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### 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/
```
Associated Topics:
High School Conic Sections/Circles
High School Linear Equations

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