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College Algebra: Finding Circle Center and Radius

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:43
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 104 MB
  • Posted: 06/27/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
College Algebra: Relations and Functions (57 lessons, $74.25)
Trigonometry: Algebra Prerequisites (60 lessons, $69.30)
College Algebra: Circles (4 lessons, $5.94)

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

Thinkwell
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Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Okay, look, let's face it. If someone gives you an equation that looks like this--look at that one over there. So what is it? x - 3^2 + y + 4^2 = 25, you can immediately rip off or whatever, the center and the radius of that circle. It's not a big deal. But what if someone took the equation for a circle, and with your back turned, actually squared out that first thing, squared out the second thing, and sort of scrambled the terms up, and then handed it to you? Well, if it's in that great form, fine, but if it's not in that great form, then you have something sort of scary.
For example, look at this. Suppose someone handed you x^2 + 6x + y^2 + 8y + 9 = 0. Now if someone handed that to you, what would you do? Well, you'd hand it back, of course. But what if someone handed it to you and said, "Okay, that's some circle, and I want you to find its center and its radius." Not as easy as if it was just given to you in that nice, factored form. So how would you do this? Well, the thing to do is to try to take this complicated-looking expression, and convert it into something of the form x minus a number squared, plus y minus a number squared equals a number, and then you can read off--h, k, and the r. So how do you do that? You complete the square--one of my favorite things.
Now, if you want to review completing the square, you might want to click somewhere around me and actually do the "complete the square" little tour. But anyway, let me just give you a quick recap of how the completing the square system works. The first thing I want to do is keep all the variables on the left, take any constants that may appear, and move them to the right. All variables to the left, constants stay to the right. So in this case I'd write the following: x^2 + 6x + y^2 + 8y = -9. Okay, now what I do is write the whole thing out again, but I'm going to put some spaces here. Remember what completing the square is all about. To complete a square it means I have a piece of a square, and now I'm going to add something else to make it a perfect square. So I have x^2 + 6x, and then I'm going to have a + _____, and then I'm going to write the rest of the stuff--y^2 + 8y + ____ = -9 + ____ + ____. Because if I add something on this side, I have to add to this side.
How does completing the square work? Well, what I do is I make sure, first of all, that my coefficient on the square terms are 1. For example, if there was like a 2 here, and a 2 here, I'd have to divide everything through by 2, because to complete the square, I want to start off with a coefficient of 1 in front of the squared terms. Happily, this is already just 1x^2 and this is already a 1y^2, so we're all set. Once we did that, I then take a look at the coefficient on the x term alone--in this case it's 6. I take half of it and square it. Half of 6 is 3, and if I square it, I get 9, and that's what I add here. So I add 9 here, and thus I have to add 9 here in order to make that thing and equation still true.
I do the same thing here. I look at 8, take half of it and get 4, and then square 4, and I get 16. And that's the term I add here. Now, if I add 16 to both sides, I don't change the truth of this equality; it's still equal. But now look what happens. This should factor to be a perfect square, and in fact, if you notice, it does. It's (x + 3)^2. You can check it if you actually foil that out. You've got to foil. x times x is x^2. Inside terms are 3x; outside terms are 3x, that's 6x, and notice that 3 times 3 is 9. So that checks. And similarly, here, we get (y + 4)^2, and that equals what? Well, -9 + 9 = 0, and then I have a 16. So this really complicated looking algebraic expression can be massaged by completing the square twice into this form. Well, now it looks just like a circle form, and we can read it off, because it's now of this nice form where you can see these numbers, so the center would equal what?
Well, you may think the center is at (3,4). If so, that's a great guess, but if so, then you're not quite remembering the formula for the circle, which is that it's going to be (x - h)^2 + (y - k)^2. So I have to write these things as minuses, which I do as saying (x - -3)^2 + (y - -4)^2. And then it's more apparent what the center is. The center is at (-3,-4), so in fact, what I should say here is a (-3,-4) and the radius, well, the radius equals--not 16. You may be tempted to say 16, but you have to take the square root of 16, because look at the formula over there--it's x - 8^2 + y - k^2 = r^2, and so in fact, I have to take the square root, and I get 4. So the radius if 4, the center is (-3,-4).
I think this is really cool, by the way, that you can take sort of garbage like this, and just by a little bit of completing the square, adding a couple of things in, all of a sudden decode exactly that it's a circle, where the center is, and where the radius is. I just think it's really cool. In fact, I think it's so cool I want you to have the thrill, the rush--it's sort of like, you know, doing any kind of rappelling. Have you ever rappelled? There's this great rush when you're sitting on the edge of a cliff, you know, and that's what you're doing here. You're sort of sitting at the end of the cliff of mathematics and rappelling.
So how about x^2 - 4x + y^2 + 12y = -4? Why don't you try right now to find out where the center and what the radius is for this circle, which is sort of all camouflaged, like this. Give it a try, and then we'll do it together.
Okay, well, we have to do some completing the square here. So I already have the constant to the right-hand side, which is good, so what I'm going to do now is write this out as x^2 - 4x + and then I'm going to leave a little space there, and then I've got y^2 + 12y + and I'm going to leave a little space, and then I have = -4 and then a + and then a +. Okay, what do I add in here? Well, I take a look at the -4 here. Notice that my coefficients are 1 in front of the squared terms. That's a thing to check. I take the -4, take half of it, which would be -2, and I square it, and that would give me 4. And that's what I add to both sides to complete the first square, taking the part of the square that's given, and adding the other part to make it a perfect square. And then here, I take 12, take half of it, which is 6, square it and get 36.
Once you get the hang of this it's not that painful. Okay, and so what do we see here? Well, this should factor beautifully into a perfect square. That's the whole point. x + 2^2 seems to work, and here I have a y + 6^2. You can check that squared out foil, and that equals -4 + 40, I just have 36. So the center is at -2, minus... Oh, oh, wait, wait, wait. There's a major typo here. Did you catch the major typo? I hope you did. This is wrong. See, this is actually a great thing for me to do. In fact, this was not an intentional mistake. But sometimes when you're completing the square you sort of feel so empowered, you're just moving along. If this were a quiz, sure I'd be moving, but I would have gotten it wrong. You know why? You have to be really careful. There's a negative sign here, so in fact, this should be negative. Right? Foil it out. I'll get a -2x and a -2x combined to give me a negative 4x. With a plus sign there, I need a plus sign here. So actually, this was wrong. This should be a minus sign, in which case, if it's a minus there, this would be a plus here. So the center is at (2,-6), not (-2,-6). Great mistake. In fact, I proved that by making it live for you. And the radius is equal to the square root of 36, which is 6.
Again, you want to go at a good clip, but you don't want to go so fast that you make sloppy mistakes as I just did. But I caught by myself by realizing there's a negative sign there. I never saw that sort of pop out here, and so I caught it. Try some of these on your own, and please, don't make the mistakes that I make here. Have fun.
Relations and Functions
Circles
Finding the Center and Radius of a Circle Page [2 of 2]

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