Diketahui dua buah lingkaran sebagai berikut. Persamaan garis singgung persekutuan yang terbentuk dari kedua lingkaran tersebut adalah ….

Pertanyaan

Diketahui dua buah lingkaran sebagai berikut.

begin mathsize 14px style straight L subscript 1 blank colon open parentheses x minus 2 close parentheses squared plus open parentheses y plus 2 close parentheses squared equals 16 end style
begin mathsize 14px style straight L subscript 2 blank colon blank open parentheses x plus 2 close parentheses squared plus open parentheses y plus 4 close parentheses squared equals 4 end style

Persamaan garis singgung persekutuan yang terbentuk dari kedua lingkaran tersebut adalah ….

  1. begin mathsize 14px style y equals negative 2 end style 

  2. begin mathsize 14px style y equals negative 6 end style 

  3. begin mathsize 14px style y equals negative 2 blank dan blank 3 y equals 4 x plus 2 end style 

  4. begin mathsize 14px style y equals negative 6 space dan space 3 y equals 4 x plus 6 space end style  

  5. size 14px 3 size 14px y size 14px equals size 14px 4 size 14px x size 14px plus size 14px 6 size 14px space size 14px dan size 14px space size 14px 6 size 14px y size 14px equals size 14px 4 size 14px x size 14px plus size 14px 2  

R. RGFLSATU

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Terlebih dahulu tentukan pusat dan jari-jari masing-masing lingkaran.

Untuk begin mathsize 14px style L subscript 1 ∶ left parenthesis x minus 2 right parenthesis squared plus left parenthesis y plus 2 right parenthesis squared equals 16 end style, didapat

begin mathsize 14px style straight P subscript 1 open parentheses 2 comma negative 2 close parentheses straight r subscript 1 equals square root of 16 equals 4 end style

Untuk begin mathsize 14px style straight L subscript 2 blank colon blank open parentheses x plus 2 close parentheses squared plus open parentheses y plus 4 close parentheses squared equals 4 end style, didapat

begin mathsize 14px style straight P subscript 2 equals open parentheses negative 2 comma negative 4 close parentheses r subscript 2 equals square root of 4 equals 2 end style

Jarak titik pusat ke garis singgung akan sama dengan jari-jari dari masing-masing lingkaran.

Oleh karena itu, jarak titik begin mathsize 14px style straight P subscript 1 equals open parentheses 2 comma negative 2 close parentheses end style ke garis singgung begin mathsize 14px style y minus m x minus c equals 0 end style adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator 1 open parentheses negative 2 close parentheses minus m open parentheses 2 close parentheses minus c over denominator square root of open parentheses negative m close parentheses squared plus 1 squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar fraction numerator negative open parentheses 2 plus 2 m plus c close parentheses over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 4 row cell 2 plus 2 m plus c end cell equals cell plus-or-minus 4 square root of 1 plus m squared end root end cell row c equals cell plus-or-minus 4 square root of 1 plus m squared end root minus 2 minus 2 m blank horizontal ellipsis blank open parentheses 1 close parentheses end cell end table end style

Selanjutnya, jarak titik begin mathsize 14px style straight P subscript 2 equals open parentheses negative 2 comma negative 4 close parentheses end style ke garis singgung begin mathsize 14px style y minus m x minus c equals 0 end style adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator 1 open parentheses negative 4 close parentheses minus m open parentheses negative 2 close parentheses minus c over denominator square root of open parentheses negative m close parentheses squared plus 1 squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar fraction numerator negative open parentheses 4 minus 2 m plus c close parentheses over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 2 row cell 4 minus 2 m plus c end cell equals cell plus-or-minus 2 square root of 1 plus m squared end root end cell row c equals cell plus-or-minus 2 square root of 1 plus m squared end root minus 4 plus 2 m blank horizontal ellipsis blank open parentheses 2 close parentheses end cell end table end style

Substitusi persamaan (1) ke persamaan (2).

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell plus-or-minus 4 square root of 1 plus m squared end root minus 2 minus 2 m end cell equals cell plus-or-minus 2 square root of 1 plus m squared end root minus 4 plus 2 m end cell end table end style

Pada bentuk di atas, terdapat 4 kemungkinan, yaitu sebagai berikut.

Kemungkinan 1.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 4 square root of 1 plus m squared end root minus 2 minus 2 m end cell equals cell 2 square root of 1 plus m squared end root minus 4 plus 2 m end cell row cell 4 square root of 1 plus m squared end root minus 2 square root of 1 plus m squared end root end cell equals cell 2 plus 2 m minus 4 plus 2 m end cell row cell 2 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell end table end style

Kemungkinan 2.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 4 square root of 1 plus m squared end root minus 2 minus 2 m end cell equals cell negative 2 square root of 1 plus m squared end root minus 4 plus 2 m end cell row cell 4 square root of 1 plus m squared end root plus 2 square root of 1 plus m squared end root end cell equals cell 2 plus 2 m minus 4 plus 2 m end cell row cell 6 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell end table end style

Kemungkinan 3.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 4 square root of 1 plus m squared end root minus 2 minus 2 m end cell equals cell 2 square root of 1 plus m squared end root minus 4 plus 2 m end cell row cell negative 4 square root of 1 plus m squared end root minus 2 square root of 1 plus m squared end root end cell equals cell 2 plus 2 m minus 4 plus 2 m end cell row cell negative 6 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell end table end style

Kemungkinan 4.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 4 square root of 1 plus m squared end root minus 2 minus 2 m end cell equals cell negative 2 square root of 1 plus m squared end root minus 4 plus 2 m end cell row cell negative 4 square root of 1 plus m squared end root plus 2 square root of 1 plus m squared end root end cell equals cell 2 plus 2 m minus 4 plus 2 m end cell row cell negative 2 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell end table end style

Dari 4 kemungkinan di atas, dapat disederhanakan menjadi 2, yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell plus-or-minus 6 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell row cell plus-or-minus 2 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell end table end style

Kemungkinan 1.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell plus-or-minus 6 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell row cell open parentheses plus-or-minus 6 square root of 1 plus m squared end root close parentheses squared end cell equals cell left parenthesis 4 m minus 2 right parenthesis end cell row cell 36 open parentheses 1 plus m squared close parentheses end cell equals cell 16 m squared minus 16 m plus 4 end cell row cell 36 plus 36 m squared end cell equals cell 16 m squared minus 16 m plus 4 end cell row cell 36 m squared minus 16 m squared plus 16 m plus 36 minus 4 end cell equals 0 row cell 20 m squared plus 16 m plus 32 end cell equals 0 row cell 5 m squared plus 4 m plus 8 end cell equals 0 end table end style

Perhatikan bahwa bentuk begin mathsize 14px style 5 m squared plus 4 m plus 8 equals 0 end style merupakan bentuk definit positif sehingga tidak ada nilai begin mathsize 14px style m end style yang memenuhi persamaan tersebut.

Kemungkinan 2.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell plus-or-minus 2 square root of 1 plus m squared end root end cell equals cell 4 m minus 2 end cell row cell open parentheses plus-or-minus 2 square root of 1 plus m squared end root close parentheses squared end cell equals cell left parenthesis 4 m minus 2 right parenthesis squared end cell row cell 4 open parentheses 1 plus m squared close parentheses end cell equals cell 16 m squared minus 16 m plus 4 end cell row cell 4 plus 4 m squared end cell equals cell 16 m squared minus 16 m plus 4 end cell row 0 equals cell 16 m squared minus 4 m squared minus 16 m plus 4 minus 4 end cell row 0 equals cell 12 m squared minus 16 m end cell row 0 equals cell 4 m left parenthesis 3 m minus 4 right parenthesis end cell end table end style

Diperoleh nilai begin mathsize 14px style m equals 0 end style dan begin mathsize 14px style m equals 4 over 3 end style.

Untuk begin mathsize 14px style m equals 0 end style, didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 1 plus m squared end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank m end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell square root of 1 plus 0 end root end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis 0 right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus-or-minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table end style

Dari bentuk di atas, didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table 2 table attributes columnalign right center left columnspacing 0px end attributes row blank blank atau end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table 6 end style

Untuk begin mathsize 14px style m equals 4 over 3 end style, didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row c equals cell plus-or-minus 2 square root of 1 plus m squared end root minus 4 plus 2 m end cell row blank equals cell plus-or-minus 2 square root of 1 plus open parentheses 4 over 3 close parentheses squared end root minus 4 plus 2 open parentheses 4 over 3 close parentheses end cell row blank equals cell plus-or-minus 2 square root of 1 plus 16 over 9 end root minus 4 plus 6 over 3 end cell row blank equals cell plus-or-minus 2 square root of 25 over 9 end root minus 12 over 3 plus 6 over 3 end cell row blank equals cell plus-or-minus 2 open parentheses 5 over 3 close parentheses minus 4 over 3 end cell end table end style

Dari bentuk di atas, didapat

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank c end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 10 over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 4 over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank atau end table table attributes columnalign right center left columnspacing 0px end attributes row c equals cell negative 10 over 3 minus 4 over 3 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 14 over 3 end cell end table end style

Cek untuk nilai undefined dan begin mathsize 14px style c equals negative 2 end style apakah memenuhi atau tidak.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator 2 plus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar fraction numerator 2 minus 2 over denominator 1 end fraction close vertical bar end cell equals 4 row cell open vertical bar 0 close vertical bar end cell equals cell 4 blank open parentheses SALAH close parentheses end cell row blank blank blank row cell open vertical bar fraction numerator 4 minus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar fraction numerator 4 minus 2 over denominator 1 end fraction close vertical bar end cell equals 2 row cell open vertical bar 2 close vertical bar end cell equals cell 2 blank open parentheses BENAR close parentheses end cell end table end style

Karena hanya salah satu saja yang bernilai benar, maka pasangan nilai undefined dan begin mathsize 14px style c equals negative 2 end style TIDAK memenuhi persamaan garis singgung persekutuan yang diinginkan.

Cek untuk nilai undefined dan begin mathsize 14px style c equals negative 6 end style apakah memenuhi atau tidak.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator 2 plus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar fraction numerator 2 plus 0 minus 6 over denominator 1 end fraction close vertical bar end cell equals 4 row cell open vertical bar negative 4 close vertical bar end cell equals cell 4 blank open parentheses bold BENAR close parentheses end cell row blank blank blank row cell open vertical bar fraction numerator 4 minus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar fraction numerator 4 minus 0 minus 6 over denominator 1 end fraction close vertical bar end cell equals 2 row cell open vertical bar negative 2 close vertical bar end cell equals cell 2 blank open parentheses bold BENAR close parentheses end cell row blank blank blank end table end style

Karena keduanya benar maka pasangan nilai undefined dan begin mathsize 14px style c equals negative 6 end style memenuhi persamaan garis singgung persekutuan yang diinginkan.

Cek untuk nilai begin mathsize 14px style m equals 4 over 3 end style dan begin mathsize 14px style c equals 2 end style apakah memenuhi atau tidak. 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator 2 plus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar fraction numerator 2 plus 2 open parentheses 4 over 3 close parentheses plus 2 over denominator square root of 1 plus open parentheses 4 over 3 close parentheses squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar 4 close vertical bar end cell equals cell 4 blank open parentheses bold BENAR close parentheses end cell row blank blank blank row cell open vertical bar fraction numerator 4 minus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar fraction numerator 4 minus 2 open parentheses 4 over 3 close parentheses plus 2 over denominator square root of 1 plus open parentheses 4 over 3 close parentheses squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar 2 close vertical bar end cell equals cell 2 blank open parentheses bold BENAR close parentheses end cell end table end style

Karena keduanya benar maka pasangan nilai begin mathsize 14px style m equals 4 over 3 end style dan begin mathsize 14px style c equals 2 end style memenuhi persamaan garis singgung persekutuan yang diinginkan.

Cek untuk nilai begin mathsize 14px style m equals 4 over 3 end style dan begin mathsize 14px style c equals negative 14 over 3 end style apakah memenuhi atau tidak. 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar fraction numerator 2 plus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar fraction numerator 2 plus 2 open parentheses 4 over 3 close parentheses minus 14 over 3 over denominator square root of 1 plus open parentheses 4 over 3 close parentheses squared end root end fraction close vertical bar end cell equals 4 row cell open vertical bar 0 close vertical bar end cell equals cell 4 blank open parentheses SALAH close parentheses end cell row blank blank blank row cell open vertical bar fraction numerator 4 minus 2 m plus c over denominator square root of 1 plus m squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar fraction numerator 4 minus 2 open parentheses 4 over 3 close parentheses minus 14 over 3 over denominator square root of 1 plus open parentheses 4 over 3 close parentheses squared end root end fraction close vertical bar end cell equals 2 row cell open vertical bar negative 2 close vertical bar end cell equals cell 2 blank open parentheses BENAR close parentheses end cell row blank blank blank row blank blank blank end table end style

Karena hanya salah satu yang benar, maka pasangan nilai begin mathsize 14px style m equals 4 over 3 end style dan begin mathsize 14px style c equals negative 14 over 3 end style TIDAK memenuhi persamaan garis singgung persekutuan yang diinginkan.

Jadi, diperoleh pasangan nilai undefined dan begin mathsize 14px style c end style yang memenuhi adalah undefined dan begin mathsize 14px style c equals negative 6 end style serta begin mathsize 14px style m equals 4 over 3 end style dan begin mathsize 14px style c equals 2 end style.

Oleh karena itu, terdapat dua persamaan garis singgung persekutuan kedua lingkaran tersebut, yaitu sebagai berikut.

begin mathsize 14px style y equals m x plus c y equals negative 6 end style

dan

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row y equals cell m x plus c end cell row y equals cell 4 over 3 x plus 2 end cell row cell 3 y end cell equals cell 4 x plus 6 end cell end table end style

Jika kita gambar pada koordinat kartesius, maka diperoleh gambar sebagai berikut.

Jadi, jawaban yang tepat adalah D.

167

0.0 (0 rating)

Pertanyaan serupa

Diketahui persamaan lingkaran sebagai berikut. Salah satu persamaan garis singgung persekutuan dua lingkaran di atas adalah ….

43

0.0

Jawaban terverifikasi

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Roboguru

Coba GRATIS Aplikasi Ruangguru

Download di Google PlayDownload di AppstoreDownload di App Gallery

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ruangguru WhatsApp

081578200000

Email info@ruangguru.com

info@ruangguru.com

Contact 02140008000

02140008000

Ikuti Kami

©2022 Ruangguru. All Rights Reserved PT. Ruang Raya Indonesia