>> Here's a hyperbola. Notice it's just two curves that are sort of facing one another or they look like reflections of one another. Well, the hyperbola can be described like the other conic sections in terms of distance. Now recall that with the ellipse, we were talking about the collection of points on the ellipse that were--all of them had the same sum of distances to two focus points. Well here, we have a couple of focus points as well, but this time we're talking about a difference in distances that give rise to these two curves. For example, a point in this position. Measure the distance to each of the focus point, these two distances. Now we wanna take the difference in the two distances and geometrically, we can show that difference like this. Just take the short distance and take it away from the long distance. Now we have the difference in the distances. Now, let's put a number on that distance. Let's just say for conversation, it's a distance of 10. Well, if we take another point on the hyperbola, and measure the distance to the two focus points, and take the difference, that difference will still be 10. And another point, measure the distance to the two focus points, take the difference, the difference is, you guessed it, 10. Points on the other curve have the same characteristic. Measure the distance to each of the two focus points, and that difference is 10. And the collection of all such points, all points in the plane whose difference in distance to those two focus points is 10, compose the hyperbola. Let's talk about terminology. We know that the points we identified here are called focus points, and we know that the plural of focus is foci. These two curves are called branches of the hyperbola and a line that contains the focus points will intersect the branches at two points called vertex points. We know the plural of vertex is vertices. Now a point right in between the two vertex points is the center of the hyperbola. Notice that the center and the vertex points and the focus points, all important aspects of the hyperbola, lie along the same horizontal line here. Now the portion of that line which connects the two vertices, that is this segment between vertices, is called the transverse axis. Sometimes, a transverse axis refers to the entire line. But we'll be using this segment as the transverse axis. Well, to understand aspects of the graph, let's consider that the center of the hyperbola is located at the origin of a coordinate plane. The equation that describes the hyperbola is this equation. Notice how closely it looks like it's related to the equation of the ellipse. It has a one on the right side of the equation or two fractions on the left side. The most important difference is this minus sign. Just like the ellipse, the A distance is a distance from the center to vertex points, and the B distance is a distance that's measured perpendicular to the A distance. Remember with the ellipse, the B distance was along the minor axis which was perpendicular to the major axis. And here, the B distance is perpendicular to the transverse axis.