Writing the equation of an ellipse from graph

0:00- [Instructor] Over here, we have an ellipse graphed, and what we wanna do is come up with the equation of this ellipse, but we're going to do it in steps. The first thing we want to do is figure out the center of this ellipse, and if you are so inspired, think about where that center is.

0:14All right, so if we think about the x coordinate of the center, we can see that the ellipse goes from x equals negative four to x equals eight, and so the x coordinate of the center should be right in between. So, from negative four to eight is a total of 12, so if we go six from either side, half of 12, then that will be the x coordinate of the center.

0:35So, six to the right of negative four would put us at two, and two is also six to the left of eight, so that would be our x coordinate of the center. And then the y coordinate of the center, we see that our y values for this ellipse go from one to seven, so it goes a total of six, or our change in y is six from both of these extremes.

0:57So, if we were to go three from the bottom or three from the top, that would be the center from a y perspective. So, one, two, three, right over here at y equals four. So, the center here is x equals 2 and y equals 4. Now, the next thing we wanna figure out, and we've already actually talked about this a little bit, is what is the minor radius or sometimes called the semi-minor axis?

1:25Well, these are very fancy words. We know if you think about a circle, you only have one radius, but an ellipse, you can think about, where does the radius hit a minimum, and where does the radius hit a maximum? And when we're talking about the minor radius, we're talking about the minimum radius, right over here.

1:44So, from the center, to say this point over here, we already figured out that that is 3. We could have also gone from the center to that point there. And then some people will also call that the length of the semi-minor axis. Now, we can do the same for the major radius.

2:00That is the longest radius that we have in this ellipse. Let me do that in blue. So, that would be from the center all the way out here. And we already figured out that that length is 6. So, now we have all of the information we need for the equation of this ellipse.

2:16When we express it, we could write, let me write it this way. First we think about x. So, x minus the coordinate of the x center. So, x minus 2 squared over the radius that is in the direction of the x. In this case, we have the major radius that is in, I guess you could say the x direction.

2:40So, that is over that squared, so that's where that 36 comes from. And then, plus, if we think about the y values, it's gonna be y, or the, I guess y part of the equation, y minus the y coordinate of the center squared, and then all of that over, in this case, the length of our semi-minor axis or our minor radius squared.

3:09So, that's 9 is equal to 1. And in other videos, we have actually proved where this comes from. But this is just a way for you to recognize, if you know the center and you know the minor and major radii, then you can figure out the equation of the ellipse.