Kedudukan Antara Dua Lingkaran

Kedudukan antara dua lingkaran atau kedudukan 2 lingkaran menujukkan posisi antara lingkaran pertama dan lingkaran kedua. Posisi tersebut dapat berupa lingkaran di dalam lingkaran, kedua lingkaran bersinggungan di dalam lingkaran, kedua lingkaran berpotongan di dua titik, kedua lingkaran bersinggungan di luar lingkaran, atau kedua lingkaran saling lepas (tidak memiliki titik potong). Untuk menentukan posisi lingkaran pertama terhadap lingkaran ke dua akan sangat mudah jika di lihat dalam gambar. Seperti halnya terlihat pada gambar di bawah.

alt="" src="https://cdn.utakatikotak.com/finder/lingkaran%20ke%20dua%20berada%20di%20dalam%20lingkaran%20pertama.png" style="height:258px; width:246px" />

Berdasarkan gambar di atas dapat dilihat bahwa posisi lingkaran ke dua berada di dalam lingkaran pertama. Namun, bagaimana jika yang diketahui hanya persamaan kedua lingkaran? Mencari tahu kedudukan 2 lingkaran dengan menggambarnya terlebih dahulu tentu bukan merupakan solusi yang baik. Cara ini sangat tidak efektif, sehingga tidak dianjurkan. Lalu, bagaimana cara untuk mengetahui kedudukan antara dua lingkaran yang baik? Caranya dapat dilakukan dengan memanfaatkan rumus jarak antara dua titik dan kriteria yang akan dibahas pada materi di bawah. Sebelumnya, akan mari kita ingat kembali rumus mengenai jarak antara dua titik.

Jarak Titik Terhadap Garis

1. Jarak antara titik  alt="P(x_{1}, \; y_{1})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-b2b271894d393c0cce81687bead8755a_l3.svg" style="height:16px; width:61px" title="Rendered by QuickLaTeX.com" /> dan  alt="Q(x_{2}, y_{2})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-82841ec8969aa9acdd008fb30e6473c2_l3.svg" style="height:16px; width:57px" title="Rendered by QuickLaTeX.com" />.
    
   alt="" src="https://cdn.utakatikotak.com/finder/Jarak%20Titik%20Terhadap%20Garis.png" style="height:240px; width:246px" />

      alt="\[ d =\sqrt{\left( x_{1} - x_{2} \right)^{2} + \left( y_{1} - y_{2} \right)^{2}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5b718c2df8f0f453d0a3df635fbf0b76_l3.svg" style="height:27px; width:186px" title="Rendered by QuickLaTeX.com" />

2. Jarak antara titik  alt="P(x_{1}, \; y_{1})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-b2b271894d393c0cce81687bead8755a_l3.svg" style="height:16px; width:61px" title="Rendered by QuickLaTeX.com" /> ke garis ax + by + c = 0.
    
   alt="" src="https://cdn.utakatikotak.com/finder/Jarak%20Titik%20Terhadap%20Garis%202.png" style="height:250px; width:246px" />

      alt="\[d = \left| \frac{ax_{1} + by_{1} + c}{\sqrt{a^{2} + b^{2}}} \right|\]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5a30f8f644e517862047264057b029bc_l3.svg" style="height:36px; width:122px" title="Rendered by QuickLaTeX.com" />

Kadua rumus di atas berguna untuk menentukan jarak antara kedua pusat lingkaran. Sehingga, kedudukan 2 lingkaran dapat diketahui melalui bentuk umum persamaan lingkarannya, tanpa harus menggambarnya terlebih dahulu. Oke, sekarang mari kita simak kriteria untuk menentukan kedudukan 2 lingkaran.

Kriteria Kedudukan Antara Dua Lingkaran

Diketahui dua buah lingkaran:

1. Lingkaran 1  alt="(L_{1})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c60701c23c46ff38ea18324c29134b5a_l3.svg" style="height:16px; width:26px" title="Rendered by QuickLaTeX.com" />

   • Pusat:  alt="P_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-3b2ec92d31a59c6a05aa750217349197_l3.svg" style="height:14px; width:14px" title="Rendered by QuickLaTeX.com" />

   • Jari-jari:  alt="r_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-40b842015f2d77183007329cefa03f59_l3.svg" style="height:10px; width:12px" title="Rendered by QuickLaTeX.com" />

2. Lingkaran 2  alt="(L_{2})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-dc3d7b02f7f77e9312363cabeea0c614_l3.svg" style="height:16px; width:26px" title="Rendered by QuickLaTeX.com" />

   • Pusat:  alt="P_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-2625b53304cbcda0cdba01c587bdb308_l3.svg" style="height:13px; width:15px" title="Rendered by QuickLaTeX.com" />

   • Jari-jari:  alt="r_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5b24038d0cefec08dce4777a19e49cac_l3.svg" style="height:9px; width:13px" title="Rendered by QuickLaTeX.com" />

Kriteria kedudukan antara dua lingkaran adalah sebagai berikut.

1. Memiliki Pusat yang Sama
   Jika  alt="P_{1} = P_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-255c70d1f2e9456fb281dae0ed98ecfe_l3.svg" style="height:14px; width:50px" title="Rendered by QuickLaTeX.com" /> dan  alt="\left| P_{1} P_{2} \right| = 0" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5a1c6309b3d9d8339ff99aa591e6b16c_l3.svg" style="height:15px; width:65px" title="Rendered by QuickLaTeX.com" />, maka  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> memiliki pusat yang sama dengan  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" />.
   alt="" src="https://cdn.utakatikotak.com/finder/lingkaran%20Memiliki%20Pusat%20yang%20Sama.png" style="height:219px; width:250px" />

2. Bersinggungan di dalam lingkaran
   Jika  alt="\left| P_{1}P_{2} \right| = (r_{1} - r_{2})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-159b1979aa9da9b483cc39dcad4cffd5_l3.svg" style="height:16px; width:112px" title="Rendered by QuickLaTeX.com" />, maka  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> dan  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> bersinggungan di dalam salah satu lingkaran.
   alt="" src="https://cdn.utakatikotak.com/finder/Bersinggungan%20di%20dalam%20lingkaran.png" style="height:276px; width:300px" />

3. Lingkaran kecil terletak di dalam lingkaran besar
   Jika  alt="\left| P_{1}P_{2} \right| \leq (r_{1} - r_{2})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-6c3f4b456d409bcdf0131d0151ae9ecd_l3.svg" style="height:16px; width:112px" title="Rendered by QuickLaTeX.com" />, maka  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> di dalam  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" />.
   alt="" src="https://cdn.utakatikotak.com/finder/Lingkaran%20kecil%20terletak%20di%20dalam%20lingkaran%20besar.png" style="height:291px; width:300px" />

4. Berpotongan di dua titik
   Jika  alt="(r_{1} - r_{2}) < \left| P_{1} P_{2} \right| < (r_{1} + r_{2})" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c9b03a7e10f843af527503939058d5d1_l3.svg" style="height:16px; width:186px" title="Rendered by QuickLaTeX.com" />, maka  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> berpotongan dengan  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> di dua titik.
   alt="" src="https://cdn.utakatikotak.com/finder/Berpotongan%20di%20dua%20titik.png" style="height:198px; width:300px" />

5. Bersinggungan di luar lingkaran (berpotongan di satu titik)
   Jika  alt="\left| P_{1} P_{2} \right| = r_{1} + r_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-9ba147f6167f281c2e90a6cc25f49d6b_l3.svg" style="height:15px; width:101px" title="Rendered by QuickLaTeX.com" />, maka  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> dan  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> bersinggungan di luar lingkaran.
   alt="" src="https://cdn.utakatikotak.com/finder/Bersinggungan%20di%20luar%20lingkaran%20(berpotongan%20di%20satu%20titik).png" style="height:192px; width:300px" />

6. Tidak Bersinggungan (Saling Lepas)
   Jika  alt="\left| P_{1} P_{2} \right| > r_{1} + r_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-0e73caba465fffacf41b72e72786e70c_l3.svg" style="height:15px; width:101px" title="Rendered by QuickLaTeX.com" />, maka  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> dan  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> tidak bersinggugan.
   alt="" src="https://cdn.utakatikotak.com/finder/Tidak%20Bersinggungan%20(Saling%20Lepas).png" style="height:187px; width:300px" />

Keterangan:  alt="\left| P_{1} P_{2} \right|" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-749e1b61ec8f74c6d0ebbdec30d911b9_l3.svg" style="height:15px; width:37px" title="Rendered by QuickLaTeX.com" /> adalah jarak antara kedua pusat lingkaran. Gunakan rumus jarak antara dua titik untuk menghitung jarak kedua pusat lingkaran.

Contoh soal menentukan kedudukan dua lingkaran

Diketahui pusat lingkaran  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> adalah (2, 6) dengan panjang jari-jari 2 cm. Sedangkan koordinat pusat lingkaran  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> adalah (10, 0) dengan jari-jari 6 cm. Selidikilah kedudukan antara lingkaran  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> dan lingkaran  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" />!

Pembahasan:
Diketahui:

   alt="\[ P_{1} (2, 6) \rightarrow r_{1} = 2 \; \textrm{cm} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-c7e3ed58c57e34d387b00ede8d77bad0_l3.svg" style="height:16px; width:133px" title="Rendered by QuickLaTeX.com" />

   alt="\[ P_{2} (10, 0) \rightarrow r_{2} = 6 \; \textrm{cm} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-9528b94308fa0ac2b8ce58aeb5aa9621_l3.svg" style="height:16px; width:139px" title="Rendered by QuickLaTeX.com" />

Akan dihitung jarak antara kedua titik pusat  alt="P_{1}(2, 6)" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-aebcef07b94297bb364a05d6e2f4c037_l3.svg" style="height:16px; width:47px" title="Rendered by QuickLaTeX.com" /> dan  alt="P_{2}(10,0)" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-140532673c2fd1871eb1548675af3d92_l3.svg" style="height:16px; width:54px" title="Rendered by QuickLaTeX.com" />

   alt="\[ \left| P_{1} P_{1}\right| = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - x_{2})^{2}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-54925cd856d6c70d415e8dd9d6f6fbab_l3.svg" style="height:18px; width:219px" title="Rendered by QuickLaTeX.com" />

   alt="\[ \left| P_{1} P_{1}\right| = \sqrt{(2- 10)^{2} + (0 - 6)^{2}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-2454a5aca5c425824b8a89317562179d_l3.svg" style="height:18px; width:197px" title="Rendered by QuickLaTeX.com" />

   alt="\[ \left| P_{1} P_{1}\right| = \sqrt{(-8)^{2} + (- 6)^{2}} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-dfb25dabafa68ede72b23e2ea11e6ca5_l3.svg" style="height:18px; width:163px" title="Rendered by QuickLaTeX.com" />

   alt="\[ \left| P_{1} P_{1}\right| = \sqrt{64 + 36} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-95cadfd6614e4e54501110571446e153_l3.svg" style="height:17px; width:117px" title="Rendered by QuickLaTeX.com" />

   alt="\[ \left| P_{1} P_{1}\right| = \sqrt{100} = 10 \; \textrm{cm} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8adff1057c268df076946348fe0edd26_l3.svg" style="height:18px; width:148px" title="Rendered by QuickLaTeX.com" />

Jumlah jari-jari adalah  alt="r_{1} + r_{2} = 2 + 6 = 8" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-798ca59ea5a1cb31670a871511d5ee74_l3.svg" style="height:14px; width:122px" title="Rendered by QuickLaTeX.com" /> cm.

Hungan antara jarak antara kedua pusat lingkaran dengan jari-jari adalah

   alt="\[ \left| P_{1} P_{2}\right| > r_{1} + r_{2} \]" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-5ffdcb16c1f56e114bfcacabadde731e_l3.svg" style="height:15px; width:101px" title="Rendered by QuickLaTeX.com" />

Kriteria di atas merupakan kondisi untuk kedua lingkaran yang saling bebas (tidak berpotongan atau bersinggungan).

alt="" src="https://cdn.utakatikotak.com/finder/Contoh%20soal%20menentukan%20kedudukan%20dua%20lingkaran.png" style="height:187px; width:300px" />

Jadi, hubungan antara  alt="L_{1}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-785dc767752a8b6c62e22f23a6798ee7_l3.svg" style="height:14px; width:15px" title="Rendered by QuickLaTeX.com" /> dan  alt="L_{2}" src="http://idschool.net/wp-content/ql-cache/quicklatex.com-8b8a9a09859529ed8a5d007d6956a3ab_l3.svg" style="height:13px; width:16px" title="Rendered by QuickLaTeX.com" /> adalah saling lepas.