Menentukan Kedudukan Suatu Garis Terhadap Lingkaran

27 Juli 09

Dalam artikel kali ini saya akan mencoba memberikan sedikit solusi mencari kedudukan sebuah garis terhadap sebuah lingkaran. Kedudukan garis tersebut ada tiga jenis yaitu memotong lingkaran (memotong lingkaran di 2 titik), menyinggung lingkaran (memotong lingkaran di 1 titik) dan diluar lingkaran (tidak memotong lingkaran sama sekali). Kedudukan garis tersebut dapat di gambarkan sebagai berikut :

Gb.1 Garis Memotong Lingkaran di Dua Titik

Gb.2 Garis Memotong Lingkaran di Satu Titik

Gb.3 Garis Tidak Memotong Lingkaran

Nah, sekarang marilah kita lanjutkan dengan contoh satu soal saja di bawah ini:
Diketahui lingkaran X² + Y² – 4X + 6Y – 12 = 0 dan garis 3X + 4Y + 19 = 0. Tentukan kedudukan garis tersebut terhadap lingkaran!
Untuk menyelesaikan soal ini kita bisa menggunakan cara diskriminan atau cara jarak titik terhadap garis. Cara pertama mungkin agak membingungkan bagi beberapa dari kalian, namun cara kedua juga boleh digunakan walaupun agaknya menyimpang dari pakem yang digunakan dalam kurikulum.

Tinggalkan sola pakem, sekarang kita mulai mengerjakan soal di atas:

  1. Misalkan jarak pusat lingkaran terhadap garis adalah S maka kita punya tiga kemungkinan yaitu:
    1. jika, S < r, maka garis memotong lingkaran di dua titik
    2. jika, S = r, maka garis memotong lingkaran di satu titik
    3. jika, S > r, maka garis tidak memotong lingkaran
  2. Dari konsep diatas maka kita cari pusat dan jari-jari lingkaran. Kita sederhanakan persamaan lingkaran menjadi:
    x^2~+~ y^2 ~-~ 4x~ +~ 6y~-~12~=~0
    doubleleftright x^2~-~ 4x~+~y^2~+~6y~=~12
    doubleleftright (x~-~2)^2 ~+~(y~ +~3)^2~=~12+~4~+~9
    doubleleftright (x~-~2)^2 ~+~(y~ +~3)^2~=~25
    Jadi, didapat pusat (2,-3) dan jari-jari = 5
  3. Sekarang kita mencari jarak titik pusat (x_1,y_1)terhadap garis 3X + 4Y + 19 = 0 dengan rumus:
    S=~delim{|}{{ax_1+by_1+c}/{sqrt{a^2+b^2}}}{|} dengan a = 3, b = 4, c = 19 , x1 = 2 dan y1 = -3
  4. Sehingga,
    S=delim{|}{{3.2+4(-3)+19}/{sqrt{3^2+4^2}}}{|}
    S=delim{|}{{6-12+19}/{sqrt{9+16}}}{|}
    S=delim{|}{{13}/{sqrt{25}}}{|}
    S=~{13/5}

  5. Karena S < r maka disimpulkan garis memotong lingkaran di dua titik.

Nah, gitu caranya kalo kita lupa dengan rumus diskriminan. Jangan bosan belajar, matematika bisa gampang kok… hehehehe

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