Now let's look at circles.
A circle is a set of points
that are a fixed distance from the center (h, k).
Here, (x, y) represents any point along that circle.
The letter r represents the radius of the circle,
which is the distance between the center and any point on the circle.
Using the distance formula, we can get an equation.
Let's look. The distance is going to be equal to
x minus h squared plus y minus k squared.
Of course in this particular example,
that distance represents the radius, so we'll write: equals r.
Now let's just look at this right here.
What happens if we square both sides?
We're going to get x minus h squared
plus y minus k squared equals r squared.
This is very important.
It is the standard form of the equation of a circle.
You need to remember this formula and what everything stands for.
Remember (h, k) is the center,
(x, y) represents any point on the circle, and r is the radius.
Now if we were to multiply all this out,
and bring all the terms to one side and leave zero on the other,
we're going to get another form of the equation of a circle.
It's going to look like this.
This is called the general form of the equation of a circle.
You need to remember these two forms.
When they ask for a particular one, pay attention to which form you put it in.
Now let's look at an example where we find the equation of a circle given information.
Here, we're given the radius and the center,
and we want to find both forms of the equation of the circle.
Let's start with the standard form.
Recall the standard form looks like this:
x minus h squared plus y minus k squared equals r squared.
Here, h and k are (-2, 1), and the radius is 3.
Plugging that in, we get x minus a negative 2, which is x plus 2,
and we get y minus 1 squared, and we get 3 squared, which is 9.
Here is the standard form of the equation of the circle.
Now to get the general form, we're going to multiply all this out
and then get zero on one side.
Let's square x plus 2.
We're going to get x squared plus 4x plus 4,
and we're going to square y minus 1.
We get y squared minus 2y plus 1 equals 9.
Now we're going to collect all the terms on the lefthand side.
We're going to write it in the form: x squared first,
then the y squared term,
then the x term, then the y term,
and then we're going to get all the constants together.
Notice on the left, you have 5,
and when you subtract 9, you're going to get a negative 4.
Here is the general form of the equation of the circle.
Let's look at another example.
Let's look at this example where we're given the equation of the circle,
and we want to find the center and the radius of the circle.
Graph the circle, and find the intercepts if there are any.
Well, notice first of all you might be tempted to think that this is in standard form,
but remember in standard form,
the coefficient of the x squared and the y squared is a 1: the x minus h.
What I want to do first is to divide everything by 3.
Now it's in standard form,
so now I can say that the center is, and remember it's (h, k),
so h must be zero, so the center, (h, k),
is 0 and y minus k, so k is 5.
The radius is the square root of 4, which is 2.
Now with that information, we can easily graph the circle.
Here's the center (0, 5), and the radius is 2,
which means we can go up 2, down 2, right 2, and left 2
to get four points on the graph.
I'm going to go up 2, I'm going to go down 2,
I'm going to to right 2, and I'm going to go left 2.
I know that these are four points on the graph.
Draw my circle, so here's the circle.
Now we can easily, just from the graph, list the intercepts.
What's the x-intercept?
Well, you can see that the graph never crosses the x-axis,
so there are no x-intercepts.
What are the y-intercepts?
Well, we can easily find them from the graph.
Since the center is on the y-axis, which is (0, 5),
and we know that the radius is 2,
so, if we go up 2, this is going to be 7,
and, if we go down 2, this is going to be 3,
so the y-intercepts are 3 and 7.
What's important in this example is that you have to get it in standard form first.
If there's any number here in front of the x and the y squared,
you want to divide by that number first.
Now let's look at another similar example.
Let's look at this example where we want to find the center and the radius
and then find the intercepts.
Notice that this particular equation is in general form,
and when it's in general form,
we don't know what the center and the radius are,
so we're going to have to work backwards.
Remember we multiplied it out to get from standard form to general form.
To go from general form to standard form,
we're going to have to complete the square.
What I'm going to do is put the x terms together, 4x,
and then put the y terms together.
I'm going to bring the constant to the other side.
Now remember when completing the square,
what you do is you take half the middle term,
so take half of 4, which is 2, and you square it.
I want to add 4 here.
Again, this is negative 4.
Half of negative 4 is negative 2,
but, when you square it, you still get 4.
Notice what I've added on the lefthand side is 8,
so, to make the equation still hold true,
I need to add 8 to both sides.
I'm going to come over here and add 8.
Now what I have are two perfect squares.
This is the perfect square of x plus 2.
This is the perfect square of y minus 2.
On the righthand side, we have 9.
Now that it's in standard form, we can easily find the center,
which is going to be (-2, 2), and the radius, the square root of 9, is 3.
Now let's find the intercepts.
Here's the standard form of the equation of the circle that we got,
and we're going to find the intercepts using this equation.
Remember to find the x-intercepts if there are any.
You're going to set y equal to 0.
When I set y equal to 0, I'm going to get 0 minus 2.
On that term, you're going to get x plus 2 squared.
Since y is 0, you get negative 2 squared,
which is 4, equals 9. Subtracting 4 from both sides, you get 5.
Now I'm going to solve this quadratic equation by using the square root method.
Remember your plus or minus, so I get x plus 2
equals plus or minus the square root of 5,
so x equals negative 2 plus or minus the square root of 5.
Here are your two x-intercepts that you can see are irrational.
Now let's see if there are any y-intercepts.
To do that, we're going to set x equal to 0.
Here, if we set x to 0, we're going to get 4
plus y minus 2 squared equals 9.
Again, we're going to subtract 4 from both sides. We get 5.
Again taking the square root of both sides,
you get plus or minus the square root of 5.
Then adding 2 to both sides,
I get y equals 2 plus or minus the square root of 5.
Here are two y-intercepts for this particular circle.
Now let's look at another example.
Alright, let's look at this example.
We want to find the general form of the equation of a circle
given the center (-3, 1),
and this circle is tangent to the y-axis.
Remember, what do we need to get the equation of a circle?
We need to find the center, and we need the radius.
Well, we have the center, so we need to find the radius.
Well, let's draw what we know so far.
We know the center is (-3, 1).
If the circle is tangent to the y-axis,
what that means is the circle touches the y-axis at exactly one point,
which means it's going to touch right here.
It can't touch anywhere else or it will intersect at two points,
so that means that the circle is like this.
It's tangent to the y-axis. It only touches at one point,
so we can easily find this radius by counting.
The radius is of course going to be 3 units.
Now that we know the center and the radius,
we can get the equation of the circle.
We're going to start with the standard form,
which is x minus h squared plus y minus k squared equals r squared.
We're going to plug in our h and k (-3, 1).
We're going to get x plus 3 squared plus y minus 1 squared.
Since the radius is 3, 3 squared gives us 9.
Now that's the standard form.
Now to get the general form, we're going to multiply this all out.
We get x squared plus 6x plus 9
plus y squared minus 2y plus 1 equals 9.
Again in general form, you want the x squared term first,
then the y squared term, then the x term,
then the y term, and then we're going to collect our constants together.
We have 10 on the left minus 9 gives us a plus 1 equals 0.
Here is the general form of the equation of the circle.
Okay, let's look at another example.
Alright, let's look at this example.
We want the general form of the equation of the circle.
This time, the information that we're given
is we're given two endpoints of a diameter of a circle.
Well, let me start off by drawing what we know so far.
I'm going to plot each of these points.
If (4, 3) and (0, 1) are two endpoints of a diameter,
the circle is something like this. Which means this is your diameter.
And what do you know?
The center is on every diameter, so am I going to find the center?
I'm going to find the midpoint of the segment
of these two points: (4, 3) and (0, 1).
Remember how to find the midpoint.
You're going to find the midpoint, which is my center.
To find the midpoint, you take the x-coordinates,
you add them up, and you divide by 2.
We're going to take 4 plus 0
divided by 2 and add the y-coordinates together.
Divide by 2. What do we get?
We're going to get 4 over 2. We get (2, 2) as the center.
Now we can see from the graph that that seems reasonable
just by plotting the points. That center seems reasonable.
We're doing good. We've got the center.
What else do we need to find the equation of the circle?
Right, we need the radius.
We need to find the distance from this center
out to either point. It doesn't matter which point we use.
I'm going to use the point (0, 1) with (2, 2).
The distance between these two points is going to be my radius.
Using the distance formula, I'm going to do 2 minus 0 squared
plus 2 minus 1 squared. Let's see what we get. We get 4 plus 1.
The radius is the square root of 5.
Okay, now we know the center: (2, 2).
We know the radius,
so we can get the standard form of the equation of the circle.
We're going to do x minus 2 squared plus y minus 2 squared.
When you square the square root of 5, you get 5.
Now to get the general form,
we need to multiply all this out.
Let's see. We're going to get x squared minus 4x plus 4
plus y squared minus 4y plus 4
equals 5. Again, get the x squared term first,
then the y squared term, then the x term,
then the y term, and, again collecting the constants,
I get 8 minus 5, which is 3.
Here's the general form of the equation of the circle
with center and with these two endpoints of a diameter.
Now circles are very important.
You need to be able to recognize when you see the equation of a circle
that it's a circle in either form,
and you need to be able to get the equation given information.
You need to practice these skills.