hopefully you now feel comfortable at least a little bit with conic sections my goal with this little video is to give you a summary of it all because you have these like four different clinic sections and then you got to kind of put them all together and spot when you have one and not the other and recognize the different ones and how to graph each of the properties of each so I'm gonna kind of summarize all of them on one page I already broke down a whole bunch of stuff under here cuz I make the pennant easier for me to describe it cuz I'm gonna say a lot of stuff here but there's a lot of stuff going on with all these clinics and it's really important to get all the big picture put together in your head so check with me through this hopefully I make some sense if you need it pause it for a second you go ahead and saying let's do it here we go the summary first of all you have this HK HK is the center or the vertex referenced here's the center is the green dots this one is the vertex where there's no center of a parabola so let's talk about each one of these shapes again that's parabola circle ellipse and hyperbola the four conic sections all one happy family all right if you look that's the vertex one negative two you're a value is negative one because of how kind of wide or narrow that is I'm not going to go into that one in particular for this situation there's the equation that gives you the parabola now this is honest aside it's x equals if it was say a vertical like that it would be y equals and the agent K would be flip-flop that'd be an X anyways it would be a little bit different but I'm not going to put every single example on this video anyway since it's on the side is x equals and if you look at the formula you have 18 K and so if you see here we have a Y and isn't K the Y value so she's offices it's the opposite of opposite of K so the office of negative to positive to the H is one and this is going the negative direction that's when negative this would be positive direction so it's negative one a value now the parabola probably the hardest to graph four people associate sideways parabola now a vertical and horizontal vertical probably up or down to be a little bit easier because you would have gone over those most likely with the quadratics but cinnamons on the side it's a different situation and so really gives people is the HK is kind of backwards on them because if that's X this is y so again the y value of H K is liquor why there's always square here anyways so probably a little bit more difficult again if it's vertical it would've been y equals okay problems circle the center is 2 4 BAM well if you can count the radius see the radius you go over right there 5 if you go up five down five anyway you go it's 5 so the formula for it is X minus H Y minus K goes squared and the radius squared so I all my information so it's X the opposite of 2 is negative 2 squared opposite of 4 is negative 4 y squared equals radius squared 25 all right circles are probably one of the easier ones to do because you just take the center square them x and y and then square the radius I like circles hopefully you do too all right next one is ellipse which I kind of call like an oval or like a smooshed circle you race the formula for an ellipse always equals 1 same with hyperbola actually look at these two equations they're very very similar the big difference is a plus and minus yes the Y is first here but there's a reason for that one sometimes the X is first okay but they're both equal to one now for an ellipse again it looks a lot like a circle see this circle here see how the top looks a lot like a circle except there's a 1 instead of a radius and there's two denominators on an ellipse and I kind of say these denominators are kind of like the radius split up in two pieces because if you look at the slips you kind of see I have a radius this way and a radius this way they're not really radii but you kind of think of it that way so a radius that way and that way in each of these are based its own radius let's talk about this three negative two BAM HK my a value they called the eighth is this radius if we can count over six three I went up three this radius is three let's be basically a is always neat a longer side B is the shorter side so sometimes a is over here in B's over here depending on if X or Y is the X or the Y X is this way why this way depending on which one is longer and depends on where they would say it is so again a is the longer we get in this situation it's the under the X cuz the X is the longest direction so again with this formula I plugged in the opposite of three the opposite of negative two now where you get 36 + 9 y square a and a square 3 and always equals one now you think about it these two values again are the radii to separate radii code you can think of this kind of a smooshed or squished or stretched circle or circles one radius every direction is the same alright so again it looks a little very similar now a hyperbola just a minus here so it's very very similar you have your center right there negative two to four is the radius a larger B is the other one the smaller actually it isn't always a larger and smaller a is actually the positive it's kind of weird it's just it's the first is always a because thing with this it is larger because if you think about this negative this technically is kind of like a negative I'm not confused you they're just this basically the first one on hyper bus always your Eddy just your B is always the second because the B is always smaller because it's kind of negative anyways um the four is how far puff here you went and the two is how far over you make this little box and the reason for this little box is you're making these things called asymptotes and what you do is you start here at the center and you go kind of through the corners of the box and that creates an asymptote and for this you go through the corners of the box and then you create your head right there by the way the ex is not part of the graph it's just boundaries that help you graph so you kind of you know Lam just little boundaries the asymptotes help you graph it really quick nice way of graphing and again the a and the B give you this box kind of parameters of your box um okay so if you look at this why why is there the Y value first and why is the X here what do you see others hyperbola is opening up and down wise first if my head perv loves over here and over here guess what X would be first it's very important on a hyperbola the first letter the X to the Y tells you which way you're opening all right over here see the X and y1 connection over here the X see how the 36 is M into DX because you went six in the x direction so the nine is under the Y because you went three in the Y direction see here Y 4 squared 16 7 4 in the Y direction at 4 and the y over 2 in the X CIA 2 squared in the X direction so wherever your a and B whichever direction your a and B are going is what letters below so if a squared 4 squared is 16 it's under the Y because you went up four you went over to 2 squared is 4 so again they both have squares there's a minus and a 1 again the first letter is the direction of which you're hyperbolas hoping these always usually X is first because with a plus it doesn't matter the order but we tend to put X's first the a value is the largest denominator here the a value is a first okay and again you square the a and the B to get the bottom parts and again here you have a circle near the HK this is your radius squared and again parabola once again same idea of H K this is K this is H you have an a value which is kind of the width of your parabola and negative positive Z which direction and this is pretty obvious there's only one squared on a parabola or the rest of you have two sets of squared pieces so hopefully you can see some connection they all have like HKS these all have A's and B's this has one radius okay this a and B makes a box they hope you graph these any B just gives you the radiuses which you then can make the dots because you make boom-boom-boom mix the dots however to graph one of these is H equal 5 5 5 5 go all the way around all right so that's the kind of a summary of quadratic sections conic sections not quadratic shalini um so good luck with them hopefully this summary kind of helps you up now I'd like to do next is a application of clinic sections so take a look at this why our clinic section is useful here we go the three conic sections I'm I talked about or miss skipped up circles have very interesting reflective properties it could be like Mira Sh reflection or even lenses like glasses reflection of light so what happens is all these conic sections have focal points which is a whole different concept I didn't put in the previous lecture video part so these are focal points these red dots what's really cool about focal points is if light say this is a lens their eyeball or like glasses if light comes directly towards that it's reflected refracted to the focal point no matter what light comes directly at it its refractive to the focal point so that's kind of like how like a lens like magnifying glasses can like make really hot like you can burn paper to be say the burnin ant or something like that anyways all the light that comes and hits this refracts into here from straight it gets refracted in now this is a mirror which you see solar cells sometimes making these in you've seen them maybe on TV or anyway so if you put a sensor here to heat something up all the light that comes in and it's reflected reflects to that focal point so anything that comes from the inside of this as a mirror is reflective of their so all this is furnace to one point we're here it comes in it is refracted into one focal plane really cool property here what's interesting about the two focal points is if you were to say make a pool table where you had an elliptical pool table if you simply hit your pool ball against any wall from this focal point it'll bounce off and go into that hole if you had a hole right there if you like the dummy-proof pool table as long as you didn't put English on it so if I hit right here anywhere here against the wall boon of go in the hole it's really cool another thing is what if you made an oval office the presidents room the Oval Office and in the Oh blossom is you have a a chair sitting here and a chair sitting here everything you've stayed will reflect off the walls and bounce to that person so you can basically have two people having a conversation in an oval room at these two points they could have a conversation what everybody else around them is talking because everything you say all the words all the noises that come through mouth BAM bounce and bounce over that spot back and forth kind of cool last thing hyperbola really cool about hyperbola is if for instance ignoring let's say ignore this one right here if if I have this situation right actually not let's not ignore this one let's say this one I here's the mirror a mirror and I'm aiming at this so if I aim at this when it hits this it will go to that so if I'm here and I aim at that it will go and reflect that focal point now if I for instance do this one let's say I look at this one right here if I focus on this point I bounce off this you know go to that one as long as this is not you're blocking it but if I aim at this focal point it'll bounce to that focal point so if I say I was right here and I aimed at that it will boom bounced to that focal point so it also has some reflective properties internal reflective designer of the game over here or external if you come and bounce off this to that one they all have really interesting focal points like mirrors even through lenses and all the other focal points which is another thing that's learned within clinic sections because each of these three have very specific unique focal points that are very useful hopefully I was kind of interesting you