Welcome back, everyone. So up to this point, we've been talking about conic sections. And in the last video, we looked at the circle shape. In this video, we're going to take a look at the ellipse. The ellipse forms when you take a three-dimensional cone and slice it with a two-dimensional plane at a slight angle. If you slightly tilt the plane into the cone, this is going to give you the ellipse shape, which we're going to talk about in this video. The ellipse is a bit more complicated than the circle because there's more that you need to keep track of when it comes to your graph and equation. But don't sweat it because in this video, we're going to figure out that there are actually some similarities between the circle and the ellipse. Even though it's a bit more complicated, the problem should be pretty straightforward once you know how the equations and graph work. So let's get right into this.

Now we've looked at a circle already, and we figured out that the circle depends on its radius. This radius tells us the distance from the center of the circle to any point on it. From this graph here, the circle has a radius of 3 units. That means any distance you travel to any point on the circle is always going to be a distance of 3 from the center. The equation for a circle looks like this:

x2 + y2 = r 2Imagine that we take this circle and stretch it horizontally. This new shape, different from the circle, does not have the same symmetry all around. In order to define this new shape, we need to keep track of two distances: the semi-major axis and the semi-minor axis. When a (4 units long) is larger than b (3 units long), a is the semi-major axis and b the semi-minor axis. That's the main idea of the horizontal ellipse. What if we stretched the same circle vertically instead of horizontally? The vertically stretched ellipse now has the major axis on the y rather than the x, switching the directions of a and b.

The equations are important to note: For the horizontal ellipse:

x2 a2 + y2 b2 = 1And for the vertical ellipse:

x2 b2 + y2 a2 = 1A simple way to remember these equations is by comparing them to the circle equation when divided by r squared:

x2 r2 + y2 r2 = 1This looks very similar to the ellipse equations, with the only main difference being a and b, due to the lack of symmetry all the way around the ellipse. This is the main idea and concepts behind the graphs of ellipses at the origin where the center is right in the middle of the graph. I hope you found this video helpful. Thanks for watching, and let's move on.