 Triangles are very important in trigonometry. Right triangles, especially. But equally important trigonometry to triangles, that is, is the study of circles. Circles are extremely important in trigonometry. And so let's remind ourselves what is a circle from an analytic geometry perspective. A circle is, in fact, the set of all points in the plane, which are a fixed distance, called r, from some fixed point, h comma k. Now you see a picture of a circle over here. This point, h comma k, is typically referred to as the center of the circle. And that's because everyone on the circle, so the points on the circle, are these points over here. All of these points are equidistant from the center. Every line, or excuse me, every point from the center is the same distance, which we call the radius of the circle. Or we sometimes call it little r for short. So the circle is the set of all equidistant points from the center in the plane. And so that's all a circle is. It's just a set of these points, which are the same distance away. Another bit of vocabulary that will come up from time to time when you talk about a circle is if you take a line segment that goes from one point on a circle to another point, this is called a chord. It's a line segment that cuts the circle into two pieces. Now, if the chord goes through the center of the circle, like this one does right here, this gives us what we call a diameter. So the diameter is, in fact, the longest possible chord you can have for a circle, in which case, if you look at it, if you take half of the diameter, that's actually just a radius. And in fact, the midpoint of the diameter is just the center of the circle. So we often have the statement that a diameter is just two radii. It's twice the length of the radius. So a little bit of vocabulary about circle centers, radii, diameters, and chords. OK. So let's go with this notation we've established. Let's say the center of the circle at the coordinate h, k. Let's say that the radius has just a positive number r. And let's say we take a fixed point on the circumference of the circle here, x comma y. Can we find an equation to represent this circle? So we know that the distance between the center of the circle and this point, x comma y, is a distance of r. So since we know the distance between, well, the center, we'll call it c. And we know this point here is p. We'll call it p, I should say. We know that the distance between p and c here, the so-called segment PC, this should be equal to r. So if we apply the distance formula to these coordinates, we're going to get that the difference of the x coordinates of p and c is going to be h minus r. We're going to take that difference and square it. Then if you take the difference of the y coordinates, y minus k, square that, then you take the square root of this is equal to r. This is just the distance formula. If you square both sides, we get the so-called standard formula, the standard form of the equation of a circle, x minus h squared plus y minus k squared is equal to r squared. You'll notice that this equation looks like very much the Pythagorean equation, a squared plus b squared equal to c squared. And that's because given any point on a circle, if we, from that point, we can construct a right triangle inside of said circle where one of the corners is going to coincide to be the center of the circle. And then the radius will coincide to the hypotenuse of this triangle. And so with that perspective in mind, this standard equation of a circle is just the Pythagorean equation with the concept that we have a fixed center, but we allow the points on the circle to be variable. So this is our equation of a circle. If we place the center of the circle at the origin, that is if h comma k is equal to 0, 0, the origin, then the h and k would disappear in the formula and we get the equation of a circle centered at the origin. If you take the radius to equal 1, we call this the unit circle and it has the equation x squared plus y squared equals 1. The unit circle will be very important in our study of circles and trigonometry, but remember this general formula for circles. For example, if you wanted to graph the circle x plus three squared plus y minus two squared is equal to 16, how would you do that? Well, let's first, let's look at the anatomy of this circle. So because we have an x plus three squared and we have a y minus two squared, we can find the center of the circle. The center is gonna be negative, well, excuse me, the center hk, this is gonna equal negative three comma two. Notice in the general formula, you're looking for x minus h squared plus y minus k squared is equal to r squared. You expect negatives to be inside of this formula. And so if you don't have negatives, you have to kind of translate it into having negatives. So this looks like x minus, x minus a negative three quantity squared. So in other words, when you look at the formula of a circle, when you look, when you go from the formula to the coordinates of the center, you're gonna switch the sign. If you see x plus three, you're gonna get a negative three for the x-coordinate. If you see a y minus two, then you're gonna get a positive two for the coordinate, the y-coordinate of the center. And so if you're graphing a circle, I would then plot that point, the center of the circle right there. Then how do you find the radius of the circle? The radius, well, since the radius should be over here, it's this constant square, you have to take the square root. So the square root of 16 is gonna give us the radius, which is equal to four. So now that we know the radius and we know the center of the circle, we can actually draw the circle. Now, if you're like me and you have a shaky hand, drawing a circle can look pretty hideous. That doesn't look like a circle at all. So one thing you can do is you can actually draw the following cross to give you a skeleton of the circle. So where do these four yellow points come from? So you'll notice this point going from the center to the right. If the radius is four, that means all points on the circle be four units away from the center. So if I just count it and go one, two, three, four, this gives me a point on the circle. That is I added four to the x-coordinate. And that's a negative three comma two becomes one comma two. That gives you a point on the circle. If you add four to the y-coordinate, that gives you a point above the circle, negative three, six. If you subtract four from the x-coordinate, that gives you the point negative seven comma two. And if you subtract four from the y-coordinate, that gives you the point negative three, negative two. This gives you this cross shape And this offers a skeleton to draw your circle. So if you use these four points, you can often draw something that at least resembles a circle, right? If looks far less hideous than I did before, it's not perfect, but unless you're gonna pull out a compass and straight edge, you know, this is a pretty good thing. If you're trying to do it by hand. If you did this on a computer, you see this perfect circular fashion. This is the graph of the circle we saw on the screen just a little bit ago. X plus three squared plus Y minus two squared is equal to 16. So we can go from the equation to the graph. But it turns out this process is also reversible. Look at another example. If you look at this circle right here, we see that the center of the circle is negative three six. And we see that the radius is equal to five. They actually told us that. What if they didn't tell us this was five and they gave us some coordinate right here, some point X comma Y, we could use the distance formula to figure out that the distance between the center and the point on the circle is in fact five. We'll see an example of that in just a second before we end this video here. So we know the radius is five. We know the center of the circle is five. So we can get the equation of the circle. We're gonna get X minus a negative three squared plus we get Y minus six squared is equal to five squared. Clean up this formula a little bit. We're gonna get X plus three quantity squared plus Y minus six quantity squared is equal to 25. And this then gives us the equation of the circle in standard form. Standard form is the form of an equation of circle we want the most because it'll be the simplest form and we can readily see the radius in the center from the standard form. Now it could be that someone actually was a jerk and multiplied all of this stuff out. You might get something like X squared plus six X plus nine plus Y squared minus 12 Y plus 36 is equal to 25. And you can add all these numbers together. It's like, what are you doing? X squared plus six X minus 12, excuse me, plus Y squared minus 12 Y, nine and 36 come together to make 45, but you have 25 on the other side. So you just tracked it. You get like a 20 equals zero. Some people write the equation of the circle like this. And this is just a rude thing to do because when it's all multiplied out like that it's not obvious what the center is. It's not obvious what the radius is. If you needed to actually put it in standard form you could do that. It would just require that you complete the square. You have to complete the square for the X coordinate or repeat the square for the Y coordinate and then go from there. So let's do one last example here. Let's find the equation of the circle that has as points, one, eight and five, negative six. And these are actually the end points of a diameter. So we wanna think about that for a moment. We have our circle right here. We have the point, let's see, one comma eight. That would be something like here, one comma eight. And then the other point, five, negative six. And this forms the diameter of the circle. So we need to find the equation of the circle which again looks like X minus H squared plus Y minus K squared equals R squared. We need to know the circle center, H comma K. We need to know the radius. Now the center of the circle, H, K this is actually gonna be the midpoint between these two points since we have a diameter here. So to find the midpoint we average together the X coordinates so we get one plus five over two. We average together the Y coordinates Y minus six over two. So one plus five is six, eight minus six is two. And simplifying the fractions we end up with three comma one. So that's gonna be the center of our circle. We have to also find the radius of the circle. There's two ways to do that. We can measure the distance between one comma eight and five comma negative six. That would give us the diameter which would be two times the radius. So that's something we could do or we could just find the distance between one of these points and the center of the circle. That would give us a radius. So the radius R will equal the distance between the point one eight and three one. So take the difference of the X coordinates we get three minus one in squared then take the difference of the Y coordinates one minus eight and then square it. Two, three minus one is a two. One minus seven is excuse me one minus eight is a negative seven got a little ahead of myself there. Two squared is four negative 49 squared is 49. And so then we get the square root of 53. Don't bother squaring it because actually I fibbed a little bit. We don't even need to know the radius to find the equation. We need to find the radius squared, right? That's what the formula requires up here. So the radius squared is actually gonna be 53. So we see that the equation of the circle here is going to be X minus three squared plus Y minus one squared is equal to 53. And so we've now found the equation of this circle. And so now in our discussion here about equations of circles and also it finishes lecture one for our lecture series Math 1060. So thanks for watching everyone. If you have any questions whatsoever during these videos, Bob means post your questions in the comments below and I'd be glad to answer any questions. If you learned anything about trigonometry feel free to hit the like button. And if you wanna see more videos like this in the future subscribe to the channel. Bye everyone, I'll see you next time.