Finding the Center of a Circle

Date: 06/06/99 at 15:37:38
From: Eddie 
Subject: Circle: given in form: Ax^2+Cy^2+Dx+Ey+F=0, find the center 
and radius and graph it.

Hi, 

I am in advanced math in Rockville Centre, Long Island, and I need to 
know how to find the center and radius of a circle that is in the 
form: Ax^2 + Cy^2 + Dx + Ey + F = 0. It would be great if you could 
show me a sample problem or two.

Thank you very much.

Date: 06/07/99 at 11:59:17
From: Doctor Rick
Subject: Re: Circle: given in form: Ax^2+Cy^2+Dx+Ey+F=0, find the 
center and radius and graph it.

Hi, Eddie.

The form you have given is actually more general than a circle; it is 
the form for an ellipse with axes parallel to the coordinate axes. You 
can see the equations on our Analytic Geometry Formulas FAQ:

http://mathforum.org/dr.math/faq/formulas/faq.analygeom_2.html#twoellipses   

Look down that page for the circle formulas. A circle will have A = C.

To find the center and radius, you want to put the equation into the 
standard form

   (x-h)^2 + (y-k)^2 = r^2 

The center is then (h, k) and the radius is r. Here is an example:

   5x^2 + 5y^2 + 10x + 6y + 6 = 0

Divide through by 5 first, so that the coefficient of x^2 and y^2 
is 1:

   x^2 + y^2 + 2x + (6/5)y + 6/5 = 0

Group the terms in x and y separately:

   (x^2 + 2x) + (y^2 + (6/5)y) + 6/5 = 0

Complete the square in each set of parentheses. If you need help with 
this, you can search the Dr. Math Archives for completing the square.

   ((x + 1)^2 - 1) + ((y + 3/5)^2 - 9/25) + 6/5 = 0

Collect the constants and move them to the right:

   (x + 1)^2 + (y + 3/5)^2 = 4/25

Now you can read off the center: (-1, -3/5). The radius is the square 
root of the constant on the right, which is 2/5. Graph the circle; you 
will see that it passes through (-1, -1). Check this against the 
original equation:

   5(-1)^2 + 5(-1)^) + 10(-1) + 6(-1) + 6 = 0
   5       + 5       - 10     - 6     + 6 = 0
                                        0 = 0

Good, this point is indeed on the circle.

I hope this example has helped you. If you don't understand why I did 
any of these steps, write back and ask!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/