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Algebra - Notes
 Solving Equations and Inequalities Previous Chapter Next Chapter Common Graphs Lines Previous Section Next Section The Definition of a Function

## Circles

In this section we are going to take a quick look at circles.  However, before we do that we need to give a quick formula that hopefully you’ll recall seeing at some point in the past.

Given two points  and  the distance between them is given by,

So, why did we remind you of this formula?  Well, let’s recall just what a circle is.  A circle is all the points that are the same distance, r  called the radius, from a point,  - called the center.  In other words, if  is any point that is on the circle then it has a distance of r from the center, .

If we use the distance formula on these two points we would get,

Or, if we square both sides we get,

This is the standard form of the equation of a circle with radius r and center .

 Example 1  Write down the equation of a circle with radius 8 and center .   Solution Okay, in this case we have ,  and  so all we need to do is plug them into the standard form of the equation of the circle.

Do not square out the two terms on the left.  Leaving these terms as they are will allow us to quickly identify the equation as that of a circle and to quickly identify the radius and center of the circle.

Graphing circles is a fairly simple process once we know the radius and center.  In order to graph a circle all we really need is the right most, left most, top most and bottom most points on the circle.  Once we know these it’s easy to sketch in the circle.

Nicely enough for us these points are easy to find.  Since these are points on the circle we know that they must be a distance of r from the center.  Therefore the points will have the following coordinates.

In other words all we need to do is add r on to the x coordinate or y coordinate of the point to get the right most or top most point respectively and subtract r from the x coordinate or y coordinate to get the left most or bottom most points.

Let’s graph some circles.

 Example 2  Determine the center and radius of each of the following circles and sketch the graph of the circle. (a)    [Solution] (b)    [Solution] (c)    [Solution] Solution In all of these all that we really need to do is compare the equation to the standard form and identify the radius and center.  Once that is done find the four points talked about above and sketch in the circle.   (a)   In this case it’s just x and y squared by themselves.  The only way that we could have this is to have both h and k be zero.  So, the center and radius is,                                               Don’t forget that the radius is the square root of the number on the other side of the equal sign.  Here is a sketch of this circle.   A circle centered at the origin with radius 1 (i.e. this circle) is called the unit circle.  The unit circle is very useful in a Trigonometry class.   (b)   In this part, it looks like the x coordinate of the center is zero as with the previous part.  However, this time there is something more with the y term and so comparing this term to the standard form of the circle we can see that the y coordinate of the center must be 3.  The center and radius of this circle is then,                                               Here is a sketch of the circle.  The center is marked with a red cross in this graph.   (c)   For this part neither of the coordinates of the center are zero.  By comparing our equation with the standard form it’s fairly easy to see (hopefully…) that the x coordinate of the center is 1.  The y coordinate isn’t too bad either, but we do need to be a little careful.  In this case the term is  and in the standard form the term is .  Note that the signs are different.  The only way that this can happen is if k is negative.  So, the y coordinate of the center must be -4.   The center and radius for this circle are,                                              Here is a sketch of this circle with the center marked with a red cross.

So, we’ve seen how to deal with circles that are already in the standard form.  However, not all circles will start out in the standard form.  So, let’s take a look at how to put a circle in the standard form.