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Interactive video lesson plan for: Master how to write the equation of a circle given different pieces of information

Activity overview:

Subscribe! Welcome, ladies and gentlemen. So what I'd like to do is show you how to write the equation of a circle, given pieces of information. So you can see for each problem we're given different pieces of information, and we want to write the equation of the circle. So what I did is I kind of wrote in some basic general information. We don't really need the standard form of a quadratic up there anymore. But I have the equation of a circle as x minus h squared, plus y minus k squared, equals r squared. As well as the center is h, k and the radius is r.

Basically, when you're given a problem, the main important thing I want to do is label and identify what we are given. So in this case, remember the center, you can see, is h,k. So I'm just going to write h,k above that. So I can say that h is equal to 0 and k is equal to 0. We know that the radius-- radius is equal to r, but if you look at the equation. In the-- oh, their equation is equal to r. OK, so, sorry about that. I was thinking ahead. So we can say radius is just r.

So now, to write the equation, I'm just going to take the equation and then plug-in the values that we're given. So we're not given any x or y. So I'm going to have an x minus h, which is 0 squared plus y minus k, which is 0, squared equals r, which is given as 4, squared. And now we just go ahead and simplify and write our equation of our circle. X minus 0 is just x, squared is just x squared, plus y minus 0 is y. That's y squared equals 4 squared, which is equal to 16. And there you go. That's our equation of our circle.

In the next example, I'm going to do the same thing. I'm just going to write in h, k, and this is r. And then again, plug in the information into our formula for our circle. So I have x minus 3 squared plus y minus-- now a lot of times, students will get stuck with this one. So what I do is just put it in parentheses, negative 6. Squared equals the square root of 7-- I'll put that in parentheses again-- squared. And a lot of time it's actually even helpful, I didn't do this in this one, but it's very helpful.

Anytime that you're plugging in a value in replacement of another value or a variable, it's something very helpful to put in parentheses. Not only does that tell me as well as my teacher that, hey, I inserted this value for this variable, it also can help you prevent mathematical errors. For instance, like in these two problems. So I didn't do it, because 0 is pretty obvious, right? You don't really need to put parentheses around the 0. But you definitely could have put it around there, but 4 squared we know is 4 squared, right? But it really becomes helpful in these next two problems, where we could easily make a mathematical mistake if we weren't careful.

X minus 3, I don't need parentheses around that, that's pretty basic to understand. So that's just x minus 3 squared plus-- however, y minus a negative 6. Minus a negative is really addition. So that's why this turns to y plus 6 squared equals-- and then the square root of 7 squared, well, the square root and squaring our inverse operations. So that's just going to leave us with 7.

All right. In the next example, now I'm not given a radius. I'm given the center, which, I'm familiar with, is h,k. And I'm given a point, 6,4. Well, let's go back to the equation. We need to know h and k, right? To write that, as well as r. But we're not given r. The only other thing is we're given a point. Well, we know that a point, when graphing points, those can be rewritten as x and ys. So if I'm given x, h, y, k, could I plug-in that information to find r? Yes, of course, we could. So we're going to plug in all the information. So I have x, which is now 6, minus h, which is 2 squared. Oh, I forget about the red here. And that's going to equal y. Whoops, sorry. Add. Plus y, which is 4. So the y-coordinate is equal to 4 minus k, which is 1, squared equals r squared. So 6 minus 2 is going to be 4. Squared plus 4 minus 2 is 3 equals r squared. 4 squared is 16, plus 3 squared is 9 equals r squared. Therefore that's 25 equals r squared. That means-- take the square root of both sides-- r is equal to 25. So now I know what--oh, actually I didn't even need to do that-- I'm sorry r is equal to 5. OK. So now I know what the radius is, as well as the center. So what I can do, but I really actually don't even need to find r, I could've just plugged in r-squared, right? So I could have actually not solved that, and just left it as r squared equals 25. But now I know what the center is. Let's go ahead and plug that in. So it's going to be x minus 2 squared plus y minus 1 squared equals r squared, which was 25. And there we go

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