 Okay, this is gonna be a quick video about one of the few things that I find cool about conics. There are other There are some cool features about conics They all have to do with the full kind But we're not talking about that today. We're talking about How to solve how to solve conics or get the general form with with a few points So in case you don't know what a conic is Basically, it's this this equation where a where a b c d e and f are coefficients and they can be any number and When you graph them, they'll basically give you a shape that you can get if you take a slice out of like a A cone as out of like a sort of weird cone like two cones a Place like that in 3d space imagine taking a slice with a 2d plane and that would be the shapes they give you In let me show them it'd be you could get either a parabola circle and ellipse or a hyperbola, which is Now we can we can find this if we just take like Some of some points I think at most you need five Yeah, and mostly need five You can get the whole the whole conic the the catch is you need to get at least You need to get a sum or one or some of the coefficients and based on how many coefficients you need you have left a Based on how many unknown coefficients you have That's how many points you're gonna need in order to solve it So let me so I'm gonna give a quick example with a circle now In this case b will equal zero, so we don't have to worry about the complexity that x y brings a equals one and c equals one so we have a circle and These are our points Finding circles is the easiest type of these problems that you could do, but you could apply this just for anything so Since we know since we know the coefficients of a and c we put them on the right hand of the equation So we now have them here And since we know that it's one we just replace it as one and we get this okay now All that's left is we plug in we plug in the x and y values here Into each into each of these so we get something like let me turn down my sensitivity We'd get something like d zero plus E negative four plus f1 because there's no really not really an x y behind that f1 and Y equals zero negative the first square to 16 Negative again it was like it goes negative 16 So we do that for every single one of these again. We get something like d2 plus e negative four you keep doing that you get three and For some of you You might know systems of equations Matrices, which is how how these are going to be solved Matrices are really useful when solving Systems of equations. So let me give you a quick example of what's of what that's like let's say we have 3a and And we add to be And we know that's gonna be eight Okay, well then a and b could be anything so long as they add up to the eight to eight But if we also have a and we add to be And let's say we get four. Yeah, I think that works out. Yeah Now we can have an exact An exact evaluation of what three of what a and b are so We would take we would put the coefficients into a matrix. Let's just put one over here and The matrix will look like this one Two two and if we take the inverse and Apply it so it's on the right-hand side of Of this of the equations We will get a and b Typically You'll solve this using Gaussian elimination and it wouldn't look like that. It was look it would be an extended matrix where you'd have Stuff like this line here and To get to get what you want there you you have to get it you have to use Gaussian elimination to turn This into a one This into a one and these two into zeros. I'll get the idea so Let me do this real quick So what you would first do is you subtract the bottom one from the top that would give you two zero Four we divide this by two to just get a one so one zero Equals two, so we know that a equals two. Oh This is this is looking bad because this sort of looks like a one. Let's Make it like that. So it's a little clear Okay, now we take the bottom and subtract this Subtract this from the bottom. So we get a zero two Equals two and then we just divide by two you get zero one one so So then it would look like One zero one Two one so So the new matrix When you apply this Oh, yeah, it would look like to one so a would equal to and b would equal one and Let's just Make sure that That's right three times two is six two times one is two six plus two equals eight This one works out One times two is two two times one is two two plus two is four. So all of that works out now we could We apply that same Logic systems of equations to the conics. So we don't really care about the shape anymore We just care how it looks when you plug this in. So remember it looks It would look like this the matrices Would look like that Now I forgot what that is doing there. So So yeah, when you plug that in you get this for the left-hand side and This for the right-hand side. We take the inverse. I don't really feel like doing that She'll annoying oh one one thing that does make it easier is that The f side will always be one since you don't have to multiply by x or y So you can always find that easily But you take the inverse and multiply it or you take the inverse and apply it to the right-hand side Which would give you negative two zero and negative sixteen Hopefully you're falling you're falling along with me. I'm not that good at explaining stuff, but So When we apply that today's that gives us our D. So D will equal negative two e Which is zero and f which is negative 16 So the equation a full a full equation would look something like this negative 2x Plus zero y minus 16 equals negative x squared Plus y squared if we We can rewrite it as into the general form as x squared plus y squared minus 2x minus 16 Equal zero and that's how you get your That's how you get your equation Yeah, there's your general form just from three points and three coefficients Please tell me how I can improve in explaining I'm not really good at that And if you know they say can't explain it you haven't learned it Something like that That's all