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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 home stretch. 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 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. So let's just factor 104. I think it'll be useful just have each of these guys factored. So 104 is the same thing 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 times-- let's factor 495. 495 is divisible by 5. Let's see. 5 goes into 500 100 times. This is five less than that, so it's going to be 99. 99 is 9 times 11. So the 495 can be factored as 5 times 9 times 11. And then 400. 400's pretty straightforward. 400 is going to be times 4 times 10 times 10. And so we can factor out-- so 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 of-- it looks like we're actually going to have to do some serious multiplication here. The square root of 4 times 169. So let's just multiply that out. This is really a painful problem. Let's see. 13 times 13 is 169. We're going to have 169, that's the 13 times 13 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 all of this right over here is going to be 676. And then we're going to have, let's see, 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 times 100. So this is going to be plus 495 and just add two 0's. , One two. So it becomes 49,500. So this is going to become-- the square root of 16 is 4, times the square root-- this will be interesting. We're going to have to do some serious factoring here-- of 49,500 plus 676 is going to be 50,176. If this was just a 500 here, then we would get to 50,000, and we have another 176. So times the square root of this crazy thing. And so let's see if we can factor this. This is really, really a nasty problem. So 50,000 looks like it's divisible by 4. So let's see. 4 goes into 50,176. We're getting some good practice here on our arithmetic. 4 goes into 50-- maybe there's a faster way to do this-- 12 times. 12 times 4 is 48. Subtract, you get 21. 4 goes into 21 five times. You get a 20. Subtract, you get a 17. 4 goes into 17 four times. 4 times 4 is 16. Subtract, bring down, then you get a 16. So it goes 12,544 times. And once again this looks like it's divisible by 4. It definitely is divisible by 4. And let's see. This is going to be-- I don't 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 three times. 3 times 4 is 12. 4 goes into 5 one time. 1 times 4 is 4. You have 14. 4 goes into 14 three times. 3 times 4 is 12. You get a 2, 24. So 3136. Once again, this looks like it's divisible by 4 again. I should have just guessed that this whole thing was 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 seven times. You get a 28. Subtract, you get a 3. 33 goes into it eight 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. 4 goes into 7 one time. 4. You get a 38. It goes here nine times. 9 times 4 is 36. You subtract, you get a 24. 196 times. And really, if you are going to take the JEE, 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 it by 4. This is 17, I believe. Let me look. My brain is fried. 196 four times. 16, 36. 4 goes into 36 nine times, so this is 4 times 49. All right. So if we're taking the square root of the square root of all of this business, this is 4 times 49 over here. So this is the square root of 4 to the fifth power. So let's just do it this way. We could just take the square root of each of these factors. So this in blue is going to become 4 times the square root of this. We could just take the square root of each of its factors. So it's going to be 1, 2, 3, 4, 5 2's. So times 2 to the fifth, which is 32, times the square root of 49, which is 7. So this term right over here simplified to-- this whole video has been just arithmetic. My brain really is fried-- so 32 times 4. We have an 8. 4 times 3 is 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. 7 times 2 is 14, plus 5 is a 19. Bring up the one. 7 times 1 is 7, so it's 896. So all that stuff in blue was 896. So let's see. All of this stuff over here was 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 negative 104 plus 896. So if we subtract 104 from this business over here, we're going to get 792 over 990. This is what m squared is equal to. These are both divisible by nine. So let's divide both of them by 9. Let me clear all of this stuff down here. It's Just a big arithmetic video, but hopefully we'll get the right answer. It would 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 hope that isn't the case. So let's divide both of these by 9. 9 goes Into 792. It goes int 79 8 times. 8 times 9 is 72. And then you get 72. So it goes 88 times. You divide that by 9 you get 88. If you divide this by 9, you get 110. Now these are both divisible by 11. So if you divide by 11 you get 8 over 10. 8 over 10 is the same thing as 4 over 5. Unbelievable. It all simplified 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 m is equal to 2 over the square root of 5. b is going to be 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? Now let me just clear all of this business out here since we don't-- 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. So this is equal to the square root of 16 over 5. I'm running out of space. So b is equal to-- 36 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, its 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 is equal to zero. So do they have that as one of the choices? So it looks like they do if we just multiply everything times negative 1. If we multiply everything times negative 1 we get 2x minus the square root of 5y plus 4 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 is B.