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About this Lesson
 Type: Video Tutorial
 Length: 8:49
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 95 MB
 Posted: 06/26/2009
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 UTAustin 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, padic 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
 2174 lessons
 Joined:
11/14/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" 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 technologybased textbooks. For more information about Thinkwell, please visit www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.
Thinkwell lessons feature a starstudded cast of outstanding university professors: Edward Burger (PreAlgebra through...
More..Recent Reviews
The circle is the most attractive, beautiful, symmetric of all the shapes. There is nothing more enticing, and the reason is because of its wonderful symmetry. It's made by fixing a particular point in space, called the center, and then finding all the points whose distance from that center are the same. Look at that. So it's fantastic. This distance, of course, is called the radius of the circle, and all these points have the property that their distance away from that very center, that epicenter, is in fact, the sameit's that radius.
So, for example, suppose I call the center right herewell, I don't know, it has coordinates. This is some sort of axis here. I'm going to draw them in invisible ink. Watch this. Okay, so there are the axes. So this has some coordinates. Let's say those coordinates are h, k. So those are the coordinates of the center of circle. Now, what does it mean for a point out here to actually be on that circle? It means that that pointlet me just call that point since I don't know what it is, and I want to think of it as a general point, just x, y. What property must x, y possess? Well, it must possess the only property that's deemed of it in order to be a member of the circle, and that is that it's distance from here to here is that fixed distance, that fixed radius. Let's call that r. So let's let r equal the radius.
Okay, well then can we find a fact out about this x and y just knowing that its distance away from this particular fixed center is equal to the radius? Well, the answer is yes. It's the distance formula. So if I connect these two things I know that length is r, but I could also compute the distance between those two things using the distance formula, or another way of thinking about it, using the Pythagorean theorem. And if I use the distance formula, what would I see? Well, what I would see is the following. I would see that the distance between these two points would be the following. The square root ofand now I take the differences here, so I'll take x, subtract off h, so that point on the circle, x part, would be x part of the center, so that would be x  h and I square thatthat's what the distance formula says to doand then I add to that the difference in the y values, so that would be (y  k)^2, and I take the square root of that whole thing, and that equals... Well, what is that length? We know it's the radius, so I call that r.
Well, that square root looks a little bit threatening, so if I square both sidesif I square this side, the radical is going to lift. So if I square both sides, then this whole use radical will actually magically lift, but then I will have an r^2 here. So what I see is the following fact. What I would see is the fact that (x^  h)^2 + (y  k)^2 would actually equal r^2. That's what I would conclude. And you know what? That's precisely the condition, if you think about it that would guarantee that you're on the circle. Every point on the circle must satisfy that, because every point's distance away from the center is r. Conversely, if you were a point whose distance away from the center is r, then you must be on the circle. So, in fact, this is an equation for the circle. It's the circle that's centerednow, where is it centered? It's not centered at h, k, it's centered at h, k and its radius is r. So if we want to figure out the equation for a circle whose radius is r, that's centered at h, k, then what I do is I see that it must be (x  h)^2 = (y  k)^2 = r^2.
Okay, let's take a look at an actual example and see this in action. For example, let's find the equation of the circle that has radius 5 and centered at the point (2,3). Well, let's think about how that would go. So the formula for a circle is (x  h)^2 = (y  k)^2 = r^2. Something you should memorize and not understand? Absolutely not. It's just the distance. All the points, x, y, whose distance away from the point h, k is r. I just took squares of both sides to clean off that square root that would be there for the distance formula. So now I just insert all the data. You see? I know the center is 2, 3. That means that this is the value for h, this is the value for k, and this is the value for r. So now if I just insert all that I'll actually find the equation for the circle. It would be x  2, because that's the value for h, and I subtract it, squared, plus y  k, which in this case is 3^2 equals the radius, which is 5^2. And so if I simplify that a little bit I can see (x + 2)^2 + (y  3)^2 = 25. And when I see this, by the way, that tells me that I must have a circle, and I can figure out its center and I can figure out its radius.
So as long as you know the center of a circle and its radius, you can always write down a formula for it. And what does it mean to be a formula for a circle? What does it mean to be an equation for a circle? It means that anytime you find a point x, y that satisfies this equation, that point will be on the circle, and conversely, any point on this circle must satisfy this when you take its x value and y value and plug it in. That's what it means to be an equation for a circle. It's all the points that can be found, but now, algebraically, for a geometric object. And so if you think about it, what does it mean? It means if I go to the point (2,3)so there's the center, and the radius is 5, so how would that look? That circle has the equation that's over there; you can see it, and therefore every point on that circle satisfies that.
Okay, now there's a special case. Suppose that center was actually right at the origin? Well, that means if the center is at (0,0) then the formula that I have written over there is actually real easy, because then h is 0 and k is 0 and so then the formula becomes what? It becomes really easy. It just becomes x^2 + y^2 = r^2. And so there's a very special case, in fact, when the circle is centered at the origin and goes around by a certain radius, the equation of it is very simple. It's just x^2 + y^2 = r^2.
Okay, well next up we'll take a look at some more circle facts and see how we can just look at the formula for a circle and figure out what its radius is and what its center is. I'll see you there.
Relations and Functions
Circles
Finding the CenterRadius Form of the Equation of a Circle Page [1 of 2]
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