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Current time:0:00Total duration:10:36

Common tangent of circle & hyperbola (4 of 5)

Video transcript

we're in the homestretch we figured out constraint on B in terms of M for that line that's tangent to both the circle and the hyperbola I want to make one clarification here here in the last video I took the principal root when I took the square root and that's because I want the y-intercept to be positive remember when we look at this drawing that I did in the first video over here this y-intercept needs to be positive in order for it to have a positive slope and be tangent to both the circle and the hyperbola in this way and so down here so down here you definitely want to take you definitely want to take the principal root now with that said this is the constraint from the hyperbola this is the constraint from the circle let's set them equal to each other and solve for M and then we'll have our slope of the line so let's set them equal to each other so we have the square root the square root of 9 m squared minus 4 is going to be equal to is going to be equal to negative 4 M plus 4 times the square root of M squared plus 1 so the first thing we can do let's square both sides of this so we get 9m squared minus 4 is equal to this term over here squared so this is 16 M squared and then plus 2 times the product of these two terms so the product is going to be negative 16 M so it's going to be negative 32 M times the square root of M squared plus 1 and then plus Square this term 16 times M squared plus 1 so let's try to simplify this a little bit get maybe the radical on one side of the equation and see if we can simplify this and see if we can simplify this even more so it's a good place to start so let's see this is going to be this is going to be this term right over here is going to be a 16 M squared plus 16 so on the right hand side let me just write it like this we have so we have 9m squared minus 4 is equal to on the right hand side we have 16 plus 16 we have 32 32 M squared plus 16 minus 32 M times the square root of M squared plus 1 and let's see we can subtract we could subtract 9m squared from both sides so we'll subtract 9m squared from both sides and then we can add 4 to both sides we can add 4 to both sides and then we are left with actually let me do it the other way because I want to isolate this on the right hand side let me do it the other way so let's subtract 32 M squared from both sides minus 32 M Squared from both sides 32 M Squared from both oh yeah might as well do this way so subtract 32 for M Squared for both sides and then subtract 16 from both sides subtract 16 from both sides the left hand side 9 minus 32 is going to be negative 23 because 23 plus 9 yep is 2 so this is negative 23 M squared minus 20 is equal to is equal to negative 32 M times the square root of M squared plus 1 now we can square both sides of this equation it's not the cleanest problem in the world but hopefully if we haven't made any mistakes we'll get someplace productive actually let's multiply both sides equation x times negative 1 just to make it positive simplify things so that becomes positive positive and positive you square the left hand side of the equation we get 23 squared let me write it this way we get 23 M Squared squared plus 2 times the products of this so this would be 40 let me just write it 2 times 20 times 23 M squared plus 20 squared which is 400 is going to be equal to 32 squared or I'll write it's 32 M squared times M squared plus one and let's see what we can do over here so this turns into this is just a big tedious problem so I'll just write this is 23 M squared squared plus so this is going to be 40 this is 40 times 23 so 23 times 4 is 92 so it's gonna be plus 920 M squared plus 9 actually let me expand everything out here let me just go straight to the number so 23 times 23 23 squared 3 times 3 is 9 3 times 2 is 6 69 put a 0 here 2 times 3 is 6 2 times 2 is 4 so you get a 9 6 plus 6 is 12 1 plus 4 is 5 so you have 529 529 m to the fourth plus 40 times 23 4 times 20 is 80 4 times 23 is 92 so there's going to be plus 920 M squared plus 400 is equal to so we're gonna have 32 squared let me figure out what 32 squared is 32 times 32 it seems like if you're taking this exam you might have your month you might as well have your multiplication tables memorized to about 50 and I think they do that on purpose so just you get a sense I mean this isn't an exam where the test takers expect everyone to get a perfect the threshold I believe I think the number one score in India is on the order of 80% right which is pretty darn amazing to be able to do roughly the math section in about an hour 80% right it's pretty amazing but I think to make the threshold of getting into these highly selective colleges I think you have to get around 40 or 50 percent correct of it you know forgive me if I got that number wrong but I think it's on that order so they need to do that because you have two or three hundred thousand people in India taking it so you need to make sure that you have a good spread of people so this is what they do to spread people out so we have 2 times 2 is 4 2 times 3 is 6 and they put a 0 here 3 times 2 is 6 3 times 3 is 9 and for example if I if I was preparing for the exam I might even go ahead and you know and you know me I hate to memorize things but if you had to do this type of problem in the order of five on the order of five minutes you might want to memorize these type of things ahead of time you know the the the y-intercept for a line that intercepts the hyperbola y-intercept for a line that intercepts a circle who knows but anyway but in general I don't think it's good to memorize in life because when you're actually doing math problems in your life what matters more as you understand the underlying meaning and you normally have plenty of time to do it in your real life these exams are kind of an artificial circumstance but anyway 4 plus 0 is 4 6 plus 6 is 12 and then you have so it's 10 24 is equal to 1024 that's the 32 M squared times M squared plus 1 or another way this is 1024 1024 M to the fourth M the fourth plus 1024 M M Squared and now we are in the homestretch so let's subtract 529 m to the fourth from both sides so minus 529 m to the four this is extremely tedious minus 529 m to the fourth frankly at this point in the problem you're better off if you just want to do it for speed just trying out the choices they gave you and figure it out figuring out which M's and B satisfy that but anyway let's just let's just move four on which M's which M's is satisfied that probably be faster but let's just solve it properly so minus 529 m to the fourth plus and then we also want to subtract 920 M Squared so let's also subtract 920 M squared and so on this side were just left let's also subtract let's also subtract a 400 just to make this a proper quadratic and also going to put a 400 here so the whole left side simplifies to 0 is equal to 1024 minus 529 let's see 1024 minus 524 would have been 500 would have been 500 and so it's going to be 5 less than that so it's going to be 495 and to the 4 did I do that right 1024 minus 524 would be 500 but I'm subtracting 5 more than that so that is going to be 4 95 and then 1024 - 9 20 seats 80 it'll be 80 plus 24 80 plus us so it's going to be plus 100 and 4m squared - 400 - 400 is equal to zero so we have just a straight-up quadratic equation over here and you might not recognize it but this is this is the same thing m to the fourth is the same thing as M Squared squared so let's just solve it so we will get M Squared we're not solving for x anymore we're gonna get M Squared using the quadratic formula is you could substitute X is equal to M Squared and this will just become a natural quadratic then M Squared is equal to negative B negative 104 plus or minus the square root more fun math for us without a calculator plus or minus square root of 104 squared minus 4 times a which is 495 times C which is negative 400 so that'll make this a positive and you have Plus you have a 400 right over here and then all of that over 2 times 495 - times 495 is what that's going to be 800 it's going to be what's going to be 10 less than a thousand so it's gonna be 990 990 so let's try to see if we can evaluate this actually I'll stop this problem here I've already crossed the 10-minute threshold in the next video we'll just grind through this mathematics it's really just arithmetic at this point and figure out what M is going to be equal to