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Current time:0:00Total duration:14:50

Proof of the hyperbola foci formula

Video transcript

in the last video I told you that if I had a hyperbola with the equation x squared over a squared minus y squared over B squared is equal to 1 that the focal distance for this hyperbola is just equal to the square root of the sum of these two numbers the square root of a squared plus B squared in this video I really just want to show you that and actually just so you know you know this this situate this equation right here this is a particular hyperbola that opens to the left on the right and that's because that's those are the asymptotes point so this would be the axes and that's because the X term is positive if the Y term was positive and the X term had a negative sign then the hyperbola would open upwards and downwards like that and the proof that I'm showing you in this video it's just a bunch of algebra really is identical in the Y case you just switch around the XS and the Y's but I just wanted to make sure you realize that that I'm just doing a particular case of a hyperbola that opens to the left and the right I could call it a horizontal hyperbola instead of a vertical one but I wanted to make it clear that there is another type of hyperbola anyway let's draw a graphical representation of all this just to make sure we understand or we aree understand or better understand what the full side points are and where they sit on the hyperbola so those are my axes the asymptotes of this hyperbola are the lines Y is equal to plus or minus B over a we've going on a whoops not using my line tool so that's one and that's the other asymptote and then so the hyperbola will look something like this it'll look something like that it's going to intersect it a comma 0 right there this is going to be a comma 0 and then intersect it - a comma 0 we saw all of this in the previous video it looks something like that and then the focus points are going to sit out here someplace there and there and the focal length this a squared plus B squared the square root of a squared plus B squared that's just this distance right here that distance is the focal length so this is going to be the point F 0 and this is going to be the point minus F 0 now we learned in the last video that one of the definitions of a hyperbola is the locus of all points or the set of all points where if I take the difference of the distances to the two foci that difference will be a constant number so if this is the point X comma Y and it could be any point that satisfies this equation its any point on the hyperbola we know or we are told that if we take this distance right here let's call that D 1 and subtract from that the distance to the other foci call that D 2 that that number is a constant regardless of where we are on the hyperbola in fact the locus of all points are the hyperbola in fact is all of the points that satisfy that condition and we learned in the last video just by taking the distance the difference of the distance we pick this point we said okay what's that distance minus that distance and we figured out that it's 2a so d1 minus d2 is equal to going off the video screen d1 minus d2 is equal to 2a so let's use this fact right here the d1 minus d2 is equal to 2a to try to prove this right there so the first thing to do is figure out what d1 and d2 are just using the distance formula so what's d1 d1 is the distance between this point and this point minus F 0 so what you do is this we just use the distance formula which is really just the Pythagorean theorem so it's the difference difference of the X's so the X distance so it's X minus minus F squared plus the Y distance is y minus 0 so that's just Y squared take the square root of that so that's d1 right there d1 and we want to subtract from that d2 right the difference of the distances and in this case d1 is definitely bigger than d2 or you could take the absolute value if you didn't want to worry about that and so here we get the square root of x minus F X minus F squared plus y squared and what is that equal to well we said that equals to 2a that equals this distance right here so that is equal to 2a now let's see if we can simplify this at all a little interesting thing to do might just be to put this on the other hand side other side of the equation and this can get hairy so I really hope I don't make any careless mistakes so this becomes and I write I might write small to save space this becomes X plus F right minus minus squared plus y squared is equal to 2a plus the square root of x minus F squared plus y squared now to get rid of these radicals to square both sides of this equation the left hand side if you were to square it just becomes X plus F squared plus y squared and then to square this we have to this we square the first term which is 4a squared then we multiply the two terms and multiply that by 2 right we're just taking this whole thing and squaring it so that's and this is just a review of kind of binomial algebra so this is equal to plus 2a times this times 2 is 4 a times the square root of x minus F squared plus y Y squared I want to lose that squared right there and then we Square this term this is just multiplying a binomial so that's equal to we just get rid of the radical sign and I'm just now let me stay in that color for now that's equal to X minus F squared plus y squared and already we it looks like there's some cancellation that we can do we can cancel out there's a y squared on both sides of this equation so let's just cancel those out subtract Y squared from both sides of the equation and let's multiply this term out so this right here is x squared plus 2 X F plus F squared and then that is equal to 4a squared plus 4a times the square root of x minus actually well let me just write X minus F squared plus y squared and then multiply this out plus x squared minus 2x F plus F squared and then let's see what can we cancel out well we have an x squared on both sides of this we subtract x squared from both sides of the equation we have an F squared on both sides of the equation so let's cancel that out and see what can we do to simplify it so we have a minus 2xf and a plus 2x F let's add 2x F to both sides of equation or bring this term over here so if you add 2x F to both sides of this equation see my phone is ringing let me turn it off if you add 2x F to both sides of this equation what do you get you get 4x F remember I just brought this term over thus this left-hand side is equal to 4a squared plus 4a times the square root of x minus f squared plus y squared it's easy to get lost in the out remember all we're doing just to kind of remind you of where we're this whole point was we're just simplifying the difference of the distances between these two points and then see how it relates to the equation of the hyperbola itself the A's and the B's so let's take this for a put it on this side so you get 4x F minus 4a squared is equal to 4a times the square root of well let's just multiply this out because we'll probably have to eventually x squared minus 2x F plus F squared plus y squared that's this just multiply it out that's the Y squared right there we could divide both sides of this by 4 all I'm trying to do is just simplify this as much as possible so then this becomes x F minus a squared is equal to a times the square root of this whole thing x squared minus 2x F plus F squared plus y squared and now we could square both sides of this equation right here and then if you square both sides this side becomes x squared f squared minus 2 a squared X F plus a to the fourth that's this side squared and that's equal to just square the right hand side a squared times the square of the square root is just that expression x squared minus 2x F plus F squared plus y squared this really is quite hairy and let's see what we can do now to simplist divide both sides of this equation by a squared and then you get x squared I'm really just trying to simplify this as much as possible over a squared minus so the a squares cancel out minus 2x F plus a a to the fourth divided by a squared well that's just a squared so a squared is equal to x squared minus 2x F plus F squared plus y squared well good there's something to cancel out there's a minus 2xf on both sides of this equation so let's cancel that out simplify our situation a little bit and let's see we have so we could do is subtract this x squared from this so you get let me rewrite it so you get you get x squared F squared over a squared minus x squared and let's bring this let's bring this Y to this side of the equation 2 so minus y squared that's all I did I just brought that to that side and then let's bring and I'm kind of skipping a couple of steps but I don't want to take too long let's take this a and put it on that side of the equation so we took the X and the y we put D we subtracted that from both sides equation so they ended up on the left-hand side and then if we subtract a squared from both sides this equation this is a fatiguing problem then you get F squared minus a squared I think we're almost there this can simplify to let's see if we can factor out the x squared this becomes F squared over a squared minus 1 times x squared I just factored out the x squared there minus y squared is equal to F squared the focal length squared minus a squared and let's see let's divide both sides of the equation by this expression right there and we get and this is this should start to look familiar we get F squared over a squared minus 1 x squared divided by F squared minus a squared minus y squared over F squared minus a squared is equal to 1 right i divided both sides by this so I just get a 1 on the right hand side let's see if I can simplify this if I multiply the numerator and the denominator by a squared right if I as long as I multiply the numerator the denominator about the same number I'm just multiplying by one so I'm not changing anything so if I do that the numerator becomes F if I multiply it it becomes F squared minus a squared I'm just multiplying that times a squared and the denominator becomes a squared times F squared minus a squared and all of that times x squared minus y squared times F squared minus a squared is equal to 1 this cancels with this and we get something that's starting to look like the equation of a hyperbola my energy is coming back it seems like I see the light at the end of the tunnel we get x squared over a squared minus y squared over F squared minus a is equal to one now this looks a lot like our original equation of the hyperbola which was x squared over a squared minus y squared over B squared is equal to one in fact this is the equation of the hyperbola but instead of writing B squared since we wrote it we we essentially said what is the locus of all points where the difference of the distances to those two foci is equal to 2a and we just played with the algebra for a while it was pretty tiring and I'm impressed if you've gotten this far into the video and we got this equation which should be the equation of the hyperbola and it is the equation of the hyperbola it is this equation so this is the same thing as that so f squared minus a squared or the focal length squared minus a squared is equal to B squared you add a squared to both sides and you get x squared is equal to B squared plus a squared or a squared plus B squared which tells us that the focal length is equal to the square root of this of a squared plus B squared and that's what we set out to figure out in the beginning so hopefully you're now satisfied that the focal length of a hyperbola is the sum of these two denominators and if this was an up and it's also true if it's an upward or a vertical hyperbola and if we're dealing with any lips it's the difference of these two the square root of the difference of these two numbers anyway I'll leave you there that was an exhausting problem I have to go get a glass of water now