Okay, well now we're into the last part of this lesson, and it's the circle. You say, how does the circle relate to all that other stuff? Well, you're going to see it's the distance formula reworked here.
Now I'm going to begin with you at the center at the origin here. So let's go ahead and got our coordinate system set up. Let's put a dot here to mark the center. Now I want you to visualize with me some circle around here. Let's go grab a point on that circle.
just one point. Can you picture the whole circle through here? All right, let's draw from the center out to the edge.
That'd be a radius measure. Okay, so let me just mark that here. This would be radius.
Now earlier we called that a D for distance formula, but now we're just calling it an R for radius. And what I'm looking for is how far is it? How do you get that r value?
You say, well, I can do that Pythagorean theorem. See, if I leave this point and just call him 0, 0, because he's at the origin, that's what we're looking at, and this is just some generic x and y, and actually this is prettier. this is one particular one, so I'll just call it x1 and y1. You say, how would you find that distance across there?
Because you'll remember that the distance formula, I'm just going to copy it over from what we had previous in the notes. It says x2 minus x1 squared and y2 minus y1 squared and take the square root of that, that sum. Okay?
So now I'm changing out my letters. Look, we've got R, and then instead of X1, or excuse me, instead of X2 and X1, I've got X1 and 0. So just to make it look pretty, I could, it doesn't matter which one you write first. It'd kind of make more sense if I'd write the Y's and the X's first.
So now we're looking at... x1 squared and y1 squared. Now we're getting close to the answer here. Let's undo this square root because we don't want to know the distance across here.
Let's just go with r squared. You say what? Well square both sides of that thing and that'll get rid of that radical out of there. So that would be an x1 squared and a y1 squared.
But you know what? I don't need it just for one little x1 and y1. Watch carefully here. I need it for any of these around here. So here's another point, here's another point, and another point, and you can see all these little radius measures out here.
So in other words, it's for any and all of these x's and y's around here. So I can get rid of this little subscript and just say we're opening up to all of them and there's a formula. Now most of the time most students will recite it as x squared plus y squared equals r squared and that's for a that's the formula for a circle centered at the origin.
Now while we're there so hold on to that thing while we're there let's look at it. look at one more screen here where we center this thing at another point called hk see we were centered at the origin back here hold on i'm going back we were centered at the origin which is zero zero now notice where these look look at where these numbers are coming in let me see if i can highlight this for you see right here there's my zero and there's my zero you see in those well i'm gonna change that and make that an H and a K. You say, why H and K? Well, in math, those letters usually represent some kind of center. They're not used for much else.
So let's go back over here and say, let's center that thing at HK. Let's move that center point. Now, hold on.
I'm going back. We're moving that center point back off somewhere over here. Call it HK.
Now, put your dot out there, and draw your... radius, draw that circle around there. Now remember that we had x squared plus y squared equals r squared. But now, since I've got this h and k, see, this was, a while ago, this was x minus zero, without taking you back another slide. But see, now, I don't want that zero anymore.
I want to replace that with an h and a k. h and k. So now...
My formula looks like x minus h squared and y minus k squared equals r squared. That's the formula. So now what I want to do is just practice that formula with you. So let's go look at several.
These are going to go fast exercises here. So let me write the formula right up above here, just so we've got it. It says x minus h squared and y minus k squared equals r squared.
So I'm trying to determine what's the center. In other words, what is h and k? on each one of these.
Let's see what we've got. Now follow with me here. x minus h.
Well the h has to be a 1. Now be real careful here. This is reading y minus a negative 4. So come come back over there. That's y minus k, k is negative 4. Now that's going to take a little bit of getting used to. But a lot of times when I'm working with students, they just say it's the opposite of what you think. So this looks like a negative 1. That means this is a 1. This looks like a 4. This is the opposite.
And you can do that over and over and over and over and over. Okay? So let's try it on this second one here.
The center is going to be at, and you would say negative 2, 7. Let's try this one. The center is going to be at negative 5, 0, because there's an understood and it's 0 squared. Just like, I'm just going to repeat it, x minus a negative 5 squared. So what do we say?
Negative 5, 0. And then this last one, the center, which is at hk, so get practice. Writing center is at hk. This one's going to be at negative 3, 2. Hopefully you're getting that really well. Now the next thing is, what's the r value? So the radius, which is r, is going to be, and you have to look at the formula here, it's going to be the square root of this number.
So square root of 25, 5. The radius for this second example, square root of 9, 3. Radius for this next example. Square root of 16, 4. Radius for this last one, square root of 7. Yeah, just square root of 7. Now if I were trying to draw it and trying to put it out there, you wanted me to go to this point, negative 3, 2, and then count over, then I would have to grab my calculator and this would be a... approximately, we don't need it for this exercise here, but it'd be approximately about 2.6. There we go, 2.6.
You say, how would you draw one of these? Well, why don't we pick on this bottom one here? It won't hurt anything. So let's look at it. So first thing I want to do is clean off some space.
it would really help if you had a decent piece of graph paper. But I'm just going to make it work just to make sure you've got the concept of how do you graph these. Well, so we're looking at this fourth example here.
So we've got a center at negative 3, 2, 3, 2. So there's my center point. Now what I want to do is count out square root of 7, which is approximately 2.6. So bear with me here. I'm going to go.
one, two, and just a little bit more. Somewhere right in there, just make myself a little dot. And I just do this north, south, east, and west.
So now I come over one, two, and just a little bit more. So it's gonna be, ah, this pencil wants to do something different, but I'm just gonna make myself a little dot there, and then I come down one, two and just a little bit, not quite three, about two and a half, close to it. And then come over the other direction, one, two, and just a little bit more, 2.6.
Alright, and then do the best job you can to draw a decent, oh yeah. I know that this one is perfect, but anyway, that's good enough. And that's how you draw it. All right.
Now, we want to take this one step further. See, I want you to think with me. What if we were to take one of these and foil it all out?
You're like, what? Foil all that stuff out? Why would you want to do that? Well, sometimes it comes. all foiled out, and you've got to put it back into this form right here.
And that's where we want to go next. So watch with me as we go to the next example. I've got two just for practice here.
x squared y squared 12x minus 6y minus 4 is 0. Now the first thing you want to do, just copy that thing down in your notes. and when I copy it down, let's clump the x's together, alright, so that takes care of that term and that term. Let's skip a little bit of space, just enough to put another term in, and let's put the y's together. There's the y squared, and I need a minus six y. Skip a little space for another term, put your equals, and then this term right here, We're gonna move it to the other side.
So that's gonna make it a positive four over here. Got that? Okay, so my next step is to complete the square. Let me write a little note here. Complete the square.
All right, so how do we do that? Well, we take half of the 12. and square it. So, let me draw a brain over here.
This is your brain and you're saying, let's take half of the 12, the way that's six, take the six and square it. So, 36 is my magic number. So I'm gonna put a 36 right here. Now what did I just add to the left hand side of this equation? You say 36. Well I gotta keep it balanced.
So I've got to put 36 over here. Once you do to one side, you've got to do the other side. Perfect. So let's pick another number here.
Let's pick on this one. So what is half? You don't have to worry about the negative.
The negative will take care of itself. What's half of 6? 3 squared 9. If you need a brain on that one, half of the six is three.
Then you square it. You get the nine. So you're going to put nine on the left. You're going to put nine. Very nice.
Now, we want to look at these three terms, the clump of x plus that constant we added. And I want to factor it. Now.
Factoring depends on how well you can factor, but that should be x plus 6 squared. You say, how do you get that so fast? Well, one thing, you practice factoring. The other is you write down the letter, copy down the sign, and what was half of that 12? Right there.
Let's do it again. This is going to be y minus 3. Copy down the letter, copy down the sign, whatever half of that was. That's part of completing the square problem. process. On the right, you just get whatever you get.
That's 40, 49. So now we have it cleaned up. So what's the center? You'd say negative 6, 3, and the radius, 7. It's a square root of that.
Good. And if you wanted to, we could graph it. So that's what we wanted to figure out on that one. Let's do this last one here on the screen for practice.
Okay, so a little bit shorter equation here. So the rule was you copy down the x's, and that's all the x's we had. Leave a little space.
Not really, you have to. But then y's, we got y squared and y. Practicing, just leave a little space and move the half over.
So I want you to understand there's nothing to do with the x. If you want to, you could write it as x minus 0 squared, but you don't have to do that. Let's focus on this, completing the square over here.
Now understand that that's a 1 right there. So my brain is saying I'm going to take half of the 1, which is just a half, and we're going to square it. Now don't make this hard here.
When you square it, you square the top. 1 squared is 1. 2 squared is 4. So there it is. So you're going to add a fourth here.
So you need to add a fourth to the other side. So now in terms of factoring, you've got copy the letter down, copy the sign down, and what was half of that? One half. Forget that little square, part of the formula.
What is a half plus a fourth? In other words, two-fourths, two-fourths same as a half, two-fourths and one-fourths makes three-fourths. And there's my formula. So you say, yeah, let's do the center. Do you have to write the minus zero?
No. Again, if you want to leave it that way, or if you'd rather just call it x squared, it doesn't matter. The center is at hk.
So that was a zero. Remember, it's the opposite of what you think here. The radius is... Take the square root of this thing. Oh my, it's a fraction.
Don't worry about that. Square root of the top and square root of the bottom. That's how you go back and find the equation. This is the key piece.
Once you've got that, the other you can figure out. All right, I have one last example in this lesson. We're looking for the equation of a circle with... endpoints of the diameter, diameter, we haven't been talking about diameter yet, diameter at negative 1, 3, and 3, 11. So let's kind of visualize what's happening here.
Let me grab my graph paper. So we don't have to be perfect with this. We're just trying to get a concept in our head. So negative 1, 3 is going to be, oh, I don't know. I'm just kind of...
saying somewhere right in there. Let me just mark it negative one three and three eleven. Let's see.
So one, two, three, way up here. I don't know. It's somewhere right in there.
So we'll mark that, call that three eleven. So I need a circle that goes through there. Now the center, now think about this with me, this is a key point. the center of my circle.
Hang on a second. If this is the diameter and that's the midpoint, I just said it, that's the midpoint. So the center is at the midpoint.
Well, let's go find the midpoint. So how do you do that? You add them up. So take the negative 1 and the 3. So take the x values, add them up, divide by 2. Take the y values, 3 and 11, add them up, divide by 2. What does that give me?
3 minus 1 is 2. 2 divided by 2, 14 divided by 2, 7. There's my center. Now remember... Remember the letters we were using?
That was H and K, just so we can practice. Okay, well anyway, where were we back over here? We were in the process of drawing this thing.
This was the diameter. So now we need to come over and picture with me. There's my circle, and I've got my H and my K.
All I need is to know... how far that radius is. Well, the radius, let me make a note here, the radius is half of the diameter. Oh, but I don't know how long it is, do I?
But I can go find it. So the diameter, this whole length across here, would be the distance formula. Let's call it, well, this is perfect, d for diameter. The distance across there is, now remember that's x2 minus x1, so subtract my x's. That's 3 minus a negative 1 squared, and then y2, so 11, I'm just calling this y2, 11 minus 3. squared.
Let's see what that is. 3 plus 1, that's 4. I'm going to square that. 11 minus 3 is 8. We're going to square that. All right, let's see.
That'd be a 16 and a 64. Let's see. That makes 70, 80. So square root of 80. That's the distance across that whole thing. Now, can we simplify that?
We really do need to break that thing down a little bit. That's going to equal, what is it? I see a 4 goes in there, 4 times 20. Or you could say 16 times 5. So that would be 4 square roots of 5. So this distance across here is four square roots of five.
So the radius would be half of that. Well, that would be two square roots of five. But my formula, because remember, here comes my final answer. Where am I gonna put that?
How about up here? My formula says x minus h squared plus y minus k squared equals r squared. So my formula is going to require an r squared. So square that thing.
2 squared is 4. Square root of 5 squared is 5. 5 times 4 makes 20. Okay? Now come over and write your answer. X minus the H, that'd be one, squared. And Y minus the K in there. And put your R squared.
And there is the puzzle. Wow, that connected. Distance formula, midpoint, circles, all together.
Fantastic work. Very good. Now, some people ask me, The radius is half the diameter.
And so I went and found the diameter. Other people have said, look, the radius is just the distance between this point, this point right here we found as 1.7 and 3.11. Hey, that'll work too.
Just go find that distance instead of taking this one and cutting it in half. It doesn't matter. It'll get you the same answer. All right.
congratulations that was good lesson for you