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Sal introduces the four conic sections and shows how they are derived by intersecting planes with cones in certain ways. Created by Sal Khan.
Video transcript
Let's see if we can learn a thing or two about conic sections. So first of all, what are they and why are they called conic sections? Actually, you probably recognize a few of them already, and I'll write them out. They're the circle, the ellipse, the parabola, and the hyperbola. That's a p. Hyperbola. And you know what these are already. When I first learned conic sections, I was like, oh, I know what a circle is. I know what a parabola is. And I even know a little bit about ellipses and hyperbolas. Why on earth are they called conic sections? So to put things simply because they're the intersection of a plane and a cone. And I draw you that in a second. But just before I do that it probably makes sense to just draw them by themselves. And I'll switch colors. Circle, we all know what that is. Actually let me see if I can pick a thicker line for my circles. so a circle looks something like that. It's all the points that are equidistant from some center, and that distance that they all are that's the radius. So if this is r, and this is the center, the circle is all the points that are exactly r away from this center. We learned that early in our education what a circle is; it makes the world go round, literally. Ellipse in layman's terms is kind of a squished circle. It could look something like this. Let me do an ellipse in another color. So an ellipse could be like that. Could be like that. It's harder to draw using the tool I'm drawing, but it could also be tilted and rotated around. But this is a general sense. And actually, circles are a special case of an ellipse. It's an ellipse where it's not stretched in one dimension more than the other. It's kind of perfectly symmetric in every way. Parabola. You've learned that if you've taken algebra two and you probably have if you care about conic sections. But a parabola-- let me draw a line here to separate things. A parabola looks something like this, kind of a U shape and you know, the classic parabola. I won't go into the equations right now. Well, I will because you're probably familiar with it. y is equal to x squared. And then, you could shift it around and then you can even have a parabola that goes like this. That would be x is equal to y squared. You could rotate these things around, but I think you know the general shape of a parabola. We'll talk more about how do you graph it or how do you know what the interesting points on a parabola actually are. And then the last one, you might have seen this before, is a hyperbola. It almost looks like two parabolas, but not quite, because the curves look a little less U-ish and a little more open. But I'll explain what I mean by that. So a hyperbola usually looks something like this. So if these are the axes, then if I were to draw-- let me draw some asymptotes. I want to go right through the-- that's pretty good. These are asymptotes. Those aren't the actual hyperbola. But a hyperbola would look something like this. They get to be right here and they get really close to the asymptote. They get closer and closer to those blue lines like that and it happened on this side too. The graphs show up here and then they pop over and they show up there. This magenta could be one hyperbola; I haven't done true justice to it. Or another hyperbola could be on, you could kind of call it a vertical hyperbola. That's not the exact word, but it would look something like that where it's below the asymptote here. It's above the asymptote there. So this blue one would be one hyperbola and then the magenta one would be a different hyperbola. So those are the different graphs. So the one thing that I'm sure you're asking is why are they called conic sections? Why are they not called bolas or variations of circles or whatever? And in fact, wasn't even the relationship. It's pretty clear that circles and ellipses are somehow related. That an ellipse is just a squished circle. And maybe it even seems that parabolas and hyperbolas are somewhat related. This is a P once again. They both have bola in their name and they both kind of look like open U's. Although a hyperbola has two of these going and kind of opening in different directions, but they look related. But what is the connection behind all these? And that's frankly where the word conic comes from. So let me see if I can draw a three-dimensional cone. So this is a cone. That's the top. I could've used an ellipse for the top. Looks like that. Actually, it has no top. It would actually keep going on forever in that direction. I'm just kind of slicing it so you see that it's a cone. This could be the bottom part of it. So let's take different intersections of a plane with this cone and see if we can at least generate the different shapes that we talked about just now. So if we have a plane that goes directly-- I guess if you call this the axis of this three-dimensional cone, so this is the axis. So if we have a plane that's exactly perpendicular to that axis-- let's see if I can draw it in three dimensions. The plane would look something like this. So it would have a line. This is the front line that's closer to you and then they would have another line back here. That's close enough. And of course, you know these are infinite planes, so it goes off in every direction. If this plane is directly perpendicular to the axis of these and this is where the plane goes behind it. The intersection of this plane and this cone is going to look like this. We're looking at it from an angle, but if you were looking straight down, if you were listening here and you look at this plane-- if you were looking at it right above. If I were to just flip this over like this, so we're looking straight down on this plane, that intersection would be a circle. Now, if we take the plane and we tilt it down a little bit, so if instead of that we have a situation like this. Let me see if I can do it justice. We have a situation where it's-- whoops. Let me undo that. Edit. Undo. Where it's like this and has another side like this, and I connect them. So that's the plane. Now the intersection of this plane, which is now not orthogonal or it's not perpendicular to the axis of this three-dimensional cone. If you take the intersection of that plane and that cone-- and in future videos, and you don't do this in your algebra two class. But eventually we'll kind of do the three-dimensional intersection and prove that this is definitely the case. You definitely do get the equations, which I'll show you in the not too far future. This intersection would look something like this. I think you can visualize it right now. It would look something like this. And if you were to look straight down on this plane, if you were to look right above the plane, this would look something-- this figure I just drew in purple-- would look something like this. Well, I didn't draw it that well. It'd be an ellipse. You know what an ellipse looks like. And if I tilted it the other way, the ellipse would squeeze the other way. But that just gives you a general sense of why both of these are conic sections. Now something very interesting. If we keep tilting this plane, so if we tilt the plane so it's-- so let's say we're pivoting around that point. So now my plane-- let me see if I can do this. It's a good exercise in three-dimensional drawing. Let's say it looks something like this. I want to go through that point. So this is my three-dimensional plane. I'm drawing it in such a way that it only intersects this bottom cone and the surface of the plane is parallel to the side of this top cone. In this case the intersection of the plane and the cone is going to intersect right at that point. You can almost view that I'm pivoting around this point, at the intersection of this point and the plane and the cone. Well this now, the intersection, would look something like this. It would look like that. And it would keep going down. So if I were to draw it, it would look like this. If I was right above the plane, if I were to just draw the plane. And there you get your parabola. So that's interesting. If you keep kind of tilting-- if you start with a circle, tilt a little bit, you get an ellipse. You get kind of a more and more skewed ellipse. And at some point, the ellipse keeps getting more and more skewed like that. It kind of pops right when you become exactly parallel to the side of this top cone. And I'm doing it all very inexact right now, but I think I want to give you the intuition. It pops and it turns into a parabola. So you can kind of view a parabola-- there is this relationship. Parabola is what happens when one side of an ellipse pops open and you get this parabola. And then, if you keep tilting this plane, and I'll do it another color-- so it intersects both sides of the cone. Let me see if I can draw that. So if this is my new plane-- whoops. That's good enough. So if my plane looks like this-- I know it's very hard to read now-- and you wanted the intersection of this plane, this green plane and the cone-- I should probably redraw it all, but hopefully you're not getting overwhelmingly confused-- the intersection would look like this. It would intersect the bottom cone there and it would intersect the top cone over there. And then you would have something like this. This would be intersection of the plane and the bottom cone. And then up here would be the intersection of the plane and the top one. Remember, this plane goes off in every direction infinitely. So that's just a general sense of what the conic sections are and why frankly they're called conic sections. And let me know if this got confusing because maybe I'll do another video while I redraw it a little bit cleaner. Maybe I can find some kind of neat 3D application that can do it better than I can do it. This is kind of just the reason why they all are conic sections, and why they really are related to each other. And will do that a little more in depth mathematically in a few videos. But in the next video, now that you know what they are and why they're all called conic sections, I'll actually talk about the formulas about these and how do you recognize the formulas. And given a formula, how do you actually plot the graphs of these conic sections? See you in the next video.