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### Equation of a Circle

```Date: 05/13/2003 at 13:52:22
From: Balbino
Subject: Equation of a circle

Find the equation of a circle with the center at point (3,-4) and
```

```
Date: 05/14/2003 at 14:23:04
From: Doctor Dotty
Subject: Re: Equation of a circle

Hi Balbino,

Thanks for the question.

Here's a circle with centre (0,0) and radius R:

|
|
,---+---.
,-'   y+    `-P(x,y)
,'       |     / `.
,'         |    /    `.
/           |  R/       \
;            |  /         :
;            | /          :
;             |/            :
-------+-------------+------+------+------
:             |      x      ;
:            |            ;
:            |            ;
\           |           /
`.         |         ,'
`.       |       ,'
`-.    |    ,-'
`---+---'
|
|

Pythagoras' Theorem tells us that R^2 = x^2 + y^2.

If you think about it, that equation is true whatever x and y are. If
they are in a negative quadrant, the radius is still positive. Can you
see why?

So, that's a circle with centre (0,0). We don't want this; we want it
at another centre.

So what if the centre is at (a,b)?

|
y+     ,-----.P(x,y)
|   ,'      /`.
|  /      R/   \
| ;       /     :
b+ |      +      |
| :       (a,b) ;
|  \           /
|   `.       ,'
|     '-----'
-----+--------+--+--------
|        a  x
|

Now, with Pythagoras' Theorem again:

R^2 = (the distance from x to a)^2 + (the distance from y to b)^2

R^2 = (x - a)^2 + (y - b)^2

That is the equation of a circle.

So for a circle of centre (-10,5) and radius 3, we get:

3^2 = (x - -10)^2 + (y - 5)^2

9 = (x + 10)^2 + (y - 5)^2

Some people prefer a general equation where the brackets have been
multiplied out, so that it is in the form:

x^2 + y^2 + 2gx + 2fy + c = 0

In our case, that can be worked out:

9 = (x + 10)^2 + (y - 5)^2

9 = x^2 + 20x + 100  +  y^2 - 10y + 25

9 = x^2 + y^2 + 20x - 10y + 125

0 = x^2 + y^2 + 20x - 10y + 116

Can you now find the equation of your circle?

Write back if I can be of any more help - on this or anything else.

- Doctor Dotty, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles

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