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Interactive video lesson plan for: Master how to write the equation of an ellipse when the center is not at the origin

Activity overview:

Subscribe! http://www.freemathvideos.com Welcome to the jungle. So what I'd like to do is show you how to write the equation of an ellipse when the center is not at the origin. So in the previous video, we worked on problems that only had the center of the ellipse at the origin. Well, you can see in this case, they give us a center. It's at negative 2-- or negative 1, 2, so obviously it's not at the center. And in this one, we don't really know, but we're going to plot the information and determine that it's not at the center.

So again, when writing the equation of an ellipse, when you're given some piece of information, the most important thing that I think you need to do is plot the information that is given. So first of all, we see that the center is at negative 2, 1. So what I'm going to do is plot negative 2-- or negative 1, positive 2. I don't know why I keep on resaying that. And I'm going to label that as the center.

Now to write the equation of the ellipse, we need to identify a squared and b squared, where, remember, a is the distance from the vertices to the center and b is the distance from the co-vertices from the center. Well, given our information, we don't really know what that length is. They don't say, your vertices are this, your co-vertices are this.

But they do tell us the length of our major axis as well as the length of our minor axis, as well as tell us if our major axis is vertical or if and our minor axis is horizontal. And that's very, very important, because we have two equations for an ellipse. This equation is for when we have a major axis that is horizontal. This equation is for when we have a major axis that's vertical.

Well, obviously, it says "major axis"-- I guess I didn't write "major axis," but-- oh, "major vertical axis." So it's a major vertical axis. Therefore we're going to use this formula.

And again, remember that really, the difference between these-- remember, a represents the distance from the center to your vertices, and b represents the distance from your center to your co-vertices. Well, your a is always going to be larger-- the distance to your vertices is always going to be larger than the distance to your co-vertices, because the major axis is always larger than the minor axis. So when a squared's under x, you have a horizontal major axis. When a squared's under y, you have a vertical major axis. So therefore we know we're going to use the [INAUDIBLE] formula. So I'm just going to write it down so I don't forget.

OK. So now we need to look into our next piece of information. We know h and k. We know that's negative 1 and 2, so I'm just going to label that. So that's h, and that's k. The next thing is, we need to figure out what is a.

Now again, I'm just going to draw a nice little-- oh, it has a vertical major axis, right? So it's going to look something like that. OK? Now from here, we know that the distance from-- we know we're going to have a major axis that's basically vertical, right? And the distance from our major axis from our vertice to our center is a.

Well, we can go in the positive direction as well as the negative direction. And what we notice is, then, the length of the whole major axis from vertice to vertice is not a. a is the distance from the vertice to the center. So if you're going from two different vertices back to the center, the length of your major axis is going to be 2a. So therefore what we can say is, 2a is equal to 10.

Then we look at the minor axis, which is perpendicular, going through the center of your major axis. And we have our two co-vertices. There's a vertice and there's a vertice.

Now remember, the distance from the center to your co-vertices or co-vertices to your center is b. Well, again, we can go in the positive direction or we can go in the negative direction. And again, what we notice is the length from co-vertice to co-vertice is not b. It's going to be 2b, right? Co-vertice to co-vertice is b plus b, which is 2b. So therefore I can also say that 2b is equal to the length of the minor axis, which is 4.

Now to find the value of b and a, I just need to solve. So I divide by 2, divide by 2-- b equals 2. Divide by 2, divide by 2-- a equals 5.

So now I know b, I know a, and I know h and k. That's all the information I need to write the equation of an ellipse. So I'm just going to plug them into the equation that I determined was correct, because this is the equation for a vertical major axis, which was given to me in the problem.

All right. So I do x minus h, which is a negative 1, squared, all over b squared, which is 2, plus y minus k, which is 2. And I'm putting them in parentheses just to remind myself that I'm inserting these values in for the variables-- in for a squared. And of course, you don't always need to put them in parentheses, but again, to me, it really, really helps to avoid mistakes.

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