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College Algebra: Finding Equations for Ellipses


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About this Lesson

  • Type: Video Tutorial
  • Length: 4:41
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 50 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 Review (30 lessons, $59.40)
Trigonometry: Conic Sections (12 lessons, $26.73)
Trigonometry: Ellipses (3 lessons, $4.95)

An ellipse is a collection of points whose combined distance from two fixed points (both called a focus) are the same. This lesson will show you how to find the equation of an ellipse if you know some information but not all of it. The equation for this type of conic section is typically written as: x^2/a^2 + y^2/b^2 = 1. We start with a situation where you know the x-intercepts and the foci (both focuses) but are looking for the equation of the ellipse. To approach this problem, we'll use triangle formulas and the Pythagorean Theorem to find the y-intercept.

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 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

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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 or visit Thinkwell's Video Lesson Store at

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I want to buy trignometry conic section,lessons not this one I spend 29.00$ and waste my time!Please be honest with billing.

Let’s take a look at how you’d actually find equations for an ellipse if you just know some information but not everything. So, here we go. Looking at ellipses, some more. So, suppose, for example, that we have the following scenario. I tell or someone tells me that there’s an ellipse that has x intercept, crosses the x-axis, at plus or minus five. And the foci are at (-3,0) and (3,0). The question is, “What is the equation for this ellipse?”
We could try to draw it. If we try to draw it, what would it look like? Well, let’s see. I’ll try to sort of draw it over here. We know that we have five here and we have minus five here. We know at three and minus three we have these foci. Now, the thing is we have to find is we have to find these things. This gives me the a’s in the formula, x²/a². So, I’m all set there. But the question is, “What is this thing here?” Where does this thing cross the y-axis? Well, we’ve got to think about it for a second. Remember, what’s happening is we have this string and the sum of those pieces are always the same. So, let’s think about that. What would it be way over here?
Well, way over here I would see the sum of a whole bunch of things. The string would go from here all the way out to here and then back to here. So, let’s see how long that string would be. That string would be all the way out to here. From here all the way out to here and back. Remember, I start the foci and it would go all the way out to here and then back. So, let’s see what that length would be. Well, that length would be what? Well, this is three and this is three and this is two. So, if I start here and go all the way out to here that’s now three and three is six. Then I go again another two. So, that would be eight. Then I’d come back and I’d see ten.
Let me say that again. Remember that I’m thinking about the points of an ellipse as points where the sum of these lengths is always the same. So, this plus that will always be the same. But what happens when I actually look at this point right here? Then I see the string goes all the way to there and then comes back. It sort of loops and then comes back around like this. So, what’s that length? That length would be three, three, two and then another two to come back. So, it’d go and you can see that really visually right here, right like this. Do you see how the string goes from here. That’s three, another three is six. Then it goes two more, but then it comes back track itself another two more. So, the total of the string is ten.
So, this thing has length ten. If the string will always have length ten, I want to find out where it crosses right here. So, what had happened here? Well, here the strings would have to be the exact same length, by symmetry. It’d be right in between. If the whole thing was ten, this must be a five. Now, I’ve got to find this height. So, you can either use Pythagorean theorem and say I’ve got a right triangle. This is three. This is five. I can solve for this. Or you might actually remember a particular triangle, the three, four, five right triangle. This would be four and check it. Three squared is nine. Four squared is 16 and 9 + 16 is 25, which equals 5². So, in fact, this must be four.
So, what I see is this must be b. So, b = 4 and a would equal 5. So, in the formula I would see x²/a² + y²/b² = 1. Well, a² is where it cuts the x-axis, in this case which is five. So, squared would be 25. So, I see x²/25 plus y² over and then the value for b. That height we already figured out was four. So 4² is 16, equals 1. If you’re just given the foci and the x intercepts, for example, you can actually figure out exactly what the b value is. Once you have the b value, using a little right triangle argument and thinking about how long that string has to be, by going up and back, then you can actually put into the formula and you see, there’s the formula for that particular ellipse. Up next we’ll take a look at an example, which actually involves satellite and the planet earth. Stay tuned.

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