Subscribe! http://www.freemathvideos.com Welcome, ladies and gentlemen. So what I'd like to do is show you how to write the equation of an ellipse given some pieces of information. So really, the main important thing about writing the equation of the ellipse is understanding our two equations. We have one equation for ellipse when the major axis is horizontal, and we have another equation for ellipse when our major axis is vertical. And really, the only difference between the two equations is a squared is under the x when the major axis is horizontal and a squared is under the y when the major axis is a vertical.
Now, basically what we need to do, then, is determine which equation we're going to use for each and every problem. And to do that, I think it's our role to identify the information we have, as well as plot the information so we can find the major axis. And once we find the major axis, we can determine is it horizontal or vertical.
One last thing I forgot to mention-- all of these problems have a center at 0, 0. So that's a good starting point for us to know, is that all of them are going to be starting at 0, 0. So the first thing we're going to do is plot the information we know.
Now, plus or minus 4, 0 can be rewritten as 4, 0, negative 4, 0. So I go over 4-- 1, 2, 3, 4-- and negative 4-- 1, 2, 3, 4. Those are my two vertices, and there's my center. So just by plotting the first piece of information I have, I have two vertices. And what that should tell us, by looking at an ellipse, we know that the major axis is horizontal. Because the vertices are the endpoints of the major axis.
So if those are the endpoints of the major axis, we know the major axis has to be horizontal. So therefore, I'm going to use this equation, because this is the equation of a horizontal major axis for an ellipse. The next thing we want to do is plot the other information, which we have co-vertices, which is going to be plus or minus 2. So there's your co-vertices.
And now what we need to do is just figure out. So if you look at the equation, we have h and k represents our center, which we know is at 0, 0 at the origin, so we've got that taken care of. All we need to figure out now is what is a and b. Remember, a is the distance from the vertice from the center. So how far am I traveling from the vertice to the center? 1, 2, 3, 4.
Now again, you could say here, oh, you're traveling negative 4? Yes, but it's the absolute distance. It's the length that we're traveling, not looking at direction. Co-vertice, or b, represents a distance from the co-vertices to the center. So you look over here, and you can say that that is 2. So all I'm simply going to do is plug in the information that I'm given into my equation.
Now remember, again, the major point, though. The major thing about this is my major axis is horizontal, so I have to use this equation. So I'm going to use x minus 0, because that's the x-coordinate, over a squared, which is 4, plus y minus 0 squared over b squared, which is 2.
And then 2 squared, and that's equal to 1. So now I just go ahead and simplify this. X minus 0 squared is just going to be x squared over 16, and that's going to be plus y squared over 4 equals 1. We have c, but we don't need to figure out c. It's not part of the equation of an ellipse. But there we go. All done.
Let's go ahead and look over to this one. Now this one, we have vertices and co-vertices, but then we have some radicals, which doesn't really look too much fun. But again, the major important thing is, again, just plotting the information. So we know we have a center at 0, 0.
Now notice I'm doing 0 plus or minus 2. Even forget about the radical. If I was just doing 0 plus or minus 2, that means I'm going down up 2 and down 2. Now squared of 2, radical 2, that's going to be square root of 8, I believe.
So square root of 8 is between 2 and 3. So that's going to be up there. But anyways, what I want you to understand is we're going up or down. So actually, even forget about plotting it. I just know 1, 2. I know the vertice is somewhere there and somewhere here.
If you even want to plug it into your calculator and figure out what exactly it is-- I think it's 2.8 and 2.84271, maybe. But just know that it's vertical. Now that I know that my two vertices are vertical, I know the major axis is vertical.
And what's so important about that is rather than using this equation, I'm going to use this equation. Rather than putting the a squared under the x, I'm going to put the a squared under the y. Now what exactly is a squared? We'll get to that.
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