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# Intro to hyperbolas

## Video transcript

let's see if we can learn a thing or two about the hyperbola hi / bola and out of all of the conic sections this is probably the one that confuses people the most because it's not quite as easy to draw as the circle and the ellipse cuz you gotta do a little bit more algebra but hopefully over the course this video you'll get pre comfortable with that and and you'll see that hyperbola is in some way are are more fun than any of the other conic sections so just as a review and I want to I want to do this just so you see the similarity in the formulas or the kind of the standard form of the different conic sections if you have a conic a circle centered at zero its equation is x squared plus y squared is equal to R squared and we saw that this could also be written as and I'm doing this because I want I wanted to show that this is really just the same thing as the standard equation for an ellipse if you divide both sides of this by R squared you get x squared over R squared plus y squared over R squared is equal to 1 and so this is a circle and once again it's just review a circle all of the points on the circle are equidistant from the center or in this case you can kind of say that the major axis and the Minor axis are the same distance that there isn't any distinction between the two you're always an equal distance away from the center so that was a circle and ellipse was pretty much this but these two numbers could be different because your distance from the center could change so it's x squared over a squared plus y squared over B squared is equal to 1 that's an ellipse and I you'll see that a and I'll skip parabola for now because parabola is kind of an interesting case and you've already touched on it so I'll go in more depth in that in a future video but in a hyperbola is very close in formula to this and so there's two ways that a hyperbola could be written and I'll do those two ways so it could either be written as x squared over a squared minus y squared over B squared is equal to 1 notice the only difference between equation and this one is that instead of a plus y squared we have a minus y squared here so that that would be one hyperbola the other one would be if the minus sign was the other way around if it was y squared over B squared minus x squared over a squared is equal to one so now the minus is in front of the x squared term instead of the y squared term and what I want to do now is try to figure out how do we graph these either of these parabolas maybe we'll do both cases and in a lot of textbooks or even if you look it up over the web they'll give you formulas but I don't like those formulas one because I'll always forget it and you'll forget it immediately after taking the test you might want to memorize it if you just want to get do be able to do the test a little bit faster but you'll forget it and the second thing is not only you forget it but you'll probably get confused because sometimes they always use the a under the X and the B under the Y or sometimes they always use the a under the positive term and the B under the the negative term so if you just memorize Oh a divided by B that's the slope of the asymptote and all of that you might be using the wrong a and B so I encourage you to always kind of reprove it to yourself and that's what we're going to do right here and it actually doesn't take too long so these are both hyperbolas and what I like to do whenever I have a hyperbola is solve for y so in this case I get C if I if I subtract x squared over a squared from both sides I get let me change the color I get minus y squared over B squared alright that stays there is equal to 1 minus x squared over a squared and then let's see I want to get rid of this minus and I want to get rid of this B squared so let's multiply both sides of this equation x minus B squared all right if you multiply the left hand side times minus B squared the minus and the B squared go away and you're just left with Y squared is equal to minus B squared and then minus B squared times a plus that becomes a plus B squared over a squared x squared we're almost there and then you get why is equal to and I'm doing this on purpose the plus or minus square root because it can be the plus or minus square root of let's switch these around just so I have this term first a positive term first V squared over a squared x squared minus B squared now what is this you said Sal you know you said this was simple I'm solving this this looks like a really complicated thing but remember we're doing this to figure out the asymptotes of the hyperbola just to kind of give you a sense of where we're going we do it here I'll do the other actually I want to do that other hyperbola so a hyperbola if that's the X that's the y axis it has two asymptotes and the asymptotes you know they're they're these lines that the hyperbola will approach so if that those are the two asymptotes and they're always the negative slope of each other.we we know that a height this hyperbola is either and we'll show it in a second which one it is it's either going to be it's going to look something like this where as we approach infinity we get closer and closer to this line and closer to close to that line and here it's either going to look like that I didn't draw it perfectly it never touches the asymptote just gets closer and closer and closer arbitrarily close to the asymptote it's either gonna look like that where it's kind of one you know opens up to the right and the left or or a hyper blues going to open up up up and down and once again as you go further and further an asymptote means it's just closer and closer to one of these lines without ever touching it but it'll get infinitely close as you get infinitely far away as X gets infinitely large so in order to figure out which one of these things is let's just think about what happens as X becomes infinitely large so if I draw so as X approaches infinity so as X approaches infinity or X approaches negative infinity so I'll say plus or minus infinity right it doesn't matter because when you take a negative this gets squared so this number becomes really huge as you approach positive or negative infinity and you'll learn more about this when we actually do limits but I think that's intuitive that this number becomes huge this number is just a constant it just stays the same so as you approach as X approaches positive or negative infinity is it gets really really large why is going to be approximately equal to that's product what I wanted is actually I think that's congruent let me I think this is I always forget notation approximately this just means not exactly but approximately equal to when X approaches infinity it's going to be approximately equal to the plus or minus the square root of b squared over a squared x squared right and that is equal to now you can take the square root of this you couldn't take a square root of this algebraically but this you can this is equal to plus or minus B over a X so that tells us essentially what the two asymptotes are where the slope of one asymptote will be B over a X this could give you this positive B over a X and the other one would be minus B over a X and I'll do this with some example so it makes it a little clearer but we still so we know what the asymptotes look like it's these two lines right because it's a plus B ax is one line y equals plus B ax let's say it's this one this asymptote right here is the y is equal to plus B over ax I know you can't read that and then the downward sloping asymptotes we could say is y is equal to minus B over ax so those are two asymptotes but we still have to figure out whether the hyperbola opens up to the left and right or doesn't open up and down and there at least there's two ways to kind of do this one you see well you know this was an approximation this is what you approach as you X approaches infinity but we see here that even when X approaches infinity we're always going to be a little bit smaller than that number because we're subtracting a positive number from this right we're subtracting a positive number that when taking a square root of whole things we're always going to be a little bit lower than the asymptote especially when I guess when we're in kind of the positive quadrant right so to me that's how I like to do it I think oh we're always at least in the positive quadrant gets a little more confusing when you went to the other quadrants we're always going to be a little bit lower than the asymptote so we're going to approach from the bottom there and since you know you're there you know it's going to be like this and approach this asymptote and then since it's opening to the right here it's also going to open to the left the other way to test it and maybe this is more intuitive for you is to figure out in the our original equation could x or y equal to 0 because when you open to the right and left notice you never get to X equal to 0 you always get you get to y equals 0 right here and here but you never get to x equals 0 and actually your teacher might want you to plot these points and there you just substitute y equals 0 and that you can just look at the original equation actually you could even look at you could even look it at this equation right here could X ever equal 0 if you look at this equation if X is equal to 0 this whole term right here would cancel out and you just be left with a minus B squared which is a you're taking B squared and you put a negative sign in front of so that's a negative number and they're taking the square root of a negative number so we're not dealing with imaginaries right now so you will you can never have X equal to 0 so what Y could be equal to 0 right you could set y equals 0 and then you could solve for it so in this case actually let's do that if Y is equal to 0 you get 0 is equal to the square root of B squared over a squared x squared minus B squared you could square both sides you get B squared over a squared x squared minus B squared is equal to 0 and now this is messy so then you get B squared over a squared x squared is equal to B squared you could divide both sides by B squared I guess you get a 1 and a 1 and then you could multiply both sides by a squared you get x squared is equal to a squared right and then you get X is equal to the plus or minus square root of a so this point right here is the point a comma 0 and this point right here is a point minus a a comma 0 let's go back to the other asset the other problem I'm a feeling I might have might be running out of time so notice that when X when the X term was positive our hyperbola opened to the right and the left and you could probably guess from deductive reasoning that when the Y term is positive which is in the case in this one we're probably going to open up and down and let's just prove that to ourselves so solve for y you get Y squared over B squared we're going to add x squared over a squared to both sides so you get equals x squared over a squared plus 1 multiply both sides by B squared y squared is equal to B squared over a squared x squared plus B squared going to distribute the B squared now take the square root I'll switch colors for that so Y is equal to the plus or minus square root of B squared over a squared x squared plus B squared and then once again oh I've run out of space we can make that same argument that as X approaches infinity as X approaches positive or negative infinity X approaches positive or negative infinity this equation this be this little constant term right here isn't going to matter as much you're just going to take the square root of this term right here which is essentially B over a X plus or minus B over a X and once again those are these same two asymptotes which I'll redraw here that line and that line but in this case we're always a little bit larger than the asymptotes we're always a little bit larger than the asymptotes especially in the positive quadrant that tells us we're going to be up here and down there and another way to think about it in this case when the hyperbola is vertical or you know it's a vertical hyperbola where you're always kind of it opens up and down you notice X could be equal to 0 but Y could never be equal to 0 and that makes sense too because if you look at our original formula right here X could be equal to 0 if X was 0 this would cancel out you could just solve for y but if Y we're equal to 0 you would have minus x squared over a squared is equal to 1 and then you would have if you solve this you would get x squared is equal to minus a squared and we're not dealing with imaginary number so you can't square something you can't get a negative number so once again this would be impossible so that's the other clue that tells you it opens up and down because in this case why couldn't equals zero anyway you might be a little confused because I stayed abstract with the bees and the A's in the next couple of videos I'll do a bunch of problems where we draw a bunch of hyperbolas ellipses and circles with actual numbers see you soon