Hey, everyone, and welcome back. So in the last video, we got introduced to this idea of conic sections. Basically, various ways you can slice a three-dimensional cone with a two-dimensional plane to get different shapes. Now today, we're going to be looking at the first shape, which is the circle. And you can get a circle from slicing a three-dimensional cone directly horizontally. Now, I think a circle is a pretty general shape that we're all familiar with. What we're going to be looking at in this video is how the equation for a circle will justify both the size and position that that shape is in. Now that might sound a bit complicated and a little bit scary, but don't sweat it. Because in this video, we're going to be taking a look at some graphs and examples that I think are going to make this concept super straightforward. So let's get right into this.

Now the thing that makes a circle unique is that a circle contains all points which are the same distance from the center of the circle. So to understand this, let's say you're looking at a circle and you have the center of it. If you want to get to this point right here on the circle, this would be the same distance as going to this point or going to that point. It's the same distance all the way around, and that's what makes this shape unique.

Now, to graph a circle, there are two things you need. You need the center and you also need this distance we just described, which is called the radius. Now, if you're given the graph already it's actually pretty straightforward to find both of these. So if you want to find the center of a circle, let's say you're dealing with a circle that's at the origin of your graph. This is a pretty common case, and when this happens the center of your circle is going to be at (0,0). This is what it means to be at the origin.

Now to find the radius of the circle, well, we could just start at the center and count. It takes 1, 2, 3, 4 units to get to this point, and so that means our radius is 4. Now let's say we have a circle that this time is not at the origin. Well, when this happens, your first step should be to find where your new center (h,k) is. H being the horizontal position, and k being the vertical position. So what we can do is start by looking at the origin of our graph, which is right here. And if I look at the origin it looks like horizontally we've gone 1, 2 units to the right. So our horizontal position is 2, and then our vertical position is going to be 1 unit up, which is 1. So (2,1) would be the center of our circle. And to find the radius, well, I just need to start at the center and go 1, 2, 3, 4 units. That would get me to this point, so the radius is also 4 for this circle.

So as you can see, finding the radius and center is pretty straightforward. But once you have these things, you can write the equation for the circle. If you have a circle at the origin, the equation is going to be x 2+y 2=r 2. And the only thing we really care about in this equation is the r value, which we already calculated to be 4. So the equation is going to be x 2+y 2=4 2. Now if the circle is not at the origin, the equation is going to look something like this. Notice it's very similar to what we had before except now we have a minus h and a minus k inside of the equation. This is because we have this new center that the circle is at, so we have to take this new position into account. So writing the equation for this, we're going to have x-2 2+y-1 2=4 2. So this would be the equation for our circle which is not at the origin.

Now, something that I'll mention is that it's actually not too common that you're going to see the graph of the circle already given to you, but rather you're going to see the equation of the circle and have to graph it from there. So let's actually see if we can try an example where we graph a circle from just the equation. Now, our first step should be to figure out what our center is h, k. Well, I can see that our h value corresponds with this 1, so our horizontal position is 1, and I can see that our k value corresponds with 2. So that would be our vertical position. So going over here on the graph, our horizontal is at 1, and our vertical is up 2. So this right here would be the center of our circle. Now step 2 tells us to find the radius of our circle. Well, notice for the equation, this last term is r 2. We can see that r 2 is going to be this 9. So that means if r 2 is equal to 9, we could find the radius by simply taking the square root on both sides of this little equation here. This will get the square to cancel, giving us that r is equal to 9, which is 3. So that means our radius is 3. Now our third step says to plot 4 points that are a distance r, and we already figured out r is equal to 3, and this is going to be to the left, right, up, and down from our center point. So if I go over here to the center point, our radius is 3 so we're going to go up 1, 2, 3 units, we're going to go down 1, 2, 3 units, we're going to go to the left 3 units, and we're going to go to the right 3 units. And these are going to be the 4 points that we need to plot. So finding these 4 points was our third step.

Now our last step is going to be to connect these 4 points with a smooth curve. So if I go ahead and draw a curve that connects all 4 of these outside points together, this will give me the circle that I desire. So this right here is the graph for our circle. Now, one more thing that I want to mention is that whenever you're dealing with a circle, the circle is not an example of a function. And this is because the circle would fail the vertical line test. We talked about in previous videos how if you're able to take a line and draw it on a graph, and that line crosses more than one point on the curve, this means you are not dealing with a function. And since the circle fails the vertical line test, it's not a function. So that's the basic idea behind the equations and graphs for a circle. Hope you found this video helpful. Thanks for watching, and let me know if you have any questions.