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Tentukan persamaan garis singgung dari kedudukan kedua lingkaran berikut. L1​≡x2+y2−2x−6y+9=0L2​≡x2+y2−4x−10y+19=0

Pertanyaan

Tentukan persamaan garis singgung dari kedudukan kedua lingkaran berikut.

L subscript 1 identical to x squared plus y squared minus 2 x minus 6 y plus 9 equals 0 L subscript 2 identical to x squared plus y squared minus 4 x minus 10 y plus 19 equals 0

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Pembahasan

Diketahui dua lingkaran berikut

L subscript 1 identical to x squared plus y squared minus 2 x minus 6 y plus 9 equals 0 L subscript 2 identical to x squared plus y squared minus 4 x minus 10 y plus 19 equals 0

Perhatikan, pada lingkaran pertama

table attributes columnalign right center left columnspacing 0px end attributes row cell P subscript 1 open parentheses negative 1 half A comma negative 1 half B close parentheses end cell equals cell P subscript 1 open parentheses negative 1 half open parentheses negative 2 close parentheses comma negative 1 half open parentheses negative 6 close parentheses close parentheses end cell row blank equals cell P subscript 1 open parentheses 1 comma 3 close parentheses end cell row cell r subscript 1 end cell equals cell square root of a squared plus b squared minus C end root end cell row blank equals cell square root of 1 squared plus 3 squared minus 9 end root end cell row blank equals cell square root of 1 end cell row blank equals 1 end table

dan, lingkaran kedua

table attributes columnalign right center left columnspacing 0px end attributes row cell P subscript 2 open parentheses negative 1 half A comma negative 1 half B close parentheses end cell equals cell P subscript 2 open parentheses negative 1 half open parentheses negative 4 close parentheses comma negative 1 half open parentheses negative 10 close parentheses close parentheses end cell row blank equals cell P subscript 2 open parentheses 2 comma 5 close parentheses end cell row cell r subscript 2 end cell equals cell square root of a squared plus b squared minus C end root end cell row blank equals cell square root of 2 squared plus 5 squared minus 19 end root end cell row blank equals cell square root of 10 end cell end table

Jarak antar pusat lingkaran adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell L subscript 1 L subscript 2 end cell rightwards double arrow cell open vertical bar P subscript 1 P subscript 2 close vertical bar equals square root of open parentheses 5 minus 3 close parentheses squared plus open parentheses 2 minus 1 close parentheses squared end root end cell row blank equals cell square root of 4 plus 1 end root end cell row blank equals cell square root of 5 end cell end table

Karena r subscript 2 plus r subscript 1 equals square root of 10 plus 1 dan open vertical bar r subscript 2 minus r subscript 1 close vertical bar equals square root of 10 minus 1 mengakibatkan

open vertical bar r subscript 2 minus r subscript 1 close vertical bar less than open vertical bar P subscript 1 P subscript 2 close vertical bar less than r subscript 2 plus r subscript 1

yang berarti bahwa lingkaran saling berpotongan dan memiliki dua garis singgung persekutuan luar. Misal persamaan garis singgungnya adalah m x plus y plus c equals 0, maka jarak antara titik pusat lingkaran satu terhadap garis singgung adalah

table attributes columnalign right center left columnspacing 0px end attributes row cell r subscript 1 end cell equals cell open vertical bar fraction numerator m plus 3 plus c over denominator square root of m squared plus 1 end root end fraction close vertical bar end cell row 1 equals cell open vertical bar fraction numerator m plus 3 plus c over denominator square root of m squared plus 1 end root end fraction close vertical bar end cell row cell plus-or-minus 1 end cell equals cell fraction numerator m plus 3 plus c over denominator square root of m squared plus 1 end root end fraction end cell row cell plus-or-minus square root of m squared plus 1 end root end cell equals cell m plus 3 plus c end cell row c equals cell negative m minus 3 plus-or-minus square root of m squared plus 1 end root end cell row cell r subscript 2 end cell equals cell open vertical bar fraction numerator 2 m plus 5 plus c over denominator square root of m squared plus 1 end root end fraction close vertical bar end cell row cell square root of 10 end cell equals cell open vertical bar fraction numerator 2 m plus 5 plus c over denominator square root of m squared plus 1 end root end fraction close vertical bar end cell row cell plus-or-minus square root of 10 end cell equals cell fraction numerator 2 m plus 5 plus c over denominator square root of m squared plus 1 end root end fraction end cell row cell plus-or-minus square root of 10 square root of m squared plus 1 end root end cell equals cell 2 m plus 5 plus c end cell row c equals cell negative 2 m minus 5 plus-or-minus square root of 10 square root of m squared plus 1 end root end cell row blank blank blank end table

Selanjutnya didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell negative m minus 3 plus-or-minus square root of m squared plus 1 end root end cell equals cell negative 2 m minus 5 plus-or-minus square root of 10 square root of m squared plus 1 end root end cell row cell pers space 1 right parenthesis space 2 plus m end cell equals cell plus-or-minus open parentheses square root of 10 plus 1 close parentheses square root of m squared plus 1 end root end cell row cell pers space 2 right parenthesis space 2 plus m end cell equals cell plus-or-minus open parentheses square root of 10 minus 1 close parentheses square root of m squared plus 1 end root end cell end table

Berdasarkan persamaan 1 didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 plus m end cell equals cell plus-or-minus open parentheses square root of 10 plus 1 close parentheses square root of m squared plus 1 end root end cell row cell open parentheses 2 plus m close parentheses squared end cell equals cell open parentheses square root of 10 plus 1 close parentheses squared open parentheses m squared plus 1 close parentheses end cell row cell m squared plus 4 m plus 4 end cell equals cell open parentheses 11 plus 2 square root of 10 close parentheses open parentheses m squared plus 1 close parentheses end cell row cell m squared plus 4 m plus 4 end cell equals cell open parentheses 17 comma 3 close parentheses open parentheses m squared plus 1 close parentheses end cell row blank equals cell 17 comma 3 m squared plus 17 comma 3 end cell row 0 equals cell 16 comma 3 m squared minus 4 m plus 13 comma 3 end cell row cell m subscript 1 comma 2 end subscript end cell equals cell fraction numerator 4 plus-or-minus square root of 16 minus 867 comma 16 end root over denominator 32 comma 6 end fraction end cell row blank equals cell fraction numerator 4 plus-or-minus square root of negative 851 comma 16 end root over denominator 32 comma 6 end fraction end cell end table

Persamaan 1 tidak memenuhi karena akar memuat unsur negatif.

Berdasarkan persamaan 2 didapatkan

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 2 plus m close parentheses squared end cell equals cell open parentheses square root of 10 minus 1 close parentheses squared open parentheses m squared plus 1 close parentheses end cell row cell m squared plus 4 m plus 4 end cell equals cell open parentheses 11 minus 2 square root of 10 close parentheses open parentheses m squared plus 1 close parentheses end cell row blank equals cell 4 comma 68 open parentheses m squared plus 1 close parentheses end cell row blank equals cell 4 comma 68 m squared plus 4 comma 68 end cell row 0 equals cell 3 comma 68 m squared minus 4 m plus 0 comma 68 end cell row cell m subscript 1 comma 2 end subscript end cell equals cell fraction numerator 4 plus-or-minus square root of 16 minus open parentheses 4 close parentheses open parentheses 3 comma 68 close parentheses open parentheses 0 comma 68 close parentheses end root over denominator 7 comma 36 end fraction end cell row blank equals cell fraction numerator 4 plus-or-minus square root of 16 minus 10 comma 01 end root over denominator 7 comma 63 end fraction end cell row blank equals cell fraction numerator 4 plus-or-minus 2 comma 45 over denominator 7 comma 63 end fraction end cell row cell m subscript 1 end cell equals cell 0 comma 85 end cell row cell m subscript 2 end cell equals cell 0 comma 203 end cell end table

Selanjutnya tentukan nilai c,

table attributes columnalign right center left columnspacing 0px end attributes row c equals cell negative 2 m minus 5 plus-or-minus square root of 10 square root of m squared plus 1 end root end cell row cell c subscript 1 comma 2 end subscript end cell equals cell negative 1 comma 7 minus 5 plus-or-minus square root of 10 square root of 1 comma 72 end root end cell row blank equals cell negative 6 comma 7 plus-or-minus 4 comma 15 end cell row blank equals cell negative 2 comma 55 logical or negative 10 comma 85 end cell row cell c subscript 3 comma 4 end subscript end cell equals cell negative 0 comma 406 minus 5 plus-or-minus square root of 10 square root of 1 comma 04 end root end cell row blank equals cell negative 5 comma 406 plus-or-minus 3 comma 23 end cell row blank equals cell negative 2 comma 18 logical or negative 8 comma 636 end cell end table

Uji nilai c yang memenuhi rumus jarak hanyalah c equals negative 2 comma 55 dan c equals negative 2 comma 18

Persamaan garis singgung persekutuannya yaitu

table attributes columnalign right center left columnspacing 0px end attributes row cell 0 comma 85 x plus y minus 2 comma 55 end cell equals 0 row y equals cell 2 comma 55 minus 0 comma 85 x end cell row cell 0 comma 2 x plus y minus 1 comma 25 end cell equals 0 row y equals cell 2 comma 18 minus 0 comma 2 x end cell end table

Dengan demikian, persamaan garis singgung dari kedudukan kedua lingkaran tersebut adalah

table attributes columnalign right center left columnspacing 0px end attributes row y equals cell 2 comma 55 minus 0 comma 85 x end cell row blank blank dan row y equals cell 2 comma 18 minus 0 comma 2 x end cell end table

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