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Current time:0:00Total duration:10:58

Video transcript

let's see if we can learn a thing or two about conic sections so first of all what what are they and why are they called conic sections actually your pot you probably recognize a few of them already and I'll write them out they're the circle circle the ellipse the parabola parabola and the hyperbola hyper hyper bola that's a P hyperbola and you know what these already and I mean you know when I first learned conic sections as I go I know what a circle is I know the parabola is and I even know a little bit about ellipses and hyperbolas why on earth are they called conic sections so it to put things simply because they're the intersection of a plane and a cone and I'll draw you down 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 that's let me pick it actually let me see if I can pick a thicker line for my circles so 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 right so if this is R and this is the center the circle is the point all the points that are exactly R away from this Center and we learned that I we learned that early in our education is what a circle is it makes the world go round literally ellipse in in layman's terms it's kind of a squished circle it could look something like this it could look like let me do it lips in another color so the lips 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 it actually circles are a special case of an ellipse it's a it's an ellipse where it's 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're if you're taking algebra 2 and MIT you probably have if you're if you care about conic sections but a parabola let me draw a lines here so you know what we're two 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 but the classic well I will because you're probably familiar with it it's y is equal to x squared and then you know it could be you could shift it around and I'm you can even have a parabola that goes like this that would be X is equal to Y squared you could also rotate rotate these things around but you I think you you 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 the curves tend to they took they look a little less Jewish and a little more open but I'll explain what I mean by that so hyperbola usually looks something like this so if these are the axes these are the axes then if I were to draw let me draw some asymptotes let me draw some asymptotes I want to go right through the right through that's pretty good these are asymptotes those aren't the actual hyperbola but a hyperbola would look something like this it could either be on it would be like you know they get they could be like right here and they get really closely asymptotes they get closer and closer to those blue lines like that and happen on this side too so it kind of the graphs show up here and then they pop over and they show up there so that would be this magenta could be one hyperbola I haven't done true justice to it or another hyperbola could be on kind of a kind of holida a vertical hyperbola and that's not the exact word but it would look something like that where it's it's below the asymptote here it's above the asymptote there so those are so this this blue one would be one hyperbola and then the magenta one would be a different hyperbola so those are the different graphs so then you know the one thing that you I'm sure you're asking is why are they called conic sections why are they not called bolas or or variations of circles or whatever and in fact what's 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're both kind of look like opened use although hyperbola has two of these going in in kind of opening in different directions where they look related but what is the connection behind all these and that's that's frankly where the word conic comes from so let me see if I can draw a three-dimensional cone so if this is a cone let me see so that's the top let me draw I could use an ellipse for the top looks like that that's 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 the of this three dimensional cone so this is the axis so if we have a plane that's exactly perpendicular that axis so let's see if I can draw it in three dimensions so the plane would look something like this so we'd have a line this is the front line that's closer to you and then they'll have another line back here it's close enough and then of course you know these are infinite planes it goes off in every direction but I'm just trying so this is if this plane is directly perpendicular to the axis of these so this is where the plane goes behind it the intersection of this plane and this cone is going to look like this and we're looking at it from an angle but if you were to look at straight down if you're really sitting here and you look at this plane if you're looking at right above if I were to just flip this over like this so this is 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 if we tilt it down a little bit so instead of that we have a situation like this let me see if I can get do it justice we have a situation where it's you know the SOT whoops let me undo that and it undo where it's like this it has another side like this and then I connect them so that's the plane now the intersection of this plane which is not which is now not orthogonal or it's not perpendicular to the axis of this of this 3-dimensional cone if you take the intersection of that plane in that cone in future videos and you don't do this in your algebra 2 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 can visualize it right now it would look something like this right and if you were to look straight down on this plane if you look right above the plane this would look something this figure I just drew in purple would look something like this no I didn't draw it that well would be an ellipse you know what an ellipse looks like and if I tilted it the other way the ellipse would be would kind of it would squeeze the other way but that could just use general sense of why it's 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 it's so let's say we're pivoting around that point so when au the put 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 well so this is my three-dimensional plane and I'm drawing it in such a way that it only intersects this bottom cone and it's 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 it's going to intersect right at that point I'm kind of you can almost view that I'm pivoting around this point and the intersection of this point and the plane and the cone well this now would look the intersection would look something like this it would look like that and 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 would 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 an ellipse you get kind of a more and more skewed ellipse more and more skewed ellipse and at some point it looks kind of you know the ellipse keeps getting more and more skewed like that it's some kind of kind of pops right when you write when you become exactly parallel to the side of this top cone and I'm doing it all vary in exact right now but I think I want to give you the tuition it pops and it turns into a parabola so you can kind of view a parabola there is this relationship it's kind of the 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 if you keep tilting the plane I'll do it in another color so the intersects both sides of the cone so let me see if I could draw on 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 in 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 intersect the top cone over there and then you would have something like this you would have this would be the intersection of the plane and the bottom cone and then up here would be the intersection of the plane in the top gonna 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 if this got confusing because maybe I'll do another video I'll redraw it a little bit cleaner maybe I can find some kind of neat 3d application that that can do it better than I can do it but these are this this kind of just the reason why they all are conic section why they really are related to each other and we'll do that in a little more 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 in the next video