Finding the Radius and Center of a Circle

Date: 06/19/2002 at 15:24:19
From: John Trikouros
Subject: Radius and center of a circle

I need help trying to figure out the center and the radius for the
following equation:

   2         2
  X  + 8X + Y  - 6Y = 11


Thanks.

Date: 06/19/2002 at 16:02:49
From: Doctor Jeremiah
Subject: Re: Radius and center of a circle

Hi John,

You need to "complete the square" to get what you want.

  2         2
 X  + 8X + Y  - 6Y = 11

Now, since we can add things to one side as long as we
add them to the other side, we can find a value that
would turn (X^2 + 8X) into a factorable square and
add it to the equation on both sides.  Then we can do
the same for (Y^2 - 6Y).  When we do that we get:

  2              2
 X  + 8X + 16 + Y  - 6Y + 9 = 11 + 16 + 9
            ^             ^
            |             |
          (8/2)^2       (6/2)^2

Now we just factor the terms into squares:

        2          2
 (X + 4)  + (Y - 3)  = 11 + 16 + 9

Now fix up the right side:

        2          2
 (X + 4)  + (Y - 3)  = 36

        2          2    2
 (X + 4)  + (Y - 3)  = 6

And you have an equation that shows the center and radius.
Let me know if you still have questions.

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