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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.