 What we think about as coordinate geometry has two sources. Fermat, who said that any equation in two variables gives a curve, and Descartes, any geometric relationship, can be expressed as an equation. Now, what we usually call coordinate geometry is closer to Fermat's ideas than to Descartes, but several important curves have equations from Descartes' idea. Suppose you have a specific point and a line. Wherever you are, you can talk about your distance to the point, which we'll call the focus, and your distance to the line, which we'll call the directrix. Now, if you walk so the ratio of the distances to the focus and the line is a constant, you describe what's called a conic section. So let's give it a name. The ratio of distances from the point to the focus and to the line is the eccentricity, and we define the conic sections by their eccentricity. So let X be a point on a conic section with focus F and directrix PQ. Let OX be the distance to the line and XF the distance to the focus. The eccentricity is XF divided by OX, the distance to the focus, divided by the distance to the line. If E is less than 1, the conic section is an ellipse. If E is equal to 1, the conic section is a parabola. And if E is greater than 1, the conic section is a hyperbola. So, for example, a conic section has a focus at 5, 3, a directrix of Y equals negative 2, and a point at 1, 6. What type of conic is it? Since the type of conic is determined by the eccentricity, we should find the eccentricity. And since the definition of eccentricity refers to distances, which is a geometric idea, we should draw a picture. So if we graph our focus, directrix, and our point, we're interested in the distance between the point and the focus and the point and the line. Now, the distance to a line is always measured perpendicular to that line. So we'll drop the perpendicular. And because we've drawn the picture, we can see that the distance to the line is going to be 8. Meanwhile, the distance to the focus, for that, we can use our distance formula. And so we find that the distance to the focus will be, we'll find the eccentricity, that's the ratio of the distances, distance to focus, divided by distance to line. And that's going to be 5 eighths. And since our eccentricity is less than 1, the conic section is an ellipse.