Hey, everyone, and welcome back. So in this video, we're going to be diving into a new topic called conic sections. Conic sections are a situation and a set of problems that can often be a bit difficult because there's a lot to remember with conic sections. But in this video and over the course of the next few videos, we're going to be going over some examples, graphs, and situations that will hopefully make the concept of conic sections super clear. So let's get right into things.

Now, when it comes to conic sections, what these are is they are shapes that can be formed by taking a three-dimensional cone and slicing it with a two-dimensional plane. And for each of these shapes, you're going to need to be able to write equations for and identify characteristics of each shape that you see. The first shape we're going to take a look at is the circle. We've seen circles just in the real world and in math-related problems. But the circle will happen for a conic section if you take a cone, and if we take this and slice it horizontally down the middle, a horizontal slice, it will create a circle. This is how you can get this conic section from the shape.

Another conic section that you can get is something called an ellipse. An ellipse occurs if you take the same cone that we had, and you slice it at a slight angle. So if you slice it at a slight angle, this will create this kind of oval shape, which is the ellipse. The ellipse is kind of like a stretched-out version of a circle, and that makes sense since if you take the plane and tilt it slightly into the cone, it would make a longer end for the circle or longer ends, I should say.

The next shape we're going to take a look at is something called the parabola, and we've seen parabolas in previous videos. But in order to get a parabola from a conic section, you need to take your two-dimensional plane and slice it at a heavily tilted angle. With the ellipse, we had a slightly tilted angle, but now we have a heavily tilted plane. And when you have a heavy tilt to it, notice how part of the plane is going to kind of pop off the end here, and this will create this parabola shape, which is the graph that we have down here.

I also want to say, keep in mind for all these conic shapes and slices that you're seeing, this is almost more of a memory tool. This is a way of just recognizing the types of shapes we deal with in conic sections. So this is why we use the slicing a cone analogy because it helps us to remember these.

The last shape we're going to take a look at is, in my opinion, the most unique shape, which is called the hyperbola. The hyperbola looks something like this. We have these two curves that basically diverge away from each other, creating the hyperbola shape. The way that you can get the hyperbola shape is if you take the plane that you're dealing with and slice it vertically into the cone. Because if you have a vertical plane that slices directly through the cone like this, this will give you the shape where the two slices of the cone diverge away from each other creating the hyperbola.

Something that I will also mention is that for each of these shapes that we looked at, there is an associated equation. Some of these may look slightly familiar if you've seen previous videos like the circle, for example. But what we're going to be doing in this series on conic sections is going over each of these equations and each of these graphs and seeing how they all relate to this idea of slicing a cone in different ways. So that's the basic idea of conic sections. I hope you found this video helpful. Let me know if you have any questions. Thanks for watching.