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:
    [pmath]x^2~+~ y^2 ~-~ 4x~ +~ 6y~-~12~=~0[/pmath]
    [pmath] doubleleftright x^2~-~ 4x~+~y^2~+~6y~=~12 [/pmath]
    [pmath] doubleleftright (x~-~2)^2 ~+~(y~ +~3)^2~=~12+~4~+~9 [/pmath]
    [pmath] doubleleftright (x~-~2)^2 ~+~(y~ +~3)^2~=~25 [/pmath]
    Jadi, didapat pusat (2,-3) dan jari-jari = 5
  3. Sekarang kita mencari jarak titik pusat [pmath](x_1,y_1)[/pmath]terhadap garis 3X + 4Y + 19 = 0 dengan rumus:
    [pmath]S=~delim{|}{{ax_1+by_1+c}/{sqrt{a^2+b^2}}}{|}[/pmath] dengan a = 3, b = 4, c = 19 , x1 = 2 dan y1 = -3
  4. Sehingga,
    [pmath]S=delim{|}{{3.2+4(-3)+19}/{sqrt{3^2+4^2}}}{|}[/pmath]
    [pmath]S=delim{|}{{6-12+19}/{sqrt{9+16}}}{|}[/pmath]
    [pmath]S=delim{|}{{13}/{sqrt{25}}}{|}[/pmath]
    [pmath]S=~{13/5}[/pmath]

  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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    Hiya! I know this is kinda off topic nevertheless I’d figured I’d ask.
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    Hello there! I know this is somewhat off topic but I was wondering if you knew
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    When I initially commented I clicked the “Notify me when new comments are added” checkbox and now each time a comment
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    on the video to make your point. You definitely know what
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    when you could be giving us something enlightening to read?

    Today, I went to the beach with my children. I found
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    She never wants to go back! LoL I know this is entirely
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    Yesterday, while I was at work, my cousin stole my iphone and tested to see if it
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    I was wondering if you ever thought of changing the page layout of your website?
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    I’m truly enjoying the design and layout of your blog. It’s a very easy on the eyes which makes it much more enjoyable for me to come here and visit more often. Did you hire out a designer to create your theme? Excellent work!
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    Please let me know. Thanks
    Hi, I think your site might be having browser compatibility issues.
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    some overlapping. I just wanted to give you a quick heads up!

    Other then that, superb blog!
    Wonderful blog! I found it while surfing around on
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    I’ve been trying for a while but I never seem to get there! Thank you
    Hey there! This is kind of off topic but I need some help from an established blog. Is it hard to set up your own blog? I’m not very techincal but I can figure things out pretty
    quick. I’m thinking about creating my own but I’m not sure where to start.
    Do you have any tips or suggestions? Appreciate
    it
    Howdy! Quick question that’s totally off topic. Do you know how to make your site mobile friendly? My site looks weird when viewing from my iphone. I’m trying to find a theme or
    plugin that might be able to resolve this issue.
    If you have any suggestions, please share. Thanks!
    I’m not that much of a online reader to be honest but your blogs really nice, keep
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    I really like your blog.. very nice colors & theme. Did you create this website yourself or did you hire someone to do it for you? Plz respond as I’m looking
    to design my own blog and would like to know where u got this from.
    thank you
    Incredible! This blog looks just like my old one! It’s on a totally different topic but it has pretty much the same page layout and design. Superb choice of colors!
    Heya just wanted to give you a quick heads up and let you know a few of the pictures aren’t loading correctly.

    I’m not sure why but I think its a linking issue. I’ve tried it in two
    different web browsers and both show the same outcome.
    Howdy are using WordPress for your blog platform? I’m new to the blog world but I’m trying to get started and create my own.
    Do you require any coding expertise to make your own blog?
    Any help would be really appreciated!
    Hello this is kind of of off topic but I was wondering if blogs
    use WYSIWYG editors or if you have to manually code with HTML.
    I’m starting a blog soon but have no coding expertise so I wanted to get advice from someone with experience. Any help would be enormously appreciated!
    Hi! I just wanted to ask if you ever have any trouble with hackers? My last blog (wordpress) was hacked and I ended up losing a few months of hard work due to no backup. Do you have any methods to stop hackers?
    Hi! Do you use Twitter? I’d like to follow you if that would be okay.
    I’m undoubtedly enjoying your blog and look forward to new updates.
    Hi there! Do you know if they make any plugins to safeguard against hackers? I’m kinda paranoid about
    losing everything I’ve worked hard on. Any tips?
    Hi! Do you know if they make any plugins to help with Search Engine Optimization? I’m trying to get my blog to rank for some targeted keywords but I’m not seeing very good results. If you know of any please share. Thank you!
    I know this if off topic but I’m looking into starting my own
    blog and was curious what all is required to get set up?
    I’m assuming having a blog like yours would cost a pretty penny? I’m not very web savvy so I’m not 100% positive. Any recommendations or advice would be greatly appreciated. Thank you
    Hmm is anyone else encountering problems with the images on this blog loading? I’m trying
    to determine if its a problem on my end or if it’s the blog. Any suggestions would be greatly appreciated.
    I’m not sure exactly why but this website is loading extremely slow for me.
    Is anyone else having this issue or is it a issue on my end?

    I’ll check back later on and see if the problem still exists.
    Hey! I’m at work surfing around your blog from my new apple iphone!
    Just wanted to say I love reading your blog and look forward to
    all your posts! Carry on the fantastic work!
    Wow that was strange. I just wrote an extremely long comment but
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