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College Algebra: Decoding the Circle Formula

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About this Lesson

  • Type: Video Tutorial
  • Length: 4:32
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 49 MB
  • Posted: 11/18/2008

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 Review (30 lessons, $59.40)
College Algebra: Circles (4 lessons, $5.94)

In this lesson, you will learn how to find the (x, y) coordinates for the center of a circle graphed on the Cartesian coordinate system as well as the the radius of the circle from a formula or expression that doesn’t easily lend itself to the standard circle formula (e.g. r^2 = (x-h)^2+(y-k)^2)). To do this, you'll generally use a math technique called, Completing the Square.

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 at http://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
Thinkwell
2174 lessons
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11/13/2008

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, well suppose someone gives us a formula and we have to figure out what it looks like? If the formula happens to look something like this--(x - 4)2+ (y + 6)2 = 49. Actually, I look at that and I say, “You know, that sort of has a circle-y kind of feel to it,” and what gives it that circle-y kind of feel is well, I see stuff with x’s and it’s squared, plus stuff with y’s in it squared, and it equals something that’s positive. That sort of conjures up the thinking of the formula for a circle, which I remind you, looks like this: (x - h)2 + (y - k)2 = r2, where the center of the circle is located at h, k and the radius is r. So I see that this formula that someone gave me actually looks very much like this general one, so in fact, this must be a circle, so we have a circle, and where is it’s center? The center would be h, k, so I have to read that off of here. Now, we have to be careful, because notice it’s x - h and y - k. So whatever the sign is here, I’ve got to flip it to actually find the appropriate value for h. So, for example, h here is not -4, but it’s just 4.
What about here? Is the y part of the center going to be 6? No, because I have to write this as -k. This is sort of tricky. So to write 6 + 6 as -k I have to write it as - (-6). Let me write that down so you can see it. What I’ve got to do here is write that out as (y - (-6)2. You see, I have to write it in exactly this form. Y, y, minus, minus, number, number. And to make it equal to y + 6, I need a double minus here. So in fact, the y coordinate of the center of the circle is not going to be 6, but -6, a really important fact and a great place where people make mistakes. Does everyone see that? So I’ve got to make sure that I’ve got a minus minus to make that a plus. So the center is at 4 - 6, and the radius--well, the radius is not 49, it’s the square root of 49, which is 7. You may say, “Wait a minute. Don’t I have to look at plus or minus 7?” No, because radius represents a distance, and we always know distance is taken to be positive, so in fact, it’s just 7.
So, in fact, just knowing that we can sketch a really accurate picture--a pretty accurate picture--of what this circle would look like. Let’s see if we can try that. The center would be at 4--1, 2, 3, 4--and then I go down 6--1, 2, 3, 4, 5, 6. So that’s the center of the circle, and the radius is 7. If I know the radius is 7, I can sort of see where it sort of shoots up in various directions. So let’s see, I have to go over 7 units this way--1, 2, 3, 4, 5, 6, 7--so it’s going to pass through there somehow--1, 2, 3, 4, 5, 6, 7--and I pass through here--1, 2, 3, 4, 5, 6, 7--and I pass through here, and then 1, 2, 3, 4, 5, 6, 7. So you can actually sort of roughly sketch--it’s sort of rough--but at least you know some points that are on it. And you can see where it crosses the axes, and you get a rough sense of how the circle looks. Very accurate picture just given the original formula, which I have written over there. You can see x - 42 + y + 62 = 49. Just given that, I now have a sense of how this thing sits in space, where its center is, and how big it is, just by the formula. And so it’s important to realize that just looking at the formula you can capture and decode all the information, in particular, the radius and the center.

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