 Another simple type of graph we might want to consider is that of a circle. So let's see what the equation of a circle is. So a circle is determined entirely by two different things. Once, when you know the center of the circle, and when you know the radius of the circle, then you know everything there is to know about the circle. So given the center and radius, how can I find the equation? Well, what we know is that the equation for any curve is going to be the thing that is true for all points that are on the curve. So we want to start off with what we know is true about any point on a circle. So for a circle, the distance between any point on the circle and the center is equal to the radius. Well, we know how to calculate distance. We have our distance formula, and so the distance between the center and any point on the circle should be equal to the radius, and that allows us to set up this equation using the distance formula. And since the radius has to be a non-negative real number, we can square both sides without introducing any extraneous solutions, and that prevents us, that allows us to avoid having to deal with a square root. And so here's our equation for any circle with center hk and radius r. For example, let's take the equation of the circle, and I know where the center is, and I know what a point on the circle is. So let's see if I can write the equation. Well, again, paper is cheap. Let's go ahead and write down our equation. Let's see. So if I want to use this equation, I need to know the center of the circle. Well, I got that. And I need to know the radius of the circle, and I don't have that, so I can't solve the problem. So, well, let's give up and go to the next problem. Well, actually, this is one of the reasons why it is so very important to know all of the definitions of mathematics. I need to know the radius. What is the definition of the radius? Well, the radius is the distance from any point on the circle to the center of the circle. Well, if only I had a point on the circle, well, how about that? And if only I knew where the center of the circle was, well, I actually know these things. So I can find the radius because I know the center of the circle and the point on the circle. So let's take a look at that. I know how to calculate distance. There's my distance formula. I'll substitute in the values that I have. My x-coordinates minus 2 and 5 by y-coordinates 3 and negative 4. I'll substitute those in, and after all the dust settles, I find the distance between the center of the circle and the point on the circle is square root of 98. So now I know the radius. I know the center of the circle, and that means I can write down the equation. So that'll be x, x minus the x-coordinate of the center, squared plus y, minus the y-coordinate of the center, squared equals to the radius squared, and I can do a little bit of the algebraic cleanup there, x minus 5 squared. That's a nice factor form. We'll leave that. x minus negative 4 squared. I can write that as x plus y plus 4 squared, and that's square root of 98. Squared is just going to be 98. Well, sometimes you have to do a little bit more. Suppose I actually have the equation of a circle. What can I say about it? Well, I want to find the center of the radius of the circle, and because the equation of a circle is of the form something of x squared, something of y squared equals r squared, then if I can get the equation into this form, I can pick off the center of the circle, hk, and then I can pick off the radius of the circle, r. So that means I need to get the equation of the circle into that form, and that means that x and y are both going to be terms that are perfect squares, so we want to complete the square on both sets of terms. So let's see. I'll ignore the constant because that just mugs things up, but here my first two terms on x are x squared plus 10x, and so in order to complete the square on x, I need 10 over 2 squared. I need to add 25, and if you do something to one side, if you want to keep an equation, you've got to do the same thing to the other side. So there's my first set of terms. These are the perfect square that make up the x terms. My y terms, likewise, if I want to complete the square on the y terms, what I need to add is going to be negative 6 over 2 squared. I'll add 9, and again I'll add 9 to both sides, and I'll let the desk settle. So over on the right-hand side, I have 50 over on the left-hand side. I have this, which is x plus 5 squared, and this, which is y minus 3 squared, and then a left over 7. Well, I'm going to take care of that. I'll subtract 7 from both sides, and there is my equation of the circle. I'll do one more step to read off the center of the radius. So again, I want x minus h plus y minus k. I want to not change anything. These two expressions should be the same. These two expressions should be the same. These two expressions should be the same, but the form will look a little bit different, and that will make it a little easier to pick off the information that I need, which is that the center is located at negative 5, 3, and the radius is square root of 43.