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Misalkan garis 3x−y−6=0 menyinggung lingkaran x2+y2+8x−4y−20=0 di titik P dan menyinggung lingkaran x2+y2+16x−26=0 di titik Q, tentukan panjang garis PQ!

Pertanyaan

Misalkan garis begin mathsize 14px style 3 x minus y minus 6 equals 0 end style menyinggung lingkaran begin mathsize 14px style x squared plus y squared plus 8 x minus 4 y minus 20 equals 0 end style di titik begin mathsize 14px style P end style dan menyinggung lingkaran begin mathsize 14px style x squared plus y squared plus 16 x minus 26 equals 0 end style di titik begin mathsize 14px style Q end style, tentukan panjang garis begin mathsize 14px style P Q end style!

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F. Kurnia

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

panjang garis begin mathsize 14px style P Q equals square root of 10 end style satuan panjang.

Pembahasan

Garis singgung: begin mathsize 14px style 3 x minus y minus 6 equals 0 rightwards arrow y equals 3 x minus 6 end style. Substitusikan Persamaan garis singgung pada lingkaran

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus left parenthesis 3 x minus 6 right parenthesis squared plus 8 x minus 4 left parenthesis 3 x minus 6 right parenthesis minus 20 end cell equals 0 row cell x squared plus 9 x squared minus 36 x plus 36 plus 8 x minus 12 x plus 24 minus 20 end cell equals 0 row cell 10 x squared minus 40 x plus 40 end cell equals 0 row cell x squared minus 4 x plus 4 end cell equals 0 row cell left parenthesis x minus 2 right parenthesis squared end cell equals cell 0 x equals 2 end cell row blank rightwards arrow cell y equals 3 left parenthesis 2 right parenthesis minus 6 end cell row blank equals cell 6 minus 6 end cell row blank equals 0 end table end style 

dan

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x squared plus left parenthesis 3 x minus 6 right parenthesis squared plus 16 x minus 26 end cell equals 0 row cell x squared plus 9 x squared minus 36 x plus 36 plus 16 x minus 26 end cell equals 0 row cell 10 x squared minus 20 x plus 10 end cell equals 0 row cell x squared minus 2 x plus 1 end cell equals 0 row cell left parenthesis x minus 1 right parenthesis squared end cell equals cell 0 x equals 1 end cell row blank rightwards arrow cell y equals 3 left parenthesis 1 right parenthesis minus 6 end cell row blank equals cell 3 minus 6 end cell row blank equals cell negative 3 end cell end table end style

Jarak antara titik begin mathsize 14px style P left parenthesis 2 comma 0 right parenthesis space dan space Q left parenthesis 1 comma negative 3 right parenthesis end style yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell P Q end cell equals cell square root of left parenthesis 2 minus 1 right parenthesis squared plus left parenthesis 0 minus left parenthesis negative 3 right parenthesis right parenthesis squared end root end cell row blank equals cell square root of 1 squared plus 3 squared end root end cell row blank equals cell square root of 1 plus 9 end root end cell row blank equals cell square root of 10 end cell end table end style

Jadi panjang garis begin mathsize 14px style P Q equals square root of 10 end style satuan panjang.

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