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,

  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

154 responses to “Menentukan Kedudukan Suatu Garis Terhadap Lingkaran”

  1. sarah says:

    kenapa gambarnya ga mau keliatan ya? (using safari, mac snow leopard)

  2. rochsid tehape says:

    gambarnya sudah saya perbaiki.. sekarang sudah terlihat kan?

  3. Reea says:

    thanks.. ini cukup beguna untk saya.. :))

  4. Wulanprasetya says:

    Tnyaata matematika m’nyenan’k qaand buangeetz,,

  5. ririn says:

    kenapa garis memotong lingkaran pada dua titik yang berbeda, D > 0
    coba anda buktikan?

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    If you ever want to take some of the load off, I’d absolutely love to write some articles for your blog in exchange for a link back to mine. Please send me an e-mail if interested. Regards!
    Have you ever considered about including a little bit more than just your articles? I mean, what you say is valuable and everything. But imagine if you added some great photos or video clips to give your posts more, “pop”! Your content is excellent but with pics and videos, this website could undeniably be one of the most beneficial in its niche. Wonderful blog!
    Fascinating blog! Is your theme custom made or did you download it from somewhere? A theme like yours with a few simple adjustements would really make my blog stand out. Please let me know where you got your design. Appreciate it
    Hello would you mind stating which blog platform you’re working with?
    I’m going to start my own blog soon but I’m having a hard time choosing between BlogEngine/Wordpress/B2evolution and Drupal.

    The reason I ask is because your design and style seems different then most blogs and I’m looking for something unique. P.S Sorry for being off-topic but I had to ask!
    Hi there just wanted to give you a quick heads up. The words in your post seem to be running off the screen in Opera. I’m not sure if
    this is a format issue or something to do with web browser compatibility but I figured I’d post to let you know. The design look great though! Hope you get the problem resolved soon. Kudos
    With havin so much content do you ever run into any problems of plagorism or copyright violation? My blog has a lot of completely unique content I’ve either authored
    myself or outsourced but it seems a lot of it is popping it up all over
    the internet without my agreement. Do you know any ways to help reduce content from
    being ripped off? I’d truly appreciate it.
    Have you ever considered writing an ebook or guest authoring on other websites? I have a blog centered on the same ideas you discuss and would love to have you share some stories/information. I know my viewers would appreciate your work. If you’re even
    remotely interested, feel free to send me an email.

    Hi there! Someone in my Facebook group shared this website with us so I came to look it over.
    I’m definitely loving the information. I’m bookmarking and will be
    tweeting this to my followers! Exceptional blog and excellent style and design.

    Very good blog! Do you have any recommendations for aspiring writers?
    I’m hoping to start my own website soon but I’m a little lost on everything.
    Would you advise starting with a free platform like WordPress or go for a paid option?
    There are so many options out there that I’m completely confused .. Any tips? Kudos!
    My coder is trying to persuade me to move to .net from PHP. I have always disliked the idea because of the expenses. But he’s tryiong none the less.
    I’ve been using Movable-type on numerous websites for about a year and am anxious about switching to another platform. I have heard very good things about Is there a way I can import all my wordpress content into it? Any help would be really appreciated!
    Does your website have a contact page? I’m having a tough time locating it but,
    I’d like to shoot you an email. I’ve got some creative ideas for your blog you might be interested in
    hearing. Either way, great website and I look forward to seeing it develop over time.

    It’s a shame you don’t have a donate button! I’d definitely donate to this fantastic blog! I guess for now i’ll settle for bookmarking and adding your RSS
    feed to my Google account. I look forward to fresh updates and will share this blog
    with my Facebook group. Talk soon!
    Greetings from Carolina! I’m bored at work so I decided to browse your site on my iphone during lunch break. I really like the info you present here and can’t wait to take
    a look when I get home. I’m shocked at how fast your blog loaded on my cell phone .. I’m not even
    using WIFI, just 3G .. Anyways, good blog!
    Hiya! I know this is kinda off topic nevertheless I’d figured I’d ask.
    Would you be interested in exchanging links or maybe guest authoring a blog article
    or vice-versa? My blog addresses a lot of the same subjects
    as yours and I believe we could greatly benefit from each other.

    If you are interested feel free to send me an e-mail.
    I look forward to hearing from you! Fantastic blog by the way!

    Currently it looks like Movable Type is the top blogging platform available
    right now. (from what I’ve read) Is that what you are using on your blog?
    Fantastic post but I was wondering if you could write a litte more on this topic? I’d be
    very grateful if you could elaborate a little bit more.
    Hello there! I know this is somewhat off topic but I was wondering if you knew
    where I could find a captcha plugin for my comment form?
    I’m using the same blog platform as yours and I’m having trouble finding one?
    Thanks a lot!
    When I initially commented I clicked the “Notify me when new comments are added” checkbox and now each time a comment
    is added I get four emails with the same comment. Is there any way you
    can remove me from that service? Cheers!
    Howdy! This is my first visit to your blog! We are a collection of volunteers and starting a new initiative in a community
    in the same niche. Your blog provided us valuable information to work on.
    You have done a outstanding job!
    Hi! I know this is kinda off topic but I was wondering which blog platform are you
    using for this site? I’m getting sick and tired of WordPress because I’ve had issues with hackers and I’m looking at options for another platform. I would be fantastic if you could point me in the direction of a good platform.
    Hey! This post couldn’t be written any better!
    Reading this post reminds me of my previous room mate!
    He always kept talking about this. I will forward this
    post to him. Pretty sure he will have a good read. Thank you for sharing!

    Write more, thats all I have to say. Literally, it seems as though you relied
    on the video to make your point. You definitely know what
    youre talking about, why waste your intelligence on just posting videos to your blog
    when you could be giving us something enlightening to read?

    Today, I went to the beach with my children. I found
    a sea shell and gave it to my 4 year old daughter and said
    “You can hear the ocean if you put this to your ear.” She placed the
    shell to her ear and screamed. There was a hermit crab inside and it pinched her ear.

    She never wants to go back! LoL I know this is entirely
    off topic but I had to tell someone!
    Yesterday, while I was at work, my cousin stole my iphone and tested to see if it
    can survive a 30 foot drop, just so she can be a youtube sensation.
    My apple ipad is now destroyed and she has 83 views. I know this is totally off topic
    but I had to share it with someone!
    I was wondering if you ever thought of changing the page layout of your website?
    Its very well written; I love what youve got to say. But maybe you
    could a little more in the way of content so people could connect with it better.
    Youve got an awful lot of text for only having one or 2 pictures.

    Maybe you could space it out better?
    Hello, i read your blog from time to time and i own a similar one and
    i was just wondering if you get a lot of spam feedback?
    If so how do you protect against it, any
    plugin or anything you can advise? I get so much lately it’s driving me insane so any help is very much appreciated.
    This design is steller! You certainly know how to keep a reader entertained. Between your wit and your videos, I was almost moved to start my own blog (well, almost…HaHa!) Fantastic job. I really loved what you had to say, and more than that, how you presented it. Too cool!
    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!
    Howdy! I could have sworn I’ve been to this site before but after browsing
    through some of the post I realized it’s new to me. Anyhow, I’m definitely happy I found it and I’ll be book-marking and checking back often!
    Howdy! Would you mind if I share your blog with my zynga group? There’s a
    lot of folks that I think would really appreciate your content.

    Please let me know. Thanks
    Hi, I think your site might be having browser compatibility issues.
    When I look at your blog site in Firefox, it looks fine but when opening in Internet Explorer, it has
    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
    Yahoo News. Do you have any suggestions on how to get listed in Yahoo News?
    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
    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
    it up! I’ll go ahead and bookmark your website to come back in the future. Many thanks
    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
    after I clicked submit my comment didn’t show up. Grrrr… well I’m not writing all that over again.
    Regardless, just wanted to say wonderful

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