaran dengan pusat begin mathsize 14px style open parentheses a comma b close parentheses end style dan jari-jari begin mathsize 14px style r end style adalah begin mathsize 14px style y minus b equals m open parentheses x minus a close parentheses plus-or-minus r square root of 1 plus m squared end root end style .

Diketahui persamaan lingkaran begin mathsize 14px style open parentheses x minus 3 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 9 end style. Lingkaran tersebut berpusat di begin mathsize 14px style open parentheses 3 comma negative 2 close parentheses end style dan berjari-jari begin mathsize 14px style r equals square root of 9 equals 3 end style . Lalu, diketahui juga garis singgung tegak lurus garis begin mathsize 14px style 5 x plus 12 y minus 4 equals 0 end style. Gradien garis begin mathsize 14px style 5 x plus 12 y minus 4 equals 0 end style begin mathsize 14px style open parentheses m subscript 1 close parentheses end style sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 5 x plus 12 y minus 4 end cell equals 0 row cell 12 y end cell equals cell negative 5 x plus 4 end cell row y equals cell fraction numerator negative 5 x plus 4 over denominator 12 end fraction end cell row blank equals cell negative 5 over 12 x plus 4 over 12 end cell row blank blank blank end table end style

begin mathsize 14px style m subscript 1 equals negative 5 over 12 end style

Garis singgung tegak lurus garis begin mathsize 14px style 5 x plus 12 y minus 4 equals 0 end style maka gradien garis singgung begin mathsize 14px style open parentheses m subscript 2 close parentheses end style sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript 1 cross times m subscript 2 end cell equals cell negative 1 end cell row cell negative 5 over 12 cross times m subscript 2 end cell equals cell negative 1 end cell row cell m subscript 2 end cell equals cell negative 1 divided by open parentheses negative 5 over 12 close parentheses end cell row blank equals cell negative 1 cross times open parentheses negative 12 over 5 close parentheses end cell row blank equals cell 12 over 5 end cell end table end style

Selanjutnya, persamaan garis singgung dengan gradien begin mathsize 14px style m subscript 2 end style pada lingkaran dengan pusat begin mathsize 14px style open parentheses 3 comma negative 2 close parentheses end style dan berjari-jari begin mathsize 14px style r equals 3 end style yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus b end cell equals cell m open parentheses x minus a close parentheses plus-or-minus r square root of 1 plus m squared end root end cell row cell y minus open parentheses negative 2 close parentheses end cell equals cell m subscript 2 open parentheses x minus 3 close parentheses plus-or-minus 3 square root of 1 plus open parentheses m subscript 2 close parentheses squared end root end cell row cell y plus 2 end cell equals cell 12 over 5 open parentheses x minus 3 close parentheses plus-or-minus 3 square root of 1 plus open parentheses 12 over 5 close parentheses squared end root end cell row cell y plus 2 end cell equals cell 12 over 5 x minus 36 over 5 plus-or-minus 3 square root of 1 plus 144 over 25 end root end cell row cell y plus 2 end cell equals cell 12 over 5 x minus 36 over 5 plus-or-minus 3 square root of 25 over 25 plus 144 over 25 end root end cell row cell y plus 2 end cell equals cell 12 over 5 x minus 36 over 5 plus-or-minus 3 square root of 169 over 25 end root end cell row cell y plus 2 end cell equals cell 12 over 5 x minus 36 over 5 plus-or-minus 3 cross times 13 over 5 end cell row y equals cell 12 over 5 x minus 36 over 5 minus 2 plus-or-minus 39 over 5 end cell row y equals cell 12 over 5 x minus 36 over 5 minus 10 over 2 plus-or-minus 39 over 5 end cell row y equals cell 12 over 5 x minus 46 over 5 plus-or-minus 39 over 5 end cell end table end style
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 12 over 5 x minus 46 over 5 plus 39 over 5 end cell row y equals cell 12 over 5 x minus 7 over 5 end cell row y equals cell fraction numerator 12 x minus 7 over denominator 5 end fraction end cell row cell 5 y end cell equals cell 12 x minus 7 end cell row cell 5 y minus 12 x end cell equals cell negative 7 end cell end table end style  

begin mathsize 14px style atau end style  

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 12 over 5 x minus 46 over 5 minus 39 over 5 end cell row y equals cell 12 over 5 x minus 85 over 5 end cell row y equals cell fraction numerator 12 x minus 85 over denominator 5 end fraction end cell row cell 5 y end cell equals cell 12 x minus 85 end cell row cell 5 y minus 12 x end cell equals cell negative 85 end cell end table end style


Jadi, persamaan garis singgung lingkaran begin mathsize 14px style open parentheses x minus 3 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 9 end style tegak lurus garis begin mathsize 14px style 5 x plus 12 y minus 4 equals 0 end style yaitu begin mathsize 14px style 5 y minus 12 x equals negative 7 end style atau begin mathsize 14px style 5 y minus 12 x equals negative 85 end style.

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