ArtsAutosBooksBusinessEducationEntertainmentFamilyFashionFoodGamesGenderHealthHolidaysHomeHubPagesPersonal FinancePetsPoliticsReligionSportsTechnologyTravel

Finding The Equation of A Circle

Updated on December 30, 2010

Finding The Equation of A Circle

A circle is a set of all points in a plane equidistant from a fixed point which is the center of the circle. The distance from the center to a point on the circle is called the radius.

Standard equation for a circle with center at the origin : X^2 + Y^2 = r^2

Standard equation for a circle with center at (h, k) : (X – h)^2 + ( Y – k)^2 = r^2

General equation of the circle : X^2 + Y^2 + DX + EY + F = 0.


Sample Problems Involving Circle :


Problem Number One :

Find the equation of a circle which passes through points (10, 2) ; (3, 9 ); (-2. 10).

Solution : Consider the general equation of a circle as X^2 + Y^2 + DX + EY + F = 0.

Using the given three points we derive our equation :

From the point ( 10, 2 ) we get the equation 100 + 4 + 10D + 2E + F = 0 or

104 + 10D + 2E + F = 0 by substituting 10 to X’s and 2 to Y’s . This is our equation one.

From the point (3, 9) we get 9 + 81 + 3D + 9E + F = 0 or 90 +3D + 9E + F = 0.

From the point (-2.10) we get 4 + 100 – 2D + 10E + F = 0 or 104 -2D+ 10E +F = 0.

We now have a system of three equations :

104 + 10D + 2E + F = 0 eqn one

90 + 3D + 9E +F = 0 eqn two

104 = 2D + 10E + F = 0 eqn three

We now use elimination method to find the value of D, E and F.

Using equation one and eqn two to eliminate F, we do this by subtracting equation two from eqn one :

104 + 10D + 2E + F = 0

-(90 +2D + 9E + F = 0)

We get 14 + 7D - 7E = 0 let this be eqn four.

Using Equation 2 and eqn 3 to eliminate F we subtract eqn 3 from eqn 2 :

90 + 3D + 9E + F = 0

- ( 104D -2D +10E + F =0)

We get -14 + 5D - E = 0 let this be equation five

Using eqn four and eqn five we can solve for D and E.

14 + 7D - 7E = 0 eqn four

-14 +5D - E = 0 eqn five

Multiple equation five by -7 in order to eliminate E

(-14 + 5D - E = 0) * (-7) =è 98 =35D + 7E = 0

Adding eqn 4 and eqn 5

14 + 7D - 7E = 0

98 - 35D + 7E = 0

112 -28D = 0

(1/28) 112 = 28D (1/28)

D = 4

Substitute D = 4 in equation four ;

14 + 7(4) - 7E = 0

14 + 28 - 7E = 0

42 = 7E

E = 6

Substitute D = 4 , E = 6 in eqn 1 to solve for F

104 + 10(4) + 2(6) + F = 0

104 + 40 + 12 + F = 0

156 + F = 0

F = - 156

The general equation of the circle we are solving is :

X^2 + Y^2 +4X + 6Y-156 = 0.

We get this general equation by substituting D = 4, E = 6 and F = -156 to the general equation of the circle X^2 + Y^2 + DX + EY + F = 0.

To solve for the standard equation of the circle we will use completing the square:

X^2 + 4X + ____ + Y^2 + 6Y + ___ = 156

To complete the square divide the coefficient of the middle term by two then square it.

X^2 + 4X + 4 + Y^2 + 6Y + 9 = 156 + 4 + 9.

The standard equation of the circle we are looking for is :

(X + 2)^2 + (Y + 3)^2 = 169

The circle has its center at (-2, -3) and has a radius equal to 13.


Problem Number Two :

Find the equation of a circle whose diameter has its endpoints at A(-3, 5) and B(1, 3).

To find the center of the circle find the midpoint of the diameter using midpoint formula.

Here is the midpoint formula :

X = (X1 + X2 )/2 is used in finding the X-coordinate of the midpoint.

Y = (Y1 + Y2 )/2 is used in finding the Y-coordinate of the midpoint.

Using the endpoints (-3, 5) and (1, 3)

X = (-3 + 1)/2 = -1 Y = (5 +3 )/2 = 4

Therefore the midpoint of the diameter which is also the center of the circle we are looking for is ( -1, 4)

To find the radius of the circle find the distance between the center and one endpoint using distance formula. The distance formula is :

D = SQRT((Y2 – Y1)^2 + (X2 – X1)^2).

Using one endpoint (1, 3) and the center of the circle ( -1, 4), let us find the distance using the distance formula :

D = SQRT( ( 3-4)^2 + (1 + 1)^2 ) = > SQRT(5)

Therefore the standard equation of the circle we are looking for is :

(X + 1 )^2 + ( Y – 4 )^2 = 5

.

Comments

    0 of 8192 characters used
    Post Comment

    • profile image

      Godspower 

      6 years ago

      Thank u very much for helping me to find asnwer to my exam

    • profile image

      zhei 

      6 years ago

      thanks :))

    • profile image

      Vincent 

      6 years ago

      this is very helpful thanks

      can you write about solving circle using equation?

    • profile image

      tashana tabawan 

      6 years ago

      thanx......it helps a lot for my assignment in advance algebra....and analytic geometry,can i ask more?jejejeje,if its ok.........

    • profile image

      nadine 

      6 years ago

      this helped me a lot my analytic geometry

      TY GBY

    • profile image

      monica_2492 

      7 years ago

      there a entries(coefficients, signs)that sometimes confuses me.,but your hub helped me a lot in my analytic geometry assignment., thanks a lot!Godbless!

    • cristina327 profile imageAUTHOR

      Maria Cristina Aquino Santander 

      7 years ago from Manila

      Hi steffer and bettybarnesb thank you for taking time to read this hub. Blessings and regards.

    • bettybarnesb profile image

      bettybarnesb 

      7 years ago from Bartlett, TN

      Ok Christina: I can add, subtract, multiply and divide then I am lost. Happy New Year.

    • steffer profile image

      Stef Verheyen 

      7 years ago from Belgium (Europe)

      ah ok

      shouldn't you work for NASA or something?

    working

    This website uses cookies

    As a user in the EEA, your approval is needed on a few things. To provide a better website experience, hubpages.com uses cookies (and other similar technologies) and may collect, process, and share personal data. Please choose which areas of our service you consent to our doing so.

    For more information on managing or withdrawing consents and how we handle data, visit our Privacy Policy at: https://hubpages.com/privacy-policy#gdpr

    Show Details
    Necessary
    HubPages Device IDThis is used to identify particular browsers or devices when the access the service, and is used for security reasons.
    LoginThis is necessary to sign in to the HubPages Service.
    Google RecaptchaThis is used to prevent bots and spam. (Privacy Policy)
    AkismetThis is used to detect comment spam. (Privacy Policy)
    HubPages Google AnalyticsThis is used to provide data on traffic to our website, all personally identifyable data is anonymized. (Privacy Policy)
    HubPages Traffic PixelThis is used to collect data on traffic to articles and other pages on our site. Unless you are signed in to a HubPages account, all personally identifiable information is anonymized.
    Amazon Web ServicesThis is a cloud services platform that we used to host our service. (Privacy Policy)
    CloudflareThis is a cloud CDN service that we use to efficiently deliver files required for our service to operate such as javascript, cascading style sheets, images, and videos. (Privacy Policy)
    Google Hosted LibrariesJavascript software libraries such as jQuery are loaded at endpoints on the googleapis.com or gstatic.com domains, for performance and efficiency reasons. (Privacy Policy)
    Features
    Google Custom SearchThis is feature allows you to search the site. (Privacy Policy)
    Google MapsSome articles have Google Maps embedded in them. (Privacy Policy)
    Google ChartsThis is used to display charts and graphs on articles and the author center. (Privacy Policy)
    Google AdSense Host APIThis service allows you to sign up for or associate a Google AdSense account with HubPages, so that you can earn money from ads on your articles. No data is shared unless you engage with this feature. (Privacy Policy)
    Google YouTubeSome articles have YouTube videos embedded in them. (Privacy Policy)
    VimeoSome articles have Vimeo videos embedded in them. (Privacy Policy)
    PaypalThis is used for a registered author who enrolls in the HubPages Earnings program and requests to be paid via PayPal. No data is shared with Paypal unless you engage with this feature. (Privacy Policy)
    Facebook LoginYou can use this to streamline signing up for, or signing in to your Hubpages account. No data is shared with Facebook unless you engage with this feature. (Privacy Policy)
    MavenThis supports the Maven widget and search functionality. (Privacy Policy)
    Marketing
    Google AdSenseThis is an ad network. (Privacy Policy)
    Google DoubleClickGoogle provides ad serving technology and runs an ad network. (Privacy Policy)
    Index ExchangeThis is an ad network. (Privacy Policy)
    SovrnThis is an ad network. (Privacy Policy)
    Facebook AdsThis is an ad network. (Privacy Policy)
    Amazon Unified Ad MarketplaceThis is an ad network. (Privacy Policy)
    AppNexusThis is an ad network. (Privacy Policy)
    OpenxThis is an ad network. (Privacy Policy)
    Rubicon ProjectThis is an ad network. (Privacy Policy)
    TripleLiftThis is an ad network. (Privacy Policy)
    Say MediaWe partner with Say Media to deliver ad campaigns on our sites. (Privacy Policy)
    Remarketing PixelsWe may use remarketing pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to advertise the HubPages Service to people that have visited our sites.
    Conversion Tracking PixelsWe may use conversion tracking pixels from advertising networks such as Google AdWords, Bing Ads, and Facebook in order to identify when an advertisement has successfully resulted in the desired action, such as signing up for the HubPages Service or publishing an article on the HubPages Service.
    Statistics
    Author Google AnalyticsThis is used to provide traffic data and reports to the authors of articles on the HubPages Service. (Privacy Policy)
    ComscoreComScore is a media measurement and analytics company providing marketing data and analytics to enterprises, media and advertising agencies, and publishers. Non-consent will result in ComScore only processing obfuscated personal data. (Privacy Policy)
    Amazon Tracking PixelSome articles display amazon products as part of the Amazon Affiliate program, this pixel provides traffic statistics for those products (Privacy Policy)