SECTION 24.1: THE EQUATION OF
A CIRCLE
This chapter will be a very brief introduction to circles. All of you have
encountered circles in geometry. Notice the circle below.
The line segment that you see inside the circle and starting from the center
of the circle is called the "radius." The distance from the
center of the circle to any point the boundary of the circle is constant. The center of
a circle is not considered part of the circle: the center is a reference point
to determine the radius of a circle. The line segment that goes through the center of a circle and touches the
boundary of the circle is called the "diameter." The diameter is equal to twice the radius. The length around a circle is called the "circumference." See the circle below.
A circle is not a function! Remember the Vertical Line Test?
You can use that test to see that a circle is not a function. Look below at
the circle. This circle would have two
points on it that would have the same xvalues but different yvalues. Hence,
the circle is NOT a function.
My goal is to teach you how to recognize the equation of a circle and
how to graph a circle. Consider the general rule below:
THE EQUATION OF THE GENERAL FORM OF A CIRCLE 
(x  h)^{2} + (y  k)^{2} = r^{2}
where h, k, and r are real numbers.
The center of the circle is (h, k) and the radius is r.
Please note that there is addition between the two binomials
and
subtraction between the binomials that are squared. 
Below are some examples of equations of circles:
ex. 1) Rewrite (x  3)^{2} + (y + 2)^{2} = 81^{} in the form of a circle. Then neatly sketch the circle.
(x 
3)^{2} + (y 
2)^{2} =
9^{2}
I rewrote the binomials as subtraction and rewrote the numerical part on the right hand side of the equation as a real number squared. I can now draw some conclusions about the circle.
The center is (3, 2) and the radius is 9.
ex. 2) Rewrite x^{2} + y^{2} = 25 in the form of a circle. Then neatly sketch the circle.
(x 
0)^{2 }+ (y 
0)^{2} =
5^{2}
Notice how I rewrote x ^{2} +
y^{2} = 25 above! I put
x^{2} +
y^{2} = 25 into the form of an equation of a circle.
The center is (0, 0) and the radius is 5.
Again, the center is NOT part of the circle, it is a reference point used to
graph the circle.
ex. 3) Rewrite (x + 4)^{2} + (y  4)^{2} = 64 in the form of a circle. Then neatly sketch the circle.
(x   4)^{2} + (y  4)^{2} = 8^{2}
Center (4, 4) with radius 8.
Here are some examples of equations that are NOT circles:
ex. 1) Is (x + 4)^{2}  (y + 5)^{2} = 4 an equation of a circle?
ans:
The above is NOT a circle since subtraction exists between the two binomials
squared, but the operation needs to be addition!
ex. 2) Is (x + 4)^{2} + (y  5)^{2} = 4 an equation of a circle?
ans:
The above is NOT a circle because, in this case, you cannot have a negative symbol in front
of one of the binomials squared. Why?
HINT: Multiply both sides of the equation by 1 and see what happens.
SECTION 24.2: MORE ON GRAPHING
THE EQUATION OF A CIRCLE
I will go over an example on graphing a circle using the general form of an
equation of a circle. I will explain what I'm doing as I go through the recipe
that I use to graph the circle.
ex 1) Graph x^{2} + y^{2} = 4 using 4 points.
ans:
Step 1) (x  0)^{ 2} + (y  0)^{2} = 4
Step 2) I am able to reveal the center of the circle, (h, k) and the radius r.
The center of the
circle is (0, 0) and the radius is 2.
Step 3) Plot center point determined by (0, 0) to use as a reference to graph
your circle. Since you know the radius is 2, we can easily get 4 points
relative to the center (0, 0).
Step
4) Remove the center of circle and sketch a circle around the 4 points.
ex. 2) Graph (x  3)^{ 2} + (y + 2)^{2} = 16 using 4 points.
ans:
Step 1) (x  3)^{ 2} + (y  2)^{2} = 4^{2}
I rewrote the original equation in the circle form to reveal the center, (3, 2), and the radius 4.
Step 2) Plot the center point (3, 2) to use as a reference to graph your
circle.
Since you know the radius is 4, we can easily get 4 points relative to the
center (3, 2).
Step 3) Remove the center of circle and sketch a circle around the 4 points.
