Introduction to Conic Sections - Video Tutorials & Practice Problems

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Geometries from Conic Sections

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Hey everyone and welcome back. So in this video, we're going to be diving into a new topic called conic sections. Now, conic sections is a situation and are a set of problems that can oftentimes 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 and 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 following shape that you see. Now the first shape we're going to take a look at is the circle and we've seen circles just in the real world and in math math related problems, but the circle will happen for a conic section. If you take a cone as you can see, we have this three dimensional cone where we have these two kind of conic shapes stacked on top of each other at their ends. And if we take this and we slice it horizontally down the middle, so a horizontal slice will create a circle and this is how you can get this conic section from the shape. Now, another conic section that you can get is something called an ellipse. And 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. And the ellipse is kind of like a stretched out version of a circle. And that makes sense since if you take the plane and you tilt it slightly into the cone, it would make a longer end for the circle or longer ends, I should say. Now 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 you need to slice it at a heavily tilted angle. So 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. Now, 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. Now, the last shape we're going to take a look at is in my opinion, the most unique shape which is called the hyperbola. And the hyperbola looks something like this. We have these two curves that basically diverge away from each other and that creates this hyperbola shape. Now, the way that you can get the hyperbola shape is if you take the plane that you're dealing with and you 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. Now, something that I will also mentioned is for each of these sheets that we looked at there is an associated equation. Now, 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 the series, our 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.

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Problem

Problem

How can you slice a vertically oriented 3D cone to get a 2D parabola?

A

Slice the cone with a horizontal plane.

B

Slice the cone with a slightly tilted plane.

C

Slice the cone with a heavily tilted plane.

D

Slice the cone with a vertical plane.

3

Problem

Problem

How can you slice a vertically oriented 3D cone with a 2D plane to get a circle?

A

Slice the cone with a horizontal plane.

B

Slice the cone with a slightly tilted plane.

C

Slice the cone with a heavily tilted plane.

D

Slice the cone with a vertical plane.

4

Problem

Problem

A vertically oriented 3D cone is sliced with a vertical 2D plane. What is the conic section that will form?