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# Common tangent of circle & hyperbola (5 of 5)

## Video transcript

all right where we left off we really were in the homestretch and now we just have this messy arithmetic that we have to solve to essentially find our slope and then we'll be able to figure out the y-intercept and since we're actually going to have to take the square root of this eventually let's just factor each of these terms and see if this hopefully will simplify into something reasonable into something reasonable so let's just look at the let's just factor 104 I think it'll be useful just to have each of these guys factors so 104 is the same thing as 4 as 4 times 4 times 26 right 4 times 25 is 100 4 times 26 which is the same thing as 2 times 2 times 13 so that's 104 so 104 squared let's just worry about the radical right now so this is going to be this squared so it's going to be 4 times 4 times 2 times 2 times 13 times 13 that's this right over here I squared it so I put each of its factors twice and then you're going to have plus 4 plus 4 times let's factor 495 for 195 it's divisible by 5 let's see 5 goes into 500 five goes into 500 100 times this is 5 less than that so it's going to be 9999 is 9 times 9 times 11 so the 495 the 495 can be factored as 5 times 9 times 11 and then 400 400 is pretty straightforward 400 is going to be times 4 times 4 times 10 times 10 times 10 times 10 and so we can factor out for this expression right here under the radical sign there is a common factor of 4 times 4 here we have a 4 times 4 and there doesn't look to be any other common factor so this is going to be the square root of 16 times the square root times the square root of is this is it looks like we're actually going to have to do some serious multiplication here the square root of 4 times 4 times 169 so let's just multiply that out so we're going to have this is really a painful problem so let's just say let's just say this 4 times let's see 13 times 13 is 169 we're gonna have 169 169 that's the 13 times 13 times 4 times 4 so 4 times 9 is 36 6 times 4 is 24 plus 3 is 27 1 times 4 is 4 plus 2 is 6 so the all of this right over here is going to be 676 676 and then we're going to have let's see 5 times well 5 times not well we know what this is right over here this number right over here is going to be 495 and we're going to multiply that we're going to multiply that times 100 so this is going to be so plus 495 and just add two zeros 1 2 so it becomes 49 thousand five hundred so this is going to become this is going to become square root of 16 is 4 times the square root this will be interesting we're going to do some serious factoring here of 49 thousand five hundred plus 676 is going to be 50 thousand 50 thousand let me see so 6 650 thousand one hundred and seventy-six right if this was just a 500 here then we would get 250 thousand we have another 176 so have the square root of this crazy thing and so let's see if we can factor this let's see if we can factor this this is really really a nasty problem so 50,000 looks like it's divisible looks like it's divisible by 4 so let's see 4 goes into 50,000 176 we're getting some good practice you're on our arithmetic 4 goes into 50 maybe there's a faster way to do this 12 times 12 times 4 12 times 4 is 48 subtract to get a 21 4 goes into 21 5 times you get a 20 subtract you get a 17 4 goes into 17 4 times 4 times 4 is 16 subtract bring down that you get a 16 so it goes twelve thousand five hundred and forty four times and once again this looks like it's divisible by four or definitely is divisible by four and let's see we could this is going to be I'm not want to make any careless mistakes so let me just divide it again I'm tempted to do some of this in my head but we've gotten so far I don't want to make mistakes so 4 goes into 12 3 times 3 times 4 is 12 4 goes into 5 one time 1 times 4 is 4 you have a 14 4 goes into 14 3 times 3 times 4 is 12 you get a to 24 so 31 3136 once again this looks like it's divisible by 4 again this is 4 times I should have just guessed that this whole thing that's going to be divisible by 16 or something so 4 goes into don't want to make a careless mistake 3136 4 goes into 31 7 times you get a 28 subtract you get a 3 33 goes into it 8 times you get a 32 subtract you get a 16 you go exactly four times no remainder so 784 once again this looks like it is divisible by 4 it's getting at least simpler now 4 goes into 784 14 to 7 one time four you get a 38 it goes here 9 times 9 times 4 is 36 you subtract you get a 24 196 times and really if you are going to take the jei I do recommend being able to do this mental arithmetic much faster because I'm clearly not capable of doing it fast enough so you have 196 and 196 this is divisible by 16 I believe let me see 16 goes into 196 it goes into it actually no it's not so let's just do by 4 this is 17 I believe let me look at just for my brain is fried 196 4 times 16 thirty-six it goes into four goes into 36 nine times so this is four times 49 all right so if we're taking the square root of the square root of all of this business this is four times 49 over here so this is the square root of four to the fifth power this is four to the fifth power so over listed let's do it this way we could just take the square root of each of these factors so this in blue so this is going to become four times the square root of this we can just take the square root of each of its factors so it's going to be one two three four five twos so times two to the fifth which is thirty-two times the square root of 49 which is 7 so this term right over here simplify to this whole Pradas whole video has been just arithmetic four times my brain really is fried so 32 times four we have an 8 4 times 3 is 120 well 4 times 3 is 12 and then we want to multiply that times 7 I think you can see my handwriting degrade 7 times 8 is 56 is 56 7 times 2 is 14 plus 5 is 19 bring up the 1 7 times 1 is 7 so it's 896 so all of that stuff in blue was 896 so let's see all of this stuff over here all of this stuff over here was 896 896 now we have to decide to do plus or minus remember we want a positive slope so we want to add the 896 so we're getting M Squared is equal to 104 plus 896 or negative 104 plus 896 so if we subtract 104 from this business over here we're going to get we're going to get 700 792 792 right 792 over over 990 this is what M Squared is equal to and then this looks this looks the visibility's divisible this these are both divisible by 9 so let's divide both of them by 9 let me clear all of this stuff down here just a big arithmetic video but hopefully we'll get the right answer it be painful if I made a careless mistake earlier on and all of this is in vain and I have to redo the video so let's let's hope that isn't the case so let's divide both of these by 9 9 goes into 792 it goes into 79 it goes into 79 8 times 8 times 9 is 72 and then you get 72 so it goes 80 eight times so this is if you divide that by nine you get 88 if you divide this by nine you get 110 now these are both divisible by 11 so if you divide by 11 you get 8 you get 8 over 8 over 10 8 over 10 is the same thing as 4 over 5 unbelievable it all simplify down to 4/5 so the slope of our line M Squared is equal to 4/5 or taking the square root of both sides M is equal to 2 over the square root of 5 so that's the slope of our line and now we just have to go back to one of these equations whichever one we think is simpler and find the y-intercept this is the simpler one so B M is equal to 2 over the square root of 5 B is going to be equal to B is equal to the square root of 9 times M Squared remember M Squared is 4/5 4/5 minus 4 so what is this going to be equal to let me just clear some of this all of this business out here since we don't since well I'll just leave it there maybe it'll be useful in some way if I made a careless mistake hopefully I haven't so this is going to be equal to 9 times 4 is 36 so it's equal to 36 over 5 minus 20 over 5 minus 20 over 5 so this is equal to the square root of 16 over 5 now I'm running out of space so this is equal to so B is equal to the square 3 6 minus 20 is 16 so square root of 16 over square root of 5 or 4 over the square root of 5 and we are done we now know the slope of that tangent line the thing that had a positive slope it's equation is going to be y is equal to 2 over the square root of 5x plus 4 over the square root of 5 and if we want to simplify it because if I remember properly that problem didn't have it in they put everything on the right-hand side let's multiply everything times square root of 5 so you get square root of 5y is equal to 2x plus 4 and then we could subtract these from both sides and we get negative 2x plus the square root of 5y minus 4 minus 4 is equal to 0 so do they have that as one of the choices so it looks like they do if we just multiply everything times negative one if we multiply everything times negative one we get 2x we get 2x minus the square root of 5y plus four is equal to zero and we are done this was probably the most painful problem I've done in my life 2x minus the square root of 5y plus 4 the answer the answer is B