 With some effort, we can show that the equation of an ellipse with a horizontal director can be expressed as x squared over a squared plus y squared over b squared equal to 1. And again, with some effort, suppose that we have the equation of an ellipse with a horizontal director, the eccentricity is given by, and what this means is that we can compute the eccentricity of an ellipse directly from its equation. So, for example, in this equation, so a squared is the denominator of the fraction with x squared as numerator, so a squared is 16, b squared is the denominator of the fraction with y squared as numerator, so b squared equals 25, and our eccentricity, 1 minus a squared divided by b squared, or 3 fifths. Now, sometimes we may have to do a little bit of work before we can get our equation into a useful form, so this equation is not in the form our theorem uses, but if we apply a vertical and horizontal translation, it will be, and a translation won't affect the eccentricity. So if we shift this ellipse five units to the left and eight units downward, we get the graph of, and again, comparing our equation to that in our theorem, we have a squared equals 10, b squared equals 15, and so our eccentricity will be, or we could take something like this, a squared equals 12, b squared equals 9, so our eccentricity will be, um, there's a problem here. And so let's check the fine print of our theorem. The supposition here is that it's an ellipse with a horizontal directrix, and so our theorem for eccentricity requires the directrix be horizontal, but remember the directrix could be vertical, or even oblique, but we won't talk about that yet, because that makes things much more complicated. Well, if our directrix is vertical, what can we do? So remember that if we swap the x and y coordinates, we reflect the graph along the line y equals x, and this turns horizontal lines vertical and vertical lines horizontal. So if the directrix is vertical, swapping x and y will make it horizontal. So back to this ellipse. This ellipse appears to have a vertical directrix, so we'll reflect across the line y equals x to obtain the graph of, and comparing, we see that a squared is equal to 9, b squared equals 12, and so our ellipse will have eccentricity, and notice that the effect of the reflection across y equals x is to switch a and b. Now while you should think about graph transformations, you could incorporate them into a single formula, namely that the ellipse with equation x squared over a squared plus y squared over b squared has eccentricity e equals square root 1 minus k, where k is either a squared divided by b squared or b squared divided by a squared, whichever is necessary to make the eccentricity a real number. And so we can find the eccentricity of the ellipse, and by a horizontal and vertical translation this becomes the ellipse with the equation, and one of these is a squared and the other one is b squared, and the only way we can divide them so our eccentricity is a real number is to divide 10 17s.