Algebra

Introduction to conic sections | Conic sections | Algebra II | Khan Academy

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Let’s see if we can learn a thing or two about Conical sectors First, what are the conical sectors and why? It was called like this? You may already know some of them I will write them They are circle, ellipse, parabola And excessive cutting It’s p From Hyperbola Indeed, you know what these are The first time I studied on conical sections, I was Know what the circle is Know the parabola Little is known about the missing and excess sectors So why are all these called conical sectors? To make things easy to say, because it is a cross The top and the surface I will draw this quickly But before doing this, it might make sense To draw them ourselves I will change the colors The circle, we all know what it is Let me see if I can choose a lift line To draw the circle So the circle looks like this They are points equal to the distance from the center This distance you are called is the radius If r, and this is the center, then the circle is all Points that move away from this center In our curriculum, we learned what a circle is Literally, to make the thing round The ellipse, as defined by Lyman, is a compressed circle Looks like this Let me draw the ellipse in a different color The ellipse can be like this Or so It is hard to use, but it could be Tilted or inverted But this in general Indeed, the Chambers are a special case of the missing sectors As it is considered an ellipse, but not extended from one side More than the other It rolls off from all directions Parabola You have learned that if you have taken Algebra 2 And perhaps also if you care about the conical sectors But parabola – let me draw a line here to separate things Parabola looks like U This is the familiar form of parabola I don’t want to go to equations now Well, I’m going to do it because it’s considered familiar to you y = x ^ 2 Then, we can move it and we get On parabola like this This shape is x = y ^ 2 We can tip this off, but I think you know Parabola shape We will talk about how to represent it or how we know What are the points of interest in the parabola And then the last cut, and I thought you saw this shape Previously, it’s a plus It’s like two equal pieces, but not quite Because the curves will look less than U and And more open But I will explain what I mean Excess cut often looks like this So if this is hubs, and if you want to draw – let me Draw some close lines I want to go right through – that’s very good Here are the fonts These are not actual redundant sectors But the extra cut looks like this Come from here and Approaching the line Get closer and closer to the blue lines like this and This side Representation appears here and then Here This purple color graphic is a plus-cut; I did not draw this Well Or other equivalent parts, you can name it Vertical parabola This is not an exact word, but it is like This is below the converged line here It is above the converging line So this blue will be parabola and Purple, too, but different So these are different representations And the thing that I’m sure of is that you are asking why Conical sectors are called by this name? Why not called balls or variables? Circles or anything? Indeed, whatever the relationship is It is clear that the departments and sectors are missing Somewhat close This is the ellipse, which is a compressed circle Equivalent sectors and extra sectors are evident Close together This is P again Both contain bola at the end of the name and both They look like the letter U Although the extra cut has two of this open In different directions, but they look close But what is the link between these things? Frankly, it is the reason for naming the conical sectors So let me see if I can plot a conical strip in three dimensions This cone The highest point I used an ellipse on top Looks like this Actually, he has no summit It will continue indefinitely in this direction I’m cutting it to see it is a cone This will be the bottom of it Let’s take different surface junctions with The cone, as you can see, if we can at least generalize a difference The shapes we talked about now So if we have a flat surface – I think you call it This is the three-dimensional axis of the cone So this is the axis If we have a surface that is completely perpendicular to Axis – let me see if I can draw it in three dimensions The surface will be like this So you will call you a line This is the front line closest to you then We have another line in the back that’s enough And of course, you know that these are endless surfaces, that is, they are Go in any direction If this surface is directly perpendicular to the axis These and this is where the roof goes behind Junction of the roof and cone It will look like this We look at it from an angle, but if you want to look Below, if you listen to this and look at This surface – if you’re looking up And if I want to turn this around like this, then we are Look at this surface, this intersection It will be a circle Now, if we take the surface and its bottom slightly down Instead we have another similar situation Let me see if I can do this well We have a situation where Let me go back addition Retreat As he looks like this and has another side like this I connected them So this is the surface Now the intersection of this surface, which is now non Orthogonal or non-vertical with axis 3D cone If we take the intersection of this surface and this cone – and In future shows, you will not do this at Algebra 2 But finally, we’re going to draw a three-dimensional intersection And we prove that it is undoubtedly the case You got the equations, which I’m going to show you In the near future This intersection will look like this I think you can see it now It will look like this And if you want to look at the bottom of the roof, and if You want to look up the roof, it will look As – the shape that I painted in purple– It will look like this Well, I didn’t draw it well It will be ellipse And you know what an ellipse looks like And if you make it tilt to the other side, then the ellipse It will press in the other direction But this will give you a general feeling for a reason Consider these conical sections Now something very interesting If we continue to tilt this surface, if we hope the surface So– let’s say we’re centered on that point Now this is our surface – let me see if that can be done This is a good example of 3D drawing Let’s say he looks like this I want to go through this point So this is the 3D surface I will draw it in such a way that this only intersects With cone bottom and surface level it is parallel To the top of the cone top In this case the surface and cone intersection will be Here on this point And you can see that I’m centering on that point, on The intersection of this point, surface and cone Now this, the intersection, will look like So It will look like this It will continue to go down If you want to draw it, it will look like this If we are on the roof, if you want Surface drawing So we got the parabola This is interesting So if we keep tilting – if it starts with The circle, tilt it a little, and we get an ellipse We get a very deflected ellipse And at some point, the ellipse will still take Perverted shape like this This is like crackers here when we are scales To the top of the cone top And I’m doing it inaccurately now, but I am I think I will give you the proof It cracks and turns into parabola So you can see the parabola – here this relationship Parabola is what happens when one side of the ellipse It opens and we get the parabola Then, if we keep deflecting this surface, I will do this In a different color – so this intersects with Sides of the cone Let me see if I can draw this If this is the new surface that’s enough So if the surface looks like this – and I know that’s hard Reading now – you want to cross this surface This green and cone surface – has to repaint All this, and I hope you did not Stay tuned – the intersection will take this shape The cone bottom will intersect The top of the cone is here Then we will get a shape like this This will be the junction of the roof and the bottom of the cone And then the top here will be pick it up The roof and top Remember, this surface will go infinitely This is a general meaning for the conical sections And why it was called so And let me know if this bothers you, because maybe I will come Another offer while repainting in a clearer way And maybe I can find a good 3D application that can Do it better This is why they are considered conical sections And the reason for their association with each other And we’re going to do this in-depth sportsmanship In several shows But in the next show, you know what they are and why They are called conical sections, and in fact I will speak About their formulas and how to perceive These formulas As given, how can representations be determined? For these conical sectors? See you in the next presentation

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