IIT JEE trigonometric constraints

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Kind: captions
Language: en

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The number of values of theta
in the interval from negative pi

00:00:04.550 --> 00:00:06.270
over 2 to pi over
2-- and it's not

00:00:06.270 --> 00:00:09.150
including those because we have
curly parentheses around it

00:00:09.150 --> 00:00:13.200
--such that theta does
not equal n pi over 5.

00:00:13.200 --> 00:00:14.870
So it's not a
multiple of pi over 5,

00:00:14.870 --> 00:00:18.880
for n equals 0 plus or
minus 1, plus or minus 2.

00:00:18.880 --> 00:00:23.330
And tangent of theta is equal
to cotangent of 5 theta,

00:00:23.330 --> 00:00:28.160
as well as sine of 2 theta is
equal to cosine of 4 theta is--

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We have to find the
number of values of theta

00:00:30.560 --> 00:00:33.176
that satisfy all
of these things.

00:00:33.176 --> 00:00:34.550
So just from the
get-go, it seems

00:00:34.550 --> 00:00:36.440
like we're going to
have to solve for thetas

00:00:36.440 --> 00:00:38.800
and convert between sine and
cosine which is essentially

00:00:38.800 --> 00:00:40.500
what you're doing between
tangent and cotangent.

00:00:40.500 --> 00:00:41.880
We'll do that in a little bit.

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And those of you who
know, if you're actually

00:00:43.280 --> 00:00:44.920
taking an exam under
time pressure, I

00:00:44.920 --> 00:00:48.530
don't recommend you reproving
your trig identities

00:00:48.530 --> 00:00:49.570
from first principles.

00:00:49.570 --> 00:00:53.290
But for education purposes,
I always like to do it.

00:00:53.290 --> 00:00:56.030
Let me just draw a right
triangle right over here.

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If you're actually going
to take the IIT-JEE exam,

00:00:59.020 --> 00:01:01.210
I recommend having
most of the trig

00:01:01.210 --> 00:01:03.160
identities at your fingertips.

00:01:03.160 --> 00:01:05.550
But let's say that's
a right triangle.

00:01:05.550 --> 00:01:08.259
And let's call that theta.

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And then, this angle
right over, here

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is going to be pi over 2 minus
theta, if we're in radians.

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90 degrees right
here is pi over 2.

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The whole triangle
right over here

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is 180 degrees, which is pi.

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So this is pi over 2.

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And so these two guys
are going to have

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to add up to the
other pi over 2.

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So this is pi over
2 minus theta.

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So let's think about
what cosine of theta is.

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Cosine of theta, "sou-cah-toa,"
cosine of theta is this side.

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Let's call this a, b, and c.

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It is going to be
the adjacent side.

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So cosine is adjacent.

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Let me write this
down, "sou-cah-toa."

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Cosine is adjacent
over hypotenuse.

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So cosine of theta is b over
c, adjacent over hypotenuse,

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is equal to b over c.

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But what's also
equal to b over c?

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If we look at this
angle over here,

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from this angle's point of
view, b is the opposite side.

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So it's opposite over hypotenuse
for this angle's point of view.

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So it's also equal to the
sine of pi over 2 minus theta.

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So we got our first
main identity here,

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that the cosine of theta
is equal to the sine of pi

00:02:23.660 --> 00:02:25.659
over 2 minus theta.

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And we could go the
other way around,

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use the exact same logic to
get that the sine of theta

00:02:31.650 --> 00:02:36.732
is equal to the cosine
of pi over 2 minus theta.

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So that'll be pretty
useful when we

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try to solve this
equation right over here.

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And we can even use it
when we solve this one.

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Because if I were to write
down the cotangent of 5 theta--

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let me write this down.

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So the cotangent of-- not
cosine-- the cotangent

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of 5 theta, this is
the exact same thing.

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The cotangent of five theta
is the exact same thing

00:03:00.020 --> 00:03:05.730
as the cosine of 5 theta
over the sine of 5 theta.

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It's just 1 over the tangent.

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And this is the same thing.

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I can convert the cosine into
sine using this identity up

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here.

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This is the sine of pi
over 2 minus 5 theta

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over the cosine of pi
over 2 minus 5 theta.

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And for here, I'm just using
this identity over here.

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And this is the exact
same thing as the tangent

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of pi over 2 minus 5 theta.

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And since we're going
to actually have

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to solve this
equation eventually,

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and we want to make sure
we get all of the solutions

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to this equation.

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One thing that you
might want to remember

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is, when you take a
tangent of an angle--

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and I'll just draw
the unit circle here--

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so if I have some angle--
let me draw my axes.

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And let's say that this
right here is theta.

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You probably remember that
this theta is essentially

00:04:01.380 --> 00:04:03.840
the slope of the line formed.

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It's opposite over adjacent.

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It's the slope of
the line formed,

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or by the radius
in the unit circle.

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So this theta is going to
have the exact same tangent

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as if we add 180 degrees
or if we add pi to this.

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So if you add pi, you
get theta plus pi.

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So this whole angle right
over here is theta plus pi.

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And its tangent is going
to be the same thing.

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It's the slope of the radius
is going to be the same.

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So you can add multiples
of pi to a tangent value,

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and you're going to get
the exact same value.

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So let me put some
multiples of pi

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over here, so that
we can make sure

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that we get all
of the solutions.

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So with that said,
with this written,

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we can now solve this
first equation over here.

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And then, we can worry about
some of these other constraints

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later.

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So tangent of theta is equal
to cotangent of 5 theta.

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Well, so we can write tangent
of theta is equal to--

00:05:00.475 --> 00:05:02.850
and instead of writing cotangent
of 5 theta, we can write

00:05:02.850 --> 00:05:08.420
is equal to the tangent
of pi over 2 minus 5

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theta plus integer
multiples of pi.

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And so the tangent of this
is equal to tangent of that.

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These things could be
equal to each other.

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So we get theta is equal to pi
over 2 minus 5 theta plus n pi.

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Or at minimum, this
could be equal,

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when you take the
tangent of this,

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it's equal to the tangent
of either the same thing,

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or we could add
multiples of pi to it.

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You could view it as either way.

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But now, we just
have to solve this.

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We can add 5 pi to both
sides of this equation.

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We get, sorry, we
could add 5 theta

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to both sides of this equation.

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We're adding this to both sides.

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So you get 6 theta is equal
to pi over 2 minus n pi,

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minus multiples of pi, or
sorry, plus multiples of pi.

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There's a plus right over there.

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Divide both sides by 6.

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I get theta is equal to pi
over 12 plus n times pi over 6.

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Or just to make the
fraction adding easier,

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this is the same thing
as pi over 12 plus 2.

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Let me write it this way.

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Plus 2 n pi over 12.

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This is the same
thing as n pi over 6.

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I just multiplied the
numerator and denominator by 2.

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And this is the
exact same thing.

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You could imagine a 1
coefficient out here.

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This is the same thing as
2n plus 1 times pi over 12.

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So that's all of the solutions
for this first equation

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right over there.

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Now, let's do the same thing
for this second equation.

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And then, we'll see where
they overlap in this range.

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And we'll take out anything that
satisfies this criteria right

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over here.

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So the second equation.

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Let me write it over here.

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We have sine of 2 theta.

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Let me write it over here first.

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Cosine of 4 theta.

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What does cosine of
4 theta is equal to?

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Cosine of 4 theta, using
the same exact logic,

00:07:06.480 --> 00:07:12.960
is equal to sine of
pi over 2 minus theta.

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And of course, when we take
a sine or cosine or we take

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really any trig identity or
any trig function of any angle,

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you're going to
get the same value

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if you add multiples of 2 pi.

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So let me just add
multiples of 2 pi

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here, just so we can
make sure that we

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get all the possible solutions.

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So this is what cosine
of 4 theta is equal to.

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So let us-- oh,
let me be careful.

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Cosine of 4 theta is not equal
to pi over 2 minus theta.

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That would be cosine of theta.

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If it's cosine of 4 theta,
it's pi over 2 minus 4 theta

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plus multiples of 2 pi.

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So plus 2 pi n.

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Because obviously you
add multiples of 2 pi,

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you're going to go back
to the same angles.

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So you go back up to
this equation up here.

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We get sine of 2 theta is
equal to cosine of 4 theta.

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Which is the same thing
as this over here,

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which is equal to sine of
pi over 2 minus 4 theta plus

00:08:13.490 --> 00:08:15.200
2 pi n.

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And so these evaluate
to the same thing.

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Let's set them
equal to each other.

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2 theta is equal to pi over
2 minus 4 pi plus 2 pi n.

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Let's add 4 pi to both
sides of this equation.

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We get 6 theta.

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6 theta is equal to pi over 2.

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So this is canceled out.

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Plus 2 pi n.

00:08:46.370 --> 00:08:48.465
Let's divide both sides by 6.

00:08:51.730 --> 00:09:02.880
We get theta is equal to
pi over 12 plus pi over 3n.

00:09:02.880 --> 00:09:08.249
2 divided by 6 is pi n over 3.

00:09:08.249 --> 00:09:10.290
Now, we can put it over
a common denominator just

00:09:10.290 --> 00:09:12.000
to simplify things.

00:09:12.000 --> 00:09:13.110
So we have over 12.

00:09:13.110 --> 00:09:14.830
We have this pi.

00:09:14.830 --> 00:09:20.800
pi n over 3 is the same
thing as plus 4n pi over 12.

00:09:20.800 --> 00:09:23.150
Or we could write
this as being, this

00:09:23.150 --> 00:09:33.230
is equal to 4n plus
1 times pi over 12.

00:09:33.230 --> 00:09:34.680
So now, we just
need to see where

00:09:34.680 --> 00:09:38.287
there is overlap between this
solution and that solution.

00:09:38.287 --> 00:09:40.620
Remember, we just have to
count the number of solutions.

00:09:40.620 --> 00:09:42.510
We actually don't have
to find the solutions.

00:09:42.510 --> 00:09:45.010
And actually if you pretty savvy
and do things in your head,

00:09:45.010 --> 00:09:48.650
just at this point, you could
think about how many of these

00:09:48.650 --> 00:09:51.700
you're going to have
between negative pi over 2

00:09:51.700 --> 00:09:52.360
and pi over 2.

00:09:52.360 --> 00:09:54.390
If you remember, that was the
range that we're working with.

00:09:54.390 --> 00:09:56.690
Between negative pi over 2
and pi over 2, not including

00:09:56.690 --> 00:09:57.700
those two.

00:09:57.700 --> 00:10:00.620
And then you can figure out,
and actually every one of these

00:10:00.620 --> 00:10:02.910
is also going to
be one of these.

00:10:02.910 --> 00:10:04.900
Because no matter what
you set n equal to,

00:10:04.900 --> 00:10:07.240
if you set n equal
to twice that,

00:10:07.240 --> 00:10:09.090
you're going to have
the equivalent thing.

00:10:09.090 --> 00:10:12.960
So any of these are
going to be any of these.

00:10:12.960 --> 00:10:14.760
Anything that
satisfies this equation

00:10:14.760 --> 00:10:17.660
will also satisfy this
equation right here.

00:10:17.660 --> 00:10:20.400
So really, we can just count
how many are in this one.

00:10:20.400 --> 00:10:22.010
But just to make
it clear, I'm going

00:10:22.010 --> 00:10:24.500
to find all of them that
satisfy this equation.

00:10:24.500 --> 00:10:26.375
And then we're going to
see how many of those

00:10:26.375 --> 00:10:27.800
satisfy this equation as well.

00:10:27.800 --> 00:10:28.890
But the fast way
to do it, just find

00:10:28.890 --> 00:10:30.010
the ones that satisfy this one.

00:10:30.010 --> 00:10:32.010
Or just count the ones
that satisfy this one.

00:10:32.010 --> 00:10:34.140
And you've done the problem.

00:10:34.140 --> 00:10:36.250
So let's just count.

00:10:36.250 --> 00:10:38.035
So let's just start
at n equals 0.

00:10:40.590 --> 00:10:44.870
If n is equal to 0-- and
I'm using this n up here.

00:10:44.870 --> 00:10:45.960
I won't even write that.

00:10:45.960 --> 00:10:49.350
If n is equal to 0, we
just have pi over 12.

00:10:49.350 --> 00:10:53.720
If n is 1, we get 3 pi over 12.

00:10:56.490 --> 00:11:00.870
If n is 2, we get 5 pi over 12.

00:11:00.870 --> 00:11:02.556
And we can't make n equal 3.

00:11:02.556 --> 00:11:03.930
Because if n was
equal to 3, this

00:11:03.930 --> 00:11:07.410
would be 7 pi over 12 which
is greater than pi over 2.

00:11:07.410 --> 00:11:09.480
That's greater
than 6 pi over 12.

00:11:09.480 --> 00:11:10.660
So we can't have that.

00:11:10.660 --> 00:11:12.440
We could also go to negative n.

00:11:12.440 --> 00:11:13.940
We could also have negative n.

00:11:13.940 --> 00:11:20.195
If n is negative 1, then this
becomes negative pi over 12.

00:11:22.700 --> 00:11:30.390
If n is negative 2, this
becomes negative 3 pi over 12.

00:11:30.390 --> 00:11:32.890
And if n is negative
3, then this

00:11:32.890 --> 00:11:35.970
becomes a negative 5 pi over 12.

00:11:35.970 --> 00:11:39.930
And once again, we can't go
to n is equal to negative 4,

00:11:39.930 --> 00:11:41.380
because then we
get negative 7 pi

00:11:41.380 --> 00:11:43.920
over 12 which is
out of our range.

00:11:43.920 --> 00:11:45.800
So this satisfies
this equation up here.

00:11:45.800 --> 00:11:49.960
Now, which of these are
overlap with this over here?

00:11:49.960 --> 00:11:52.480
So let's set n equal to 0.

00:11:52.480 --> 00:11:54.720
We get pi over 12.

00:11:54.720 --> 00:12:00.700
If n is equal to 1,
we get 5 pi over 12.

00:12:00.700 --> 00:12:02.760
We can't set n equal to
2, because then we'll

00:12:02.760 --> 00:12:05.240
get out of our range,
we'll go above pi over 2.

00:12:05.240 --> 00:12:09.740
If n is equal to negative 1,
we get negative 3 pi over 12.

00:12:09.740 --> 00:12:14.450
If n is equal to negative
2, this is negative 8.

00:12:14.450 --> 00:12:16.400
Then, we get to
negative 7 pi over 12,

00:12:16.400 --> 00:12:18.850
which is smaller than
negative pi over 2.

00:12:18.850 --> 00:12:20.250
So that doesn't count.

00:12:20.250 --> 00:12:21.480
So there are three solutions.

00:12:21.480 --> 00:12:23.720
Or if we look over here,
we have three solutions.

00:12:23.720 --> 00:12:26.200
We have one, two,
three solutions.

00:12:26.200 --> 00:12:29.080
And none of them satisfy this.

00:12:29.080 --> 00:12:32.330
None of them are
multiples of n pi over 5.

00:12:32.330 --> 00:12:36.869
So the answer to
our problem is 3.

00:12:36.869 --> 00:12:39.410
And you really could have just
done the second equation right

00:12:39.410 --> 00:12:39.910
over here.

00:12:39.910 --> 00:12:42.576
And realize that anything that's
a solution in this top equation

00:12:42.576 --> 00:12:44.660
would be a solution to
this bottom equation.

00:12:44.660 --> 00:12:46.660
And you would have just
counted these solutions.

00:12:46.660 --> 00:12:48.760
And you would have been done.


IIT JEE circle hyperbola common tangent part 5

WEBVTT
Kind: captions
Language: en

00:00:00.561 --> 00:00:01.060
All right.

00:00:01.060 --> 00:00:03.770
Where we left off we really
were in the home stretch.

00:00:03.770 --> 00:00:05.840
And now we just have
this messy arithmetic

00:00:05.840 --> 00:00:08.350
that we have to solve to
essentially find our slope.

00:00:08.350 --> 00:00:10.780
And then we'll be able to
figure out the y-intercept.

00:00:10.780 --> 00:00:13.130
And since we're actually going
to take the square root of this

00:00:13.130 --> 00:00:15.171
eventually, let's just
factor each of these terms

00:00:15.171 --> 00:00:18.070
and see if this
hopefully will simplify

00:00:18.070 --> 00:00:21.670
into something reasonable.

00:00:21.670 --> 00:00:23.915
So let's just factor 104.

00:00:23.915 --> 00:00:25.290
I think it'll be
useful just have

00:00:25.290 --> 00:00:26.510
each of these guys factored.

00:00:26.510 --> 00:00:35.550
So 104 is the same thing as
4 times 4 times 26, right?

00:00:35.550 --> 00:00:38.040
4 times 25 is 100.

00:00:38.040 --> 00:00:45.970
4 times 26, which is the same
thing as 2 times 2 times 13.

00:00:45.970 --> 00:00:47.650
So that's 104.

00:00:47.650 --> 00:00:48.296
So 104 squared.

00:00:48.296 --> 00:00:50.170
Let's just worry about
the radical right now.

00:00:50.170 --> 00:00:51.669
So this is going
to be this squared.

00:00:51.669 --> 00:00:58.080
So it's going to be 4 times 4
times 2 times 2 times 13 times

00:00:58.080 --> 00:00:58.670
13.

00:00:58.670 --> 00:01:00.140
That's this right over here.

00:01:00.140 --> 00:01:03.340
I squared it, so I put
each of its factors twice.

00:01:03.340 --> 00:01:06.590
And then you're
going to have plus 4

00:01:06.590 --> 00:01:09.432
times-- let's factor 495.

00:01:09.432 --> 00:01:12.880
495 is divisible by 5.

00:01:12.880 --> 00:01:15.386
Let's see.

00:01:15.386 --> 00:01:17.950
5 goes into 500 100 times.

00:01:17.950 --> 00:01:21.380
This is five less than that,
so it's going to be 99.

00:01:21.380 --> 00:01:26.150
99 is 9 times 11.

00:01:26.150 --> 00:01:33.340
So the 495 can be factored
as 5 times 9 times 11.

00:01:33.340 --> 00:01:35.592
And then 400.

00:01:35.592 --> 00:01:36.800
400's pretty straightforward.

00:01:36.800 --> 00:01:46.920
400 is going to be times
4 times 10 times 10.

00:01:46.920 --> 00:01:50.200
And so we can factor out--
so this expression right here

00:01:50.200 --> 00:01:54.610
under the radical sign, there
is a common factor of 4 times

00:01:54.610 --> 00:01:55.750
4 here.

00:01:55.750 --> 00:01:59.510
We have a 4 times
4, and there doesn't

00:01:59.510 --> 00:02:01.457
look to be any
other common factor.

00:02:01.457 --> 00:02:03.040
So this is going to
be the square root

00:02:03.040 --> 00:02:09.865
of 16 times the
square root of-- it

00:02:09.865 --> 00:02:11.490
looks like we're
actually going to have

00:02:11.490 --> 00:02:13.370
to do some serious
multiplication here.

00:02:13.370 --> 00:02:18.240
The square root of 4 times 169.

00:02:18.240 --> 00:02:21.196
So let's just multiply that out.

00:02:21.196 --> 00:02:22.570
This is really a
painful problem.

00:02:25.180 --> 00:02:25.680
Let's see.

00:02:25.680 --> 00:02:29.720
13 times 13 is 169.

00:02:29.720 --> 00:02:38.320
We're going to have 169,
that's the 13 times 13 times 4.

00:02:38.320 --> 00:02:41.410
So 4 times 9 is 36.

00:02:41.410 --> 00:02:46.830
6 times 4 is 24, plus 3 is 27.

00:02:46.830 --> 00:02:50.190
1 times 4 is 4, plus 2 is 6.

00:02:50.190 --> 00:02:57.140
So all of this right over
here is going to be 676.

00:02:57.140 --> 00:03:01.210
And then we're going
to have, let's see,

00:03:01.210 --> 00:03:04.080
5 times not-- well we know
what this is right over here.

00:03:04.080 --> 00:03:09.090
This number right over
here is going to be 495.

00:03:09.090 --> 00:03:14.730
And we're going to
multiply that times 100.

00:03:14.730 --> 00:03:21.390
So this is going to be plus
495 and just add two 0's. ,

00:03:21.390 --> 00:03:22.040
One two.

00:03:22.040 --> 00:03:24.660
So it becomes 49,500.

00:03:24.660 --> 00:03:28.290
So this is going to become--
the square root of 16

00:03:28.290 --> 00:03:32.149
is 4, times the square root--
this will be interesting.

00:03:32.149 --> 00:03:34.440
We're going to have to do
some serious factoring here--

00:03:34.440 --> 00:03:49.900
of 49,500 plus 676 is
going to be 50,176.

00:03:49.900 --> 00:03:53.070
If this was just a 500 here,
then we would get to 50,000,

00:03:53.070 --> 00:03:55.020
and we have another 176.

00:03:55.020 --> 00:03:58.460
So times the square root
of this crazy thing.

00:03:58.460 --> 00:04:02.130
And so let's see if
we can factor this.

00:04:02.130 --> 00:04:05.660
This is really, really
a nasty problem.

00:04:05.660 --> 00:04:11.149
So 50,000 looks like
it's divisible by 4.

00:04:11.149 --> 00:04:11.690
So let's see.

00:04:11.690 --> 00:04:15.316
4 goes into 50,176.

00:04:15.316 --> 00:04:17.649
We're getting some good
practice here on our arithmetic.

00:04:17.649 --> 00:04:19.440
4 goes into 50-- maybe
there's a faster way

00:04:19.440 --> 00:04:21.310
to do this-- 12 times.

00:04:21.310 --> 00:04:26.510
12 times 4 is 48.

00:04:26.510 --> 00:04:28.170
Subtract, you get 21.

00:04:28.170 --> 00:04:30.640
4 goes into 21 five times.

00:04:30.640 --> 00:04:31.400
You get a 20.

00:04:31.400 --> 00:04:33.130
Subtract, you get a 17.

00:04:33.130 --> 00:04:36.260
4 goes into 17 four times.

00:04:36.260 --> 00:04:38.080
4 times 4 is 16.

00:04:38.080 --> 00:04:40.290
Subtract, bring down,
then you get a 16.

00:04:40.290 --> 00:04:44.860
So it goes 12,544 times.

00:04:44.860 --> 00:04:48.330
And once again this looks
like it's divisible by 4.

00:04:48.330 --> 00:04:51.130
It definitely is divisible by 4.

00:04:51.130 --> 00:04:52.600
And let's see.

00:04:52.600 --> 00:04:54.590
This is going to
be-- I don't want

00:04:54.590 --> 00:04:57.640
to make any careless mistakes,
so let me just divide it again.

00:04:57.640 --> 00:04:59.390
I'm tempted to do some
of this in my head,

00:04:59.390 --> 00:05:02.440
but we've gotten so far I
don't want to make mistakes.

00:05:02.440 --> 00:05:04.430
So 4 goes into 12 three times.

00:05:04.430 --> 00:05:05.960
3 times 4 is 12.

00:05:05.960 --> 00:05:08.080
4 goes into 5 one time.

00:05:08.080 --> 00:05:10.370
1 times 4 is 4.

00:05:10.370 --> 00:05:11.430
You have 14.

00:05:11.430 --> 00:05:13.220
4 goes into 14 three times.

00:05:13.220 --> 00:05:15.140
3 times 4 is 12.

00:05:15.140 --> 00:05:17.180
You get a 2, 24.

00:05:17.180 --> 00:05:20.480
So 3136.

00:05:20.480 --> 00:05:23.540
Once again, this looks like
it's divisible by 4 again.

00:05:25.484 --> 00:05:27.900
I should have just guessed
that this whole thing was going

00:05:27.900 --> 00:05:30.070
to be divisible by
16 or something.

00:05:30.070 --> 00:05:35.370
So 4 goes into-- don't want to
make a careless mistake-- 3136.

00:05:35.370 --> 00:05:37.780
4 goes into 31 seven times.

00:05:37.780 --> 00:05:39.360
You get a 28.

00:05:39.360 --> 00:05:40.752
Subtract, you get a 3.

00:05:40.752 --> 00:05:42.440
33 goes into it eight times.

00:05:42.440 --> 00:05:43.450
You get a 32.

00:05:43.450 --> 00:05:45.560
Subtract, you get a 16.

00:05:45.560 --> 00:05:48.010
You go exactly four
times, no remainder.

00:05:48.010 --> 00:05:50.390
So 784.

00:05:50.390 --> 00:05:54.194
Once again, this looks
like it is divisible by 4.

00:05:54.194 --> 00:05:55.610
It's getting at
least simpler now.

00:05:55.610 --> 00:05:58.230
4 goes into 784.

00:05:58.230 --> 00:05:59.280
4 goes into 7 one time.

00:05:59.280 --> 00:06:00.460
4.

00:06:00.460 --> 00:06:01.910
You get a 38.

00:06:01.910 --> 00:06:03.270
It goes here nine times.

00:06:03.270 --> 00:06:04.830
9 times 4 is 36.

00:06:04.830 --> 00:06:06.850
You subtract, you get a 24.

00:06:06.850 --> 00:06:11.026
196 times.

00:06:11.026 --> 00:06:12.900
And really, if you are
going to take the JEE,

00:06:12.900 --> 00:06:16.540
I do recommend being able to
do this mental arithmetic much

00:06:16.540 --> 00:06:20.250
faster because I'm clearly
not capable of doing it

00:06:20.250 --> 00:06:21.240
fast enough.

00:06:21.240 --> 00:06:22.920
So you have 196.

00:06:22.920 --> 00:06:26.671
And 196, this is divisible
by 16, I believe.

00:06:26.671 --> 00:06:27.170
Let me see.

00:06:27.170 --> 00:06:30.030
16 goes into 196.

00:06:30.030 --> 00:06:32.490
It goes into it--
actually no it's not.

00:06:32.490 --> 00:06:33.670
So let's just do it by 4.

00:06:33.670 --> 00:06:34.740
This is 17, I believe.

00:06:34.740 --> 00:06:36.500
Let me look.

00:06:36.500 --> 00:06:37.440
My brain is fried.

00:06:37.440 --> 00:06:39.030
196 four times.

00:06:39.030 --> 00:06:41.970
16, 36.

00:06:41.970 --> 00:06:48.910
4 goes into 36 nine times,
so this is 4 times 49.

00:06:48.910 --> 00:06:49.940
All right.

00:06:49.940 --> 00:06:54.270
So if we're taking the square
root of the square root of all

00:06:54.270 --> 00:06:57.860
of this business, this
is 4 times 49 over here.

00:06:57.860 --> 00:07:01.575
So this is the square root
of 4 to the fifth power.

00:07:04.800 --> 00:07:06.200
So let's just do it this way.

00:07:06.200 --> 00:07:09.430
We could just take the square
root of each of these factors.

00:07:09.430 --> 00:07:11.790
So this in blue
is going to become

00:07:11.790 --> 00:07:14.730
4 times the square root of this.

00:07:14.730 --> 00:07:17.160
We could just take the square
root of each of its factors.

00:07:17.160 --> 00:07:21.200
So it's going to be
1, 2, 3, 4, 5 2's.

00:07:21.200 --> 00:07:24.510
So times 2 to the fifth,
which is 32, times the square

00:07:24.510 --> 00:07:27.370
root of 49, which is 7.

00:07:27.370 --> 00:07:29.560
So this term right
over here simplified

00:07:29.560 --> 00:07:33.870
to-- this whole video
has been just arithmetic.

00:07:33.870 --> 00:07:37.900
My brain really is
fried-- so 32 times 4.

00:07:37.900 --> 00:07:38.986
We have an 8.

00:07:38.986 --> 00:07:43.100
4 times 3 is 4 times 3 is 12.

00:07:43.100 --> 00:07:45.820
And then we want to
multiply that times 7.

00:07:45.820 --> 00:07:48.100
I think you can see my
handwriting degrade.

00:07:48.100 --> 00:07:50.795
7 times 8 is 56.

00:07:53.740 --> 00:07:58.780
7 times 2 is 14, plus 5 is a 19.

00:07:58.780 --> 00:08:00.210
Bring up the one.

00:08:00.210 --> 00:08:03.380
7 times 1 is 7, so it's 896.

00:08:03.380 --> 00:08:07.190
So all that stuff
in blue was 896.

00:08:07.190 --> 00:08:08.110
So let's see.

00:08:08.110 --> 00:08:15.970
All of this stuff
over here was 896.

00:08:15.970 --> 00:08:17.720
Now we have to decide
to do plus or minus.

00:08:17.720 --> 00:08:19.800
Remember, we want
a positive slope.

00:08:19.800 --> 00:08:22.050
So we want to add the 896.

00:08:22.050 --> 00:08:32.059
So we're getting m squared is
equal to negative 104 plus 896.

00:08:32.059 --> 00:08:36.409
So if we subtract 104 from
this business over here,

00:08:36.409 --> 00:08:47.760
we're going to get 792 over 990.

00:08:47.760 --> 00:08:50.100
This is what m
squared is equal to.

00:08:54.060 --> 00:08:57.270
These are both
divisible by nine.

00:08:57.270 --> 00:08:59.170
So let's divide
both of them by 9.

00:08:59.170 --> 00:09:00.910
Let me clear all of
this stuff down here.

00:09:00.910 --> 00:09:03.342
It's Just a big
arithmetic video,

00:09:03.342 --> 00:09:05.050
but hopefully we'll
get the right answer.

00:09:05.050 --> 00:09:07.590
It would be painful if I made
a careless mistake earlier on

00:09:07.590 --> 00:09:09.930
and all of this is in vain
and I have to redo the video.

00:09:09.930 --> 00:09:12.480
So let's hope that
isn't the case.

00:09:12.480 --> 00:09:14.153
So let's divide
both of these by 9.

00:09:14.153 --> 00:09:17.090
9 goes Into 792.

00:09:17.090 --> 00:09:22.020
It goes int 79 8 times.

00:09:22.020 --> 00:09:24.142
8 times 9 is 72.

00:09:24.142 --> 00:09:24.975
And then you get 72.

00:09:24.975 --> 00:09:28.020
So it goes 88 times.

00:09:28.020 --> 00:09:30.640
You divide that by 9 you get 88.

00:09:30.640 --> 00:09:35.350
If you divide this
by 9, you get 110.

00:09:35.350 --> 00:09:37.890
Now these are both
divisible by 11.

00:09:37.890 --> 00:09:43.520
So if you divide by
11 you get 8 over 10.

00:09:43.520 --> 00:09:48.080
8 over 10 is the same
thing as 4 over 5.

00:09:48.080 --> 00:09:48.950
Unbelievable.

00:09:48.950 --> 00:09:51.240
It all simplified down to 4/5.

00:09:51.240 --> 00:09:55.410
So the slope of our line
m squared is equal to 4/5.

00:09:55.410 --> 00:09:57.790
Or taking the square
root of both sides,

00:09:57.790 --> 00:10:03.510
m is equal to 2 over
the square root of 5.

00:10:03.510 --> 00:10:05.387
So that's the slope of our line.

00:10:05.387 --> 00:10:06.970
And now we just have
to go back to one

00:10:06.970 --> 00:10:09.790
of these equations, whichever
one we think is simpler,

00:10:09.790 --> 00:10:11.220
and find the y-intercept.

00:10:11.220 --> 00:10:12.490
This is the simpler one.

00:10:12.490 --> 00:10:17.030
So m is equal to 2 over
the square root of 5.

00:10:17.030 --> 00:10:20.860
b is going to be equal
to the square root

00:10:20.860 --> 00:10:22.960
of 9 times m squared.

00:10:22.960 --> 00:10:25.700
Remember m squared is 4/5.

00:10:25.700 --> 00:10:27.570
4/5 minus 4.

00:10:27.570 --> 00:10:29.930
So what is this
going to be equal to?

00:10:29.930 --> 00:10:33.010
Now let me just clear all of
this business out here since we

00:10:33.010 --> 00:10:35.476
don't-- well I'll
just leave it there.

00:10:35.476 --> 00:10:36.850
Maybe it'll be
useful in some way

00:10:36.850 --> 00:10:38.100
if I made a careless mistake.

00:10:38.100 --> 00:10:39.310
Hopefully I haven't.

00:10:39.310 --> 00:10:43.440
So this is going to be
equal to 9 times 4 is 36,

00:10:43.440 --> 00:10:47.795
so it's equal to 36
over 5 minus 20 over 5.

00:10:50.330 --> 00:10:53.780
So this is equal to the
square root of 16 over 5.

00:10:53.780 --> 00:10:56.050
I'm running out of space.

00:10:56.050 --> 00:10:59.350
So b is equal to--
36 minus 20 is 16.

00:10:59.350 --> 00:11:01.940
So square root of 16
over square root of 5.

00:11:01.940 --> 00:11:05.950
Or 4 over the square root of 5.

00:11:05.950 --> 00:11:07.290
And we are done.

00:11:07.290 --> 00:11:11.000
We now know the slope of that
tangent line, the thing that

00:11:11.000 --> 00:11:12.790
had a positive
slope, its equation

00:11:12.790 --> 00:11:17.680
is going to be y is equal to
2 over the square root of 5x

00:11:17.680 --> 00:11:21.819
plus 4 over the
square root of 5.

00:11:21.819 --> 00:11:24.360
And if we want to simplify it,
because if I remember properly

00:11:24.360 --> 00:11:25.630
that problem didn't
have it in-- they

00:11:25.630 --> 00:11:27.010
put everything on
the right hand side.

00:11:27.010 --> 00:11:29.080
Let's multiply everything
times square root of 5.

00:11:29.080 --> 00:11:35.450
So you get square root of
5y is equal to 2x plus 4.

00:11:35.450 --> 00:11:37.770
And then we could subtract
these from both sides

00:11:37.770 --> 00:11:45.340
and we get negative 2x plus
the square root of 5y minus 4

00:11:45.340 --> 00:11:47.500
is equal to zero.

00:11:47.500 --> 00:11:50.710
So do they have that
as one of the choices?

00:11:50.710 --> 00:11:53.440
So it looks like
they do if we just

00:11:53.440 --> 00:11:55.180
multiply everything
times negative 1.

00:11:55.180 --> 00:11:56.930
If we multiply everything
times negative 1

00:11:56.930 --> 00:12:05.170
we get 2x minus the square root
of 5y plus 4 is equal to zero.

00:12:05.170 --> 00:12:06.200
And we are done.

00:12:06.200 --> 00:12:08.180
This was probably the
most painful problem

00:12:08.180 --> 00:12:09.400
I've done in my life.

00:12:09.400 --> 00:12:13.160
2x minus the square
root of 5y plus 4.

00:12:13.160 --> 00:12:15.734
The answer is B.


IIT JEE circle hyperbola common tangent part 4

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:01.880
We're in the home stretch.

00:00:01.880 --> 00:00:05.380
We figured out a constraint
on b in terms of m

00:00:05.380 --> 00:00:07.770
for the line that's
tangent to both

00:00:07.770 --> 00:00:09.591
the circle and the hyperbola.

00:00:09.591 --> 00:00:11.340
And I want to make one
clarification here.

00:00:11.340 --> 00:00:13.860
Here in the last video,
I took the principal root

00:00:13.860 --> 00:00:15.620
when I took the square
root, and that's

00:00:15.620 --> 00:00:18.090
because I want the
y-intercept to be positive.

00:00:18.090 --> 00:00:20.424
Remember, when we
look at this drawing

00:00:20.424 --> 00:00:22.090
that I did in the
first video over here,

00:00:22.090 --> 00:00:24.540
this y-intercept needs
to be positive in order

00:00:24.540 --> 00:00:26.550
for it to have a
positive slope and be

00:00:26.550 --> 00:00:32.130
tangent to both the circle
and the hyperbola in this way.

00:00:32.130 --> 00:00:36.640
So down here, you
definitely want

00:00:36.640 --> 00:00:38.527
to take the principal root.

00:00:38.527 --> 00:00:41.110
Now with that said, this is the
constraint from the hyperbola,

00:00:41.110 --> 00:00:43.210
this is the constraint
from the circle.

00:00:43.210 --> 00:00:45.560
Let's set them equal to
each other and solve for m,

00:00:45.560 --> 00:00:48.037
and then we'll have
our slope of the line.

00:00:48.037 --> 00:00:49.620
So let's set them
equal to each other.

00:00:49.620 --> 00:00:56.950
So we have the square
root of 9m squared minus 4

00:00:56.950 --> 00:01:05.560
is going to be equal to
negative 4m plus 4 times

00:01:05.560 --> 00:01:09.912
the square root of
m squared plus 1.

00:01:09.912 --> 00:01:11.370
So the first thing
we can do, let's

00:01:11.370 --> 00:01:13.070
square both sides of this.

00:01:13.070 --> 00:01:20.240
So we get 9m squared minus 4
is equal to this term over here

00:01:20.240 --> 00:01:21.380
squared.

00:01:21.380 --> 00:01:27.220
So this is 16m squared and
then plus 2 times the product

00:01:27.220 --> 00:01:28.870
of these two terms.

00:01:28.870 --> 00:01:31.740
So the product is going
to be negative 16m.

00:01:31.740 --> 00:01:37.320
So it's going to be negative
32m times the square root of m

00:01:37.320 --> 00:01:41.340
squared plus 1, and then
plus-- square this term--

00:01:41.340 --> 00:01:46.770
16 times m squared plus 1.

00:01:46.770 --> 00:01:48.570
So let's try to simplify
this a little bit,

00:01:48.570 --> 00:01:50.740
get maybe the radical on
one side of the equation,

00:01:50.740 --> 00:01:56.290
and see if we can
simplify this even more.

00:01:56.290 --> 00:01:58.370
So a good place to
start, so let's see.

00:02:01.010 --> 00:02:02.450
This term right
over here is going

00:02:02.450 --> 00:02:07.320
to be 16m squared plus 16.

00:02:07.320 --> 00:02:10.100
So on the right-hand side--
let me just write it like this.

00:02:10.100 --> 00:02:15.420
So we have 9m squared
minus 4 is equal to--

00:02:15.420 --> 00:02:18.420
on the right-hand side,
we have 16 plus 16.

00:02:18.420 --> 00:02:20.470
We have 32.

00:02:20.470 --> 00:02:28.060
32m squared plus 16, minus
32m, times the square root

00:02:28.060 --> 00:02:30.477
of m squared plus 1.

00:02:30.477 --> 00:02:31.060
Now let's see.

00:02:31.060 --> 00:02:35.730
We can subtract 9m
squared from both sides,

00:02:35.730 --> 00:02:39.130
so we'll subtract 9m
squared from both sides.

00:02:39.130 --> 00:02:41.375
And then we can add
4 to both sides.

00:02:44.370 --> 00:02:46.580
And then we are left with--
actually, let me do it

00:02:46.580 --> 00:02:48.246
the other way, because
I want to isolate

00:02:48.246 --> 00:02:50.500
this on the right-hand side.

00:02:50.500 --> 00:02:52.600
Let me do it the other way.

00:02:52.600 --> 00:02:55.970
So let's subtract 32m
squared from both sides.

00:03:02.222 --> 00:03:03.930
Well, yeah, might as
well do it this way.

00:03:03.930 --> 00:03:06.020
So subtract 32m squared
from both sides,

00:03:06.020 --> 00:03:07.960
and then subtract
16 from both sides.

00:03:10.870 --> 00:03:15.960
The left-hand
side, 9 minus 32 is

00:03:15.960 --> 00:03:19.730
going to be negative
23, because 23 plus 9,

00:03:19.730 --> 00:03:26.170
yup, is-- so this is
negative 23m squared minus 20

00:03:26.170 --> 00:03:33.850
is equal to negative 32m
times the square root

00:03:33.850 --> 00:03:36.210
of m squared plus 1.

00:03:36.210 --> 00:03:39.310
Now we can square both
sides of this equation.

00:03:39.310 --> 00:03:41.302
It's not the cleanest
problem in the world,

00:03:41.302 --> 00:03:43.260
but hopefully, if we
haven't made any mistakes,

00:03:43.260 --> 00:03:44.680
we'll get someplace productive.

00:03:44.680 --> 00:03:46.138
Actually, let's
multiply both sides

00:03:46.138 --> 00:03:49.020
of this equation times negative
1 just to make it positive

00:03:49.020 --> 00:03:50.110
and simplify things.

00:03:50.110 --> 00:03:53.110
So that becomes positive,
positive, and positive.

00:03:53.110 --> 00:03:55.480
If you square the left-hand
side of the equation,

00:03:55.480 --> 00:03:58.919
we get 23 squared.

00:03:58.919 --> 00:03:59.960
Let me write it this way.

00:03:59.960 --> 00:04:08.130
We get 23m squared squared plus
2 times the products of this.

00:04:08.130 --> 00:04:10.120
So this would be 40--
let me just write it

00:04:10.120 --> 00:04:17.980
2 times 20 times 23m squared,
plus 20 squared, which is 400,

00:04:17.980 --> 00:04:21.290
is going to be
equal to 32 squared,

00:04:21.290 --> 00:04:28.760
or I'll write it as 32m
squared, times m squared plus 1.

00:04:28.760 --> 00:04:32.230
Now let's see what
we can do over here.

00:04:32.230 --> 00:04:35.860
So this turns into-- this is
just a big tedious problem.

00:04:35.860 --> 00:04:40.990
So I'll just write this
is 23m squared squared

00:04:40.990 --> 00:04:44.070
plus-- so this is
going to be 40.

00:04:44.070 --> 00:04:46.480
This is 40 times 23.

00:04:46.480 --> 00:04:54.780
So 23 times 4 is 92, so it's
going to be plus 920m squared.

00:04:54.780 --> 00:04:57.690
Actually, let me expand
everything out here.

00:04:57.690 --> 00:05:00.500
Let me just go straight to
the numbers, so 23 times 23,

00:05:00.500 --> 00:05:02.980
23 squared.

00:05:02.980 --> 00:05:04.380
3 times 3 is 9.

00:05:04.380 --> 00:05:05.753
3 times 2 is 6.

00:05:05.753 --> 00:05:07.460
69, put a 0 here.

00:05:07.460 --> 00:05:09.170
2 times 3 is 6.

00:05:09.170 --> 00:05:11.190
2 times 2 is 4.

00:05:11.190 --> 00:05:12.080
So you get a 9.

00:05:12.080 --> 00:05:13.880
6 plus 6 is 12.

00:05:13.880 --> 00:05:16.480
1 plus 4 is 5.

00:05:16.480 --> 00:05:25.360
So you have 529m to the
fourth plus 40 times 23.

00:05:25.360 --> 00:05:27.175
4 times 20 is 80.

00:05:27.175 --> 00:05:30.050
4 times 23 is 92.

00:05:30.050 --> 00:05:34.240
So it's going to be
plus 920m squared

00:05:34.240 --> 00:05:41.080
plus 400 is equal to-- so
we're going to have 32 squared.

00:05:41.080 --> 00:05:42.980
Let me figure out
what 32 squared is.

00:05:42.980 --> 00:05:45.360
32 times 32.

00:05:45.360 --> 00:05:47.280
It seems like if you're
taking this exam,

00:05:47.280 --> 00:05:49.910
you might as well have your
multiplication tables memorized

00:05:49.910 --> 00:05:51.255
to about 50.

00:05:51.255 --> 00:05:52.787
And I think they
do that on purpose.

00:05:52.787 --> 00:05:54.870
So just to get a sense, I
mean, this isn't an exam

00:05:54.870 --> 00:05:56.580
where the test takers
expect everyone

00:05:56.580 --> 00:06:01.180
to get a perfect-- I think
the number one score in India

00:06:01.180 --> 00:06:03.810
is on the order of 80%,
which is pretty darn amazing.

00:06:03.810 --> 00:06:06.620
To be able to do roughly the
math section in about an hour,

00:06:06.620 --> 00:06:08.547
to 80% right is pretty amazing.

00:06:08.547 --> 00:06:10.380
But I think to make the
threshold of getting

00:06:10.380 --> 00:06:12.360
into these highly
selective colleges,

00:06:12.360 --> 00:06:14.707
I think you have to get
around 40% or 50% correct.

00:06:14.707 --> 00:06:16.290
Forgive me if I got
that number wrong,

00:06:16.290 --> 00:06:17.150
but I think it's on that order.

00:06:17.150 --> 00:06:18.330
So they need to do
that because you

00:06:18.330 --> 00:06:20.490
have 200,000 or 300,000
people in India taking it,

00:06:20.490 --> 00:06:21.350
so you need to
make sure that you

00:06:21.350 --> 00:06:22.558
have a good spread of people.

00:06:22.558 --> 00:06:25.710
So this is what they do
to spread people out.

00:06:25.710 --> 00:06:28.290
So we have 2 times 2 is 4.

00:06:28.290 --> 00:06:30.730
2 times 3 is 6.

00:06:30.730 --> 00:06:31.910
And then put a 0 here.

00:06:31.910 --> 00:06:34.290
3 times 2 is 6.

00:06:34.290 --> 00:06:37.010
3 times 3 is 9.

00:06:37.010 --> 00:06:39.910
And for example, if I was
preparing for the exam,

00:06:39.910 --> 00:06:42.569
I might even go ahead
and-- you know me,

00:06:42.569 --> 00:06:44.110
I hate to memorize
things, but if you

00:06:44.110 --> 00:06:47.437
had to do this type of problem
on the order of five minutes,

00:06:47.437 --> 00:06:50.020
you might want to memorize these
type of things ahead of time.

00:06:50.020 --> 00:06:53.110
The y-intercept for a
line that intercepts

00:06:53.110 --> 00:06:55.710
a hyperbola, y-intercept for a
line that intercepts a circle,

00:06:55.710 --> 00:06:56.490
who knows?

00:06:56.490 --> 00:06:57.450
But anyway.

00:06:57.450 --> 00:06:59.420
But in general, I don't think
it's good to memorize in life.

00:06:59.420 --> 00:07:01.070
Because when you're
actually doing math problems

00:07:01.070 --> 00:07:02.980
in your life, what
matters more is you

00:07:02.980 --> 00:07:04.471
understand the
underlying meaning.

00:07:04.471 --> 00:07:06.970
You normally have plenty of
time to do it in your real life.

00:07:06.970 --> 00:07:09.540
These exams are kind of an
artificial circumstance.

00:07:09.540 --> 00:07:11.740
But anyway, 4 plus 0 is 4.

00:07:11.740 --> 00:07:15.640
6 plus 6 is 12.

00:07:15.640 --> 00:07:19.230
So it's 1,024, is
equal to 1,024.

00:07:19.230 --> 00:07:24.800
That's the 32m squared
times m squared plus 1.

00:07:24.800 --> 00:07:34.930
Or another way, this is 1,024m
to the fourth plus 1,024,

00:07:34.930 --> 00:07:36.020
squared.

00:07:36.020 --> 00:07:38.550
And now we are in
the home stretch.

00:07:38.550 --> 00:07:42.820
So let's subtract 529m to
the fourth from both sides.

00:07:42.820 --> 00:07:47.240
So minus 529m to the fourth,
this is extremely tedious.

00:07:47.240 --> 00:07:51.447
Minus 529m to the fourth.

00:07:51.447 --> 00:07:53.030
Frankly, at this
point in the problem,

00:07:53.030 --> 00:07:55.450
you're better off if you
just want to do it for speed,

00:07:55.450 --> 00:07:57.180
just trying out the
choices they gave you

00:07:57.180 --> 00:07:59.920
and figuring out which
m's and b's satisfy that.

00:07:59.920 --> 00:08:01.970
But anyway, let's
just move forward--

00:08:01.970 --> 00:08:04.030
or which m's satisfy it.

00:08:04.030 --> 00:08:05.280
That would probably be faster.

00:08:05.280 --> 00:08:06.655
But let's just
solve it properly.

00:08:06.655 --> 00:08:10.420
So minus 529m to
the fourth plus--

00:08:10.420 --> 00:08:14.360
and then we also want to
subtract 920m squared.

00:08:14.360 --> 00:08:18.470
So let's also
subtract 920m squared.

00:08:18.470 --> 00:08:22.310
And so on this side,
we're just left--

00:08:22.310 --> 00:08:24.150
let's also subtract
a 400 just to make

00:08:24.150 --> 00:08:25.410
this a proper quadratic.

00:08:25.410 --> 00:08:27.310
And also we're going
to put a 400 here.

00:08:27.310 --> 00:08:29.410
So the whole left
side simplifies to 0

00:08:29.410 --> 00:08:33.842
is equal to-- 1,024 minus 529.

00:08:33.842 --> 00:08:34.341
Let's see.

00:08:34.341 --> 00:08:40.150
1,024 minus 524
would have been 500.

00:08:40.150 --> 00:08:42.570
And so it's going to
be 5 less than that.

00:08:42.570 --> 00:08:46.927
So it's going to be
495m to the fourth.

00:08:46.927 --> 00:08:47.760
Did I do that right?

00:08:47.760 --> 00:08:52.060
1,024 minus 524
would be 500, but I'm

00:08:52.060 --> 00:08:54.080
subtracting 5 more than that.

00:08:54.080 --> 00:08:56.260
So that is going to be 495.

00:08:56.260 --> 00:09:04.470
And then 1,024 minus
920, it'll be 80 plus 24.

00:09:04.470 --> 00:09:15.270
So it's going to be plus 104m
squared minus 400 is equal to 0

00:09:15.270 --> 00:09:19.660
So we have just a straight-up
quadratic equation over here.

00:09:19.660 --> 00:09:22.720
And you might not recognize
it, but this is the same thing.

00:09:22.720 --> 00:09:26.810
M to the fourth is the same
thing as m squared squared.

00:09:26.810 --> 00:09:28.180
So let's just solve it.

00:09:28.180 --> 00:09:30.397
So we'll get m squared.

00:09:30.397 --> 00:09:31.730
We're not solving for x anymore.

00:09:31.730 --> 00:09:34.566
We're going to get m squared
using the quadratic formula.

00:09:34.566 --> 00:09:36.440
You could substitute x
is equal to m squared,

00:09:36.440 --> 00:09:38.900
and this will just become
a natural quadratic then.

00:09:38.900 --> 00:09:43.980
m squared is equal to negative
b, negative 104, plus or minus

00:09:43.980 --> 00:09:47.380
the square root-- more fun math
for us without a calculator--

00:09:47.380 --> 00:09:51.160
plus or minus the square
root of 104 squared

00:09:51.160 --> 00:09:58.230
minus 4 times a, which is 495,
times c, which is negative 400.

00:09:58.230 --> 00:09:59.570
So that'll make this a positive.

00:09:59.570 --> 00:10:03.600
And you have a 400
right over here,

00:10:03.600 --> 00:10:08.430
and then all of that
over 2 times 495.

00:10:08.430 --> 00:10:10.500
2 times 495 is what?

00:10:13.690 --> 00:10:15.870
Well, it's going to
be 10 less than 1,000,

00:10:15.870 --> 00:10:20.500
so it's going to be 990.

00:10:20.500 --> 00:10:22.680
So let's try to see if
we can evaluate this.

00:10:22.680 --> 00:10:24.710
Actually, I'll stop
this problem here.

00:10:24.710 --> 00:10:26.460
I'm already cross the
10-minute threshold.

00:10:26.460 --> 00:10:27.918
In the next video,
we'll just grind

00:10:27.918 --> 00:10:29.610
through this
mathematics-- It's really

00:10:29.610 --> 00:10:32.230
just arithmetic at this
point-- and figure out

00:10:32.230 --> 00:10:35.640
what m is going to be equal to.


IIT JEE circle hyperbola common tangent part 3

WEBVTT
Kind: captions
Language: en

00:00:00.637 --> 00:00:02.720
In this video we're going
to do with the hyperbola

00:00:02.720 --> 00:00:05.350
the exact same thing
we did with the circle.

00:00:05.350 --> 00:00:09.030
We're going to find
constraints on the y-intercept

00:00:09.030 --> 00:00:11.072
for the tangent
line in terms of m.

00:00:11.072 --> 00:00:13.030
But this time we're going
to use the hyperbola.

00:00:13.030 --> 00:00:14.904
And then we can set them
equal to each other,

00:00:14.904 --> 00:00:16.530
and solve for the m.

00:00:16.530 --> 00:00:19.880
So let's remind ourselves what
the equation of the hyperbola

00:00:19.880 --> 00:00:20.740
is.

00:00:20.740 --> 00:00:22.239
They give it to us
right over there.

00:00:22.239 --> 00:00:25.930
It's x squared over 9 minus y
squared over 4 is equal to 1.

00:00:25.930 --> 00:00:27.470
Let me write this over here.

00:00:27.470 --> 00:00:34.040
So it is x squared over
9 minus y squared over 4

00:00:34.040 --> 00:00:38.100
is equal to-- I'll
write the y squared--

00:00:38.100 --> 00:00:41.100
minus y squared over
4 is equal to 1.

00:00:41.100 --> 00:00:43.360
But now we can
substitute the y squared

00:00:43.360 --> 00:00:46.530
with the y is equal
to mx plus b that we

00:00:46.530 --> 00:00:47.620
got from the last video.

00:00:47.620 --> 00:00:50.010
And we actually figured out
what y squared is equal to.

00:00:50.010 --> 00:00:53.100
Y squared is equal to all
of this business up here.

00:00:53.100 --> 00:00:54.280
Because it's the same line.

00:00:54.280 --> 00:00:55.790
Remember, this is the
whole point of it.

00:00:55.790 --> 00:00:58.081
We're trying to find two
constraints on that same line.

00:00:58.081 --> 00:00:59.842
So we can rewrite
this same thing.

00:00:59.842 --> 00:01:01.300
And actually one
thing I want to do

00:01:01.300 --> 00:01:03.466
is I'm going to multiply
both sides of this equation

00:01:03.466 --> 00:01:06.270
by 36, the common
multiple of 9 and 4,

00:01:06.270 --> 00:01:08.230
so that I can get rid
of these fractions.

00:01:08.230 --> 00:01:14.000
And so this is going to become
36-- let me write it this way--

00:01:14.000 --> 00:01:16.690
so 36 divided by 9 is 4.

00:01:16.690 --> 00:01:22.820
So it's 4x squared minus--
36 divided by 4 is 9--

00:01:22.820 --> 00:01:24.660
and I would put a y
squared here but we

00:01:24.660 --> 00:01:26.550
know that y squared is
the exact same thing

00:01:26.550 --> 00:01:28.120
as this thing over here.

00:01:28.120 --> 00:01:30.990
So y squared is the
same thing as m squared.

00:01:30.990 --> 00:01:37.000
x squared plus 2
mbx plus b squared.

00:01:37.000 --> 00:01:42.042
And then this is going to
be equal to-- remember,

00:01:42.042 --> 00:01:43.750
we multiplied both
sides of this equation

00:01:43.750 --> 00:01:49.210
by 36-- so this is
going to be equal to 36.

00:01:49.210 --> 00:01:50.839
And let me simplify this.

00:01:50.839 --> 00:01:52.630
And we're going to do
the exact same thing.

00:01:52.630 --> 00:02:00.340
We know that the m and
b have to be such that,

00:02:00.340 --> 00:02:03.040
or the line has to have a slope
and y-intercept where it'll

00:02:03.040 --> 00:02:06.390
only intersect with the
hyperbola at one point.

00:02:06.390 --> 00:02:09.410
It'll only have one solution to
this quadratic in terms of x.

00:02:09.410 --> 00:02:11.890
But let's simplify it first
before we worry about that.

00:02:11.890 --> 00:02:18.900
So this is equal to 4 x
squared minus 9 m squared

00:02:18.900 --> 00:02:30.530
x squared minus 18
mbx, minus 9 b squared.

00:02:30.530 --> 00:02:31.650
Just multiply it there.

00:02:31.650 --> 00:02:33.730
And let me subtract
this 36 from both sides,

00:02:33.730 --> 00:02:36.880
so then we have minus
36 is equal to 0.

00:02:36.880 --> 00:02:39.080
So this is a quadratic
in terms of x.

00:02:39.080 --> 00:02:43.600
But let me combine the
various degree terms.

00:02:43.600 --> 00:02:48.390
So these are the x squared
terms right over here.

00:02:48.390 --> 00:02:55.940
So this is the same thing as
4 minus 9 m squared times x

00:02:55.940 --> 00:02:56.900
squared.

00:02:56.900 --> 00:03:00.460
And then our only x term
is this right over here.

00:03:00.460 --> 00:03:01.800
That's our only x term.

00:03:01.800 --> 00:03:05.620
So this is minus 18 mbx.

00:03:05.620 --> 00:03:09.370
And then our constant
terms are right over here.

00:03:09.370 --> 00:03:18.260
So this over here is minus
9 b squared-- and let

00:03:18.260 --> 00:03:25.230
me write it this way, let
me write it as minus 9--

00:03:25.230 --> 00:03:28.842
well I'll just write it,
minus 9 b squared minus 36.

00:03:28.842 --> 00:03:30.050
I'll just write it like that.

00:03:30.050 --> 00:03:32.710
And of course that is
going to be equal to 0.

00:03:32.710 --> 00:03:34.340
And remember, quadratic
formula, if we

00:03:34.340 --> 00:03:36.840
wanted to solve for the Xs,
we'd have the quadratic formula,

00:03:36.840 --> 00:03:39.110
but we only want to
have one solution.

00:03:39.110 --> 00:03:41.240
So the discriminant part
of the quadratic formula

00:03:41.240 --> 00:03:42.150
is going to equal 0.

00:03:42.150 --> 00:03:45.230
The b squared minus 4 ac
is going to be equal to 0.

00:03:45.230 --> 00:03:47.790
This is exactly what we
did in the last video.

00:03:47.790 --> 00:03:49.876
So let's take the
b squared minus 4ac

00:03:49.876 --> 00:03:51.500
and set that equal
to 0, and then we'll

00:03:51.500 --> 00:03:55.740
have our constraints on m and b.

00:03:55.740 --> 00:03:59.985
So b squared is-- and
remember don't get the squared

00:03:59.985 --> 00:04:01.360
in the quadratic
formula confused

00:04:01.360 --> 00:04:02.860
with the b in the y-intercept.

00:04:02.860 --> 00:04:05.120
But this term
squared is-- so this

00:04:05.120 --> 00:04:11.870
is going to be-- 18 squared
m squared b squared, right?

00:04:11.870 --> 00:04:14.250
Negative 18 squared
is just positive 18.

00:04:14.250 --> 00:04:17.416
Minus 4 times a.

00:04:17.416 --> 00:04:23.620
a is 4 minus 9 m
squared times c.

00:04:23.620 --> 00:04:38.197
So I can rewrite c as negative
9 times b squared plus 4.

00:04:38.197 --> 00:04:39.030
Did I do that right?

00:04:39.030 --> 00:04:41.220
Negative 9 times b squared
is negative 9 b squared.

00:04:41.220 --> 00:04:43.200
Negative 9 times
4 is negative 36.

00:04:43.200 --> 00:04:45.760
I want to make sure I don't
make any careless mistakes.

00:04:45.760 --> 00:04:47.950
And so this becomes--
so if we just

00:04:47.950 --> 00:04:50.120
take the negative 9
and the negative 4--

00:04:50.120 --> 00:04:52.620
they become a positive 36.

00:04:52.620 --> 00:04:55.260
They become a positive 36.

00:04:55.260 --> 00:04:58.880
And then we can actually,
just to simplify things

00:04:58.880 --> 00:05:02.270
so we don't have to do
too much fancy math,

00:05:02.270 --> 00:05:05.190
this 18 squared--
let's remember--

00:05:05.190 --> 00:05:09.910
18 squared is the same
thing as-- well 18 squared

00:05:09.910 --> 00:05:12.350
is going to be divisible--
well I won't worry too much

00:05:12.350 --> 00:05:13.840
about that just yet.

00:05:13.840 --> 00:05:15.600
I just want to make
sure that-- Actually

00:05:15.600 --> 00:05:17.160
let me write it this way.

00:05:17.160 --> 00:05:23.560
So 18 squared is 2
times 9 times 2 times 9.

00:05:23.560 --> 00:05:29.280
Or another way to think about
it is 4 times 9 times 9.

00:05:29.280 --> 00:05:32.200
That's the same
thing as 18 squared.

00:05:32.200 --> 00:05:34.600
4 times 9 times 9.

00:05:34.600 --> 00:05:38.430
Now we can divide both of these
terms-- and this whole thing,

00:05:38.430 --> 00:05:41.220
remember we want this whole
thing to be equal to 0.

00:05:41.220 --> 00:05:43.150
The discriminant has
to be equal to 0.

00:05:43.150 --> 00:05:47.430
So we can divide both
sides of this equation

00:05:47.430 --> 00:05:52.540
by 36, which is the same thing
as dividing by 4 times 9.

00:05:52.540 --> 00:05:54.870
So this term right over here,
we could get rid of one 4

00:05:54.870 --> 00:05:55.960
and one 4.

00:05:55.960 --> 00:06:03.750
And we're going to get a 9 m
squared b squared over here.

00:06:03.750 --> 00:06:06.250
And then we divided by
36, so these all go away.

00:06:06.250 --> 00:06:08.400
So it's going to be
9 m squared b squared

00:06:08.400 --> 00:06:13.720
plus this thing
times this thing.

00:06:13.720 --> 00:06:15.530
So let's see what that is.

00:06:15.530 --> 00:06:18.100
So we have a 4
times a b squared.

00:06:18.100 --> 00:06:20.560
So we have plus 4 b squared.

00:06:20.560 --> 00:06:23.190
Let me do this in
a different color.

00:06:23.190 --> 00:06:24.490
I'll do it in blue.

00:06:24.490 --> 00:06:26.954
So plus 4 b squared.

00:06:26.954 --> 00:06:28.120
And then you have 4 times 4.

00:06:28.120 --> 00:06:30.000
So plus 16.

00:06:30.000 --> 00:06:33.500
And you have negative 9 m
squared times b squared.

00:06:33.500 --> 00:06:37.720
So it's negative 9
m squared b squared.

00:06:37.720 --> 00:06:39.710
And then you have negative
9 m squared times 4.

00:06:39.710 --> 00:06:44.410
So negative 36 m squared.

00:06:44.410 --> 00:06:47.110
And that's going
to be equal to 0.

00:06:47.110 --> 00:06:50.980
Lucky for us, that
and that cancel out.

00:06:50.980 --> 00:06:55.252
And then we are
left with something.

00:06:55.252 --> 00:06:56.710
And actually, what
we're left with,

00:06:56.710 --> 00:06:58.020
everything is divisible by 4.

00:06:58.020 --> 00:07:00.390
So let's divide what
we're left with by 4.

00:07:00.390 --> 00:07:04.790
So then we're left with b
squared-- that's that term.

00:07:04.790 --> 00:07:06.490
And then minus 9 m squared.

00:07:11.500 --> 00:07:12.810
That's that term over there.

00:07:12.810 --> 00:07:14.580
Just divided it by 4.

00:07:14.580 --> 00:07:20.240
And then plus 4 is equal to 0.

00:07:20.240 --> 00:07:23.852
And once again, we could
use-- well actually here,

00:07:23.852 --> 00:07:25.060
we don't have to do anything.

00:07:25.060 --> 00:07:26.160
We don't have to use
quadratic formula.

00:07:26.160 --> 00:07:27.360
We could just solve for b.

00:07:27.360 --> 00:07:30.960
We could subtract
this from both sides.

00:07:30.960 --> 00:07:38.720
And so we'll get b squared
is equal to the square root

00:07:38.720 --> 00:07:50.280
of 9 m squared minus 4.

00:07:50.280 --> 00:07:53.090
Sorry-- let me
just write this-- I

00:07:53.090 --> 00:07:54.830
don't want to skip steps here.

00:07:54.830 --> 00:08:01.057
b squared is equal to
9 m squared minus 4.

00:08:01.057 --> 00:08:02.640
They're painting the
office right now,

00:08:02.640 --> 00:08:05.400
so maybe it's making me a
little bit-- making my brain not

00:08:05.400 --> 00:08:06.070
work properly.

00:08:06.070 --> 00:08:11.790
Then b is equal to the square
root of 9 m squared minus 4.

00:08:11.790 --> 00:08:12.640
Did I do that right?

00:08:12.640 --> 00:08:13.490
Let's see.

00:08:13.490 --> 00:08:14.820
I got the 4.

00:08:14.820 --> 00:08:17.044
Yeah, it looks right.

00:08:17.044 --> 00:08:18.710
And so we're left
with a situation where

00:08:18.710 --> 00:08:20.770
the b-- if we're saying
that if the line is

00:08:20.770 --> 00:08:23.030
tangent to the
hyperbola it to be this,

00:08:23.030 --> 00:08:26.570
and if the line is
tangent to the circle b

00:08:26.570 --> 00:08:30.855
has to be equal to this,
this business over here.

00:08:30.855 --> 00:08:35.630
So let me copy it and
then let me paste it.

00:08:35.630 --> 00:08:37.480
Let me paste it.

00:08:37.480 --> 00:08:38.549
Just like that.

00:08:38.549 --> 00:08:40.950
And so we now have two
equations with two unknowns.

00:08:40.950 --> 00:08:44.049
We can set these equal to
each other and solve for m.

00:08:44.049 --> 00:08:48.070
And this will give us the m or
the slope of that tangent line.

00:08:48.070 --> 00:08:52.420
And then we can go
ahead and solve for b.

00:08:52.420 --> 00:08:54.790
I'll do that in the next video.


IIT JEE circle hyperbola common tangent part 2

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:03.100
Now that we have a
visual sense of what

00:00:03.100 --> 00:00:05.380
this common tangent
with a positive slope

00:00:05.380 --> 00:00:08.180
would look like, let's see if
we get some constraints on it,

00:00:08.180 --> 00:00:10.610
especially constraints on
its slope and y-intercept.

00:00:10.610 --> 00:00:13.590
So this line that I
drew in the last video

00:00:13.590 --> 00:00:17.770
here in pink-- it would have the
form y is equal to mx plus b.

00:00:17.770 --> 00:00:23.560
It's a line where m is the
slope and b is the y-intercept.

00:00:26.510 --> 00:00:30.060
Now, let's think about
what constraints there

00:00:30.060 --> 00:00:37.530
would have to be on m and b if
this is tangent to the circle.

00:00:37.530 --> 00:00:39.960
And you might be tempted
to break out some calculus

00:00:39.960 --> 00:00:43.500
and figure out the slope at
any point alongside a circle,

00:00:43.500 --> 00:00:45.080
but there's an
easier way to do it.

00:00:45.080 --> 00:00:48.610
You just have to realize that if
a line is tangent to a circle,

00:00:48.610 --> 00:00:52.325
it will only intersect
that circle at one point.

00:00:52.325 --> 00:00:53.950
Let me show you what
I'm talking about.

00:00:53.950 --> 00:00:55.100
So this is the line.

00:00:55.100 --> 00:00:56.641
What I want to do
is figure out where

00:00:56.641 --> 00:01:00.416
this equation and the equation
of the circle intersect.

00:01:00.416 --> 00:01:01.790
That's we'll focus
on this video.

00:01:01.790 --> 00:01:04.330
And then, we'll do the same
thing for the hyperbola.

00:01:04.330 --> 00:01:08.540
So we have y is equal to mx plus
b is the equation of the line.

00:01:08.540 --> 00:01:10.890
The circle-- they give us
the equation up here. x

00:01:10.890 --> 00:01:14.340
squared plus y squared
minus 8x is equal to 0.

00:01:14.340 --> 00:01:22.020
So the circle is x squared
plus y squared minus 8x is

00:01:22.020 --> 00:01:24.300
equal to 0.

00:01:24.300 --> 00:01:27.320
So what we can do
is we can substitute

00:01:27.320 --> 00:01:32.230
this expression over here-- we
can substitute this in for y.

00:01:32.230 --> 00:01:36.540
And then we can figure out what
are the constraints on m and b

00:01:36.540 --> 00:01:40.500
so that we only have one
solution to the intersection

00:01:40.500 --> 00:01:43.136
where we only
intersect at one point.

00:01:43.136 --> 00:01:45.510
So to do that-- actually,
let's substitute for y squared.

00:01:45.510 --> 00:01:48.160
So if y is equal to
this, let's square that.

00:01:48.160 --> 00:01:49.734
We'll get y squared.

00:01:49.734 --> 00:01:51.650
I'm just squaring the
expression for the line.

00:01:51.650 --> 00:01:53.980
y squared is going
to equal m squared

00:01:53.980 --> 00:02:00.690
x squared plus 2mbx
plus b squared.

00:02:00.690 --> 00:02:02.590
All I did is I squared
this expression here.

00:02:02.590 --> 00:02:05.770
And I did that so that now we
can substitute this whole thing

00:02:05.770 --> 00:02:09.960
right in here for y squared.

00:02:09.960 --> 00:02:16.360
And the expression for the
x point for our intersection

00:02:16.360 --> 00:02:20.280
is going to be x squared
plus all of this business.

00:02:20.280 --> 00:02:21.630
That's the y squared.

00:02:21.630 --> 00:02:29.990
Plus m squared x squared
plus 2mbx plus b squared

00:02:29.990 --> 00:02:35.080
minus 8x is equal to 0.

00:02:35.080 --> 00:02:38.560
And if we wanted to write this
as a quadratic in terms of x,

00:02:38.560 --> 00:02:42.850
this would be-- so our x squared
terms are these two terms.

00:02:42.850 --> 00:02:44.800
We could write this
as-- let's see.

00:02:44.800 --> 00:02:49.410
Let's write this as m squared
plus 1 times x squared, right?

00:02:49.410 --> 00:02:52.420
m squared times 1
times x squared.

00:02:52.420 --> 00:02:56.960
And then our x terms are
this one and this one.

00:02:56.960 --> 00:03:03.640
So then we have plus
2mb minus 8 times x.

00:03:03.640 --> 00:03:05.590
And then, we just
have this b term--

00:03:05.590 --> 00:03:07.890
this b squared, the constant
term right over here.

00:03:07.890 --> 00:03:09.990
And I'll do that in orange.

00:03:09.990 --> 00:03:13.710
So plus b squared is equal to 0.

00:03:13.710 --> 00:03:17.560
So if we knew m and b, if we
knew the equation of this line,

00:03:17.560 --> 00:03:19.372
this would just be a
straight up quadratic.

00:03:19.372 --> 00:03:20.830
You could use the
quadratic formula

00:03:20.830 --> 00:03:24.170
to figure out the x values--
where they intersect.

00:03:24.170 --> 00:03:27.260
Now, what's neat about this
is we know that they only

00:03:27.260 --> 00:03:28.690
have to intersect in one point.

00:03:28.690 --> 00:03:31.290
Remember, the quadratic
formula-- negative

00:03:31.290 --> 00:03:35.840
b plus or minus the square
root of b squared minus 4ac.

00:03:35.840 --> 00:03:37.170
All of that over 2a.

00:03:37.170 --> 00:03:39.620
And don't get this b confused
with the y-intercept b.

00:03:39.620 --> 00:03:42.200
This is just from the
quadratic formula.

00:03:42.200 --> 00:03:44.220
That's the quadratic
formula over there.

00:03:44.220 --> 00:03:50.150
This will only have one
solution if this over here

00:03:50.150 --> 00:03:54.270
is equal to 0.

00:03:54.270 --> 00:03:56.780
Because you're just
adding and subtracting 0,

00:03:56.780 --> 00:03:59.790
so you're only going
to get one solution.

00:03:59.790 --> 00:04:03.250
So when a line is
tangent to a circle,

00:04:03.250 --> 00:04:05.050
it can only intersect
in one point.

00:04:05.050 --> 00:04:06.466
Or another way to
think about it--

00:04:06.466 --> 00:04:08.900
this will only
have one solution.

00:04:08.900 --> 00:04:12.000
If a line intersects any other
type of non-tangent line,

00:04:12.000 --> 00:04:14.510
a non-tangent line would
do something like that.

00:04:14.510 --> 00:04:17.930
It would either have two
solutions, in which case, this

00:04:17.930 --> 00:04:23.360
is a positive value, or
it won't intersect at all.

00:04:23.360 --> 00:04:25.020
And then, this will
have no solutions,

00:04:25.020 --> 00:04:28.930
which means that b squared minus
4ac would be a negative number.

00:04:28.930 --> 00:04:31.860
So we know that this
is a tangent line.

00:04:31.860 --> 00:04:34.540
So we only have one solution,
or b squared minus 4ac

00:04:34.540 --> 00:04:36.010
is equal to 0.

00:04:36.010 --> 00:04:38.640
So what's b squared
minus 4ac over here?

00:04:38.640 --> 00:04:40.984
Well, this is our
b when we think

00:04:40.984 --> 00:04:42.400
in terms of the
quadratic formula.

00:04:42.400 --> 00:04:44.140
And remember, don't
get that b confused

00:04:44.140 --> 00:04:45.520
with the b of the y-intercept.

00:04:45.520 --> 00:04:47.520
I'm just thinking of the
quadratic formula here.

00:04:47.520 --> 00:04:48.320
So let's do this.

00:04:48.320 --> 00:04:49.687
Let's take this squared.

00:04:49.687 --> 00:04:51.520
So I'm just going to
rewrite this expression

00:04:51.520 --> 00:04:52.895
and set it equal
to 0, because we

00:04:52.895 --> 00:04:54.780
know there's only one solution.

00:04:54.780 --> 00:04:59.400
So we have 2mb minus 8 squared.

00:04:59.400 --> 00:05:04.940
And then you have
minus 4 times a, which

00:05:04.940 --> 00:05:11.100
is m squared plus 1,
times c, is b squared.

00:05:13.970 --> 00:05:16.940
And so this is going to
have to be equal to 0

00:05:16.940 --> 00:05:19.175
if this is truly a tangent line.

00:05:22.230 --> 00:05:24.770
So let's see any type
of interesting things

00:05:24.770 --> 00:05:26.760
that we can get out here.

00:05:26.760 --> 00:05:30.940
If we can express b
as a function of m,

00:05:30.940 --> 00:05:32.820
that's a good place to start.

00:05:32.820 --> 00:05:33.999
So let's try to do that.

00:05:33.999 --> 00:05:34.540
So let's see.

00:05:34.540 --> 00:05:38.917
If we expand this out,
this becomes 4m squared.

00:05:38.917 --> 00:05:40.500
Let me do that in
the same blue so you

00:05:40.500 --> 00:05:42.120
know what I'm expanding out.

00:05:42.120 --> 00:05:45.880
This part over here
becomes 4m squared

00:05:45.880 --> 00:05:52.650
b squared minus 2 times
8 is-- 2 times negative 8

00:05:52.650 --> 00:05:53.960
is negative 16.

00:05:53.960 --> 00:05:57.630
Multiply that times 2, so
it's negative 32mb-- I'm just

00:05:57.630 --> 00:06:00.970
squaring this over
here-- plus 64.

00:06:00.970 --> 00:06:10.200
So that is that term over there,
expanded, minus 4 times-- well,

00:06:10.200 --> 00:06:11.660
I could just expand
everything out.

00:06:11.660 --> 00:06:20.040
Minus 4 times m squared b
squared minus 4 times 1 times b

00:06:20.040 --> 00:06:24.649
squared is all going
to be equal to 0.

00:06:24.649 --> 00:06:26.940
And then, lucky for us, some
of these terms cancel out.

00:06:26.940 --> 00:06:28.020
4mb squared.

00:06:28.020 --> 00:06:30.287
Negative 4mb squared.

00:06:30.287 --> 00:06:30.870
And let's see.

00:06:30.870 --> 00:06:34.920
We could actually divide both
sides of this equation by 4,

00:06:34.920 --> 00:06:43.160
and we get negative
8mb plus-- we're

00:06:43.160 --> 00:06:48.650
dividing everything by 4--
so plus 16 minus b squared

00:06:48.650 --> 00:06:51.080
is equal to 0.

00:06:51.080 --> 00:06:53.790
And now, we can solve
for b in terms of m

00:06:53.790 --> 00:06:56.274
using the quadratic
formula again.

00:06:56.274 --> 00:06:57.690
So now we would
have a constraint,

00:06:57.690 --> 00:07:00.360
or we would essentially
know what our y-intercept is

00:07:00.360 --> 00:07:02.230
going to be in
terms of our slope.

00:07:02.230 --> 00:07:04.532
And then, we can do
that for the hyperbola.

00:07:04.532 --> 00:07:06.240
And then, we could
essentially say, well,

00:07:06.240 --> 00:07:08.010
it's the same line
so the y-intercepts

00:07:08.010 --> 00:07:08.870
have to be the same.

00:07:08.870 --> 00:07:11.202
And then, we can
solve for the slope.

00:07:11.202 --> 00:07:11.910
So let's do that.

00:07:11.910 --> 00:07:13.805
And you'll see that over
the next few videos.

00:07:13.805 --> 00:07:16.180
Let me just write this in a
form that we would recognize.

00:07:16.180 --> 00:07:17.150
This is the same thing.

00:07:19.720 --> 00:07:21.380
Let me just multiply
this equation

00:07:21.380 --> 00:07:23.270
right here-- both
sides by negative 1.

00:07:23.270 --> 00:07:25.540
So then it would
become b squared

00:07:25.540 --> 00:07:31.870
plus 8mb minus 16 is equal to 0.

00:07:31.870 --> 00:07:33.570
I just multiplied
this by negative 1

00:07:33.570 --> 00:07:35.150
and just rearranged the terms.

00:07:35.150 --> 00:07:37.800
Now let's solve for
b in terms of m.

00:07:37.800 --> 00:07:45.320
So b is going to be equal
to negative 8m plus or minus

00:07:45.320 --> 00:07:51.190
the square root of
this term squared.

00:07:51.190 --> 00:07:57.140
So it's 8 squared m squared
minus 4 times a, which

00:07:57.140 --> 00:08:03.690
is just 1, times c, which is
16-- minus 4 times negative 16.

00:08:03.690 --> 00:08:06.335
So you could view this
as plus 4 times 16.

00:08:12.000 --> 00:08:15.940
All of that over 2a.

00:08:15.940 --> 00:08:17.280
Well, a here just 2.

00:08:17.280 --> 00:08:18.860
All of that over 2.

00:08:18.860 --> 00:08:25.500
Now, this is going to be equal
to negative 8m plus or minus--

00:08:25.500 --> 00:08:27.350
now, this is 64.

00:08:27.350 --> 00:08:28.740
This is 64.

00:08:28.740 --> 00:08:31.513
So you could factor
out the 64 from here,

00:08:31.513 --> 00:08:33.179
but when you take the
square root of it,

00:08:33.179 --> 00:08:39.354
it's going to be 8 times
m squared plus 1, right?

00:08:39.354 --> 00:08:41.270
If you took the 8 in,
you'd have to square it.

00:08:41.270 --> 00:08:42.309
So it becomes 64.

00:08:42.309 --> 00:08:45.250
And the 64 times
m squared plus 64,

00:08:45.250 --> 00:08:47.190
which is exactly what
you had up there.

00:08:47.190 --> 00:08:50.460
All of that over 2.

00:08:50.460 --> 00:08:51.690
And then, we can simplify it.

00:08:51.690 --> 00:08:56.700
This is equal to negative
4m plus or minus 4 times

00:08:56.700 --> 00:09:00.420
the square root of
m squared plus 1.

00:09:00.420 --> 00:09:04.090
So this is a possible b,
given that the line is

00:09:04.090 --> 00:09:08.940
tangent to the circle.

00:09:08.940 --> 00:09:11.740
Now, let's just think
about this a little bit.

00:09:11.740 --> 00:09:14.890
If we add 4, we're
definitely going

00:09:14.890 --> 00:09:17.740
to have-- well, let's
think about it for second.

00:09:17.740 --> 00:09:21.030
If we look at the line up
here, the way I drew it-- we

00:09:21.030 --> 00:09:23.805
want a positive slope.

00:09:23.805 --> 00:09:25.680
And in order to do that--
the way I drew it--

00:09:25.680 --> 00:09:27.490
you have to have a
positive y-intercept.

00:09:30.700 --> 00:09:34.130
Let me just write it this way.

00:09:34.130 --> 00:09:36.480
This is a positive b,
a positive y-intercept.

00:09:36.480 --> 00:09:38.000
So we want this value.

00:09:38.000 --> 00:09:41.070
We want to think about the
y-intercept that is positive.

00:09:41.070 --> 00:09:42.592
Now, m is going to be positive.

00:09:42.592 --> 00:09:44.050
We know from the
problem that we're

00:09:44.050 --> 00:09:45.460
looking for positive slope.

00:09:45.460 --> 00:09:46.530
So m is positive.

00:09:46.530 --> 00:09:49.720
So negative 4, this whole term
here, is going to be negative.

00:09:49.720 --> 00:09:51.220
So our only chance
of being positive

00:09:51.220 --> 00:09:55.242
is if we add 4 times this
expression right over here.

00:09:55.242 --> 00:09:56.700
And actually, if
you look at it, it

00:09:56.700 --> 00:10:00.780
will be positive, because this
is greater than m squared.

00:10:00.780 --> 00:10:03.970
So the square root of that's
going to be greater than m,

00:10:03.970 --> 00:10:06.550
so 4 times this is going
to be greater than 4m.

00:10:06.550 --> 00:10:08.760
So if we add, it's
going to be positive.

00:10:08.760 --> 00:10:14.100
So we want to only look at b
is equal to negative 4m plus 4

00:10:14.100 --> 00:10:18.180
times the square root
of m squared plus 1.

00:10:18.180 --> 00:10:19.680
I'll leave you there
for this video.

00:10:19.680 --> 00:10:22.260
In the next video, we're going
to do the exact same thing

00:10:22.260 --> 00:10:24.760
for the hyperbola realizing
that the line will only

00:10:24.760 --> 00:10:26.520
intersect at one point.

00:10:26.520 --> 00:10:28.400
And then, since
it's the same line,

00:10:28.400 --> 00:10:30.370
we know that their b's
have to be the same.

00:10:30.370 --> 00:10:31.911
In the next video,
we're going to get

00:10:31.911 --> 00:10:35.120
b is equal to some
other function of m.

00:10:35.120 --> 00:10:36.870
We're going to get
that in the next video.

00:10:36.870 --> 00:10:38.745
And then, we can set
them equal to each other

00:10:38.745 --> 00:10:39.780
and solve for our m.

00:10:39.780 --> 00:10:40.780
And then, when once
you solve for an m,

00:10:40.780 --> 00:10:42.300
you also have solved for the b.

00:10:42.300 --> 00:10:45.070
And we'll have our line.


IIT JEE circle hyperbola common tangent part 1

WEBVTT
Kind: captions
Language: en

00:00:00.930 --> 00:00:04.210
The circle x squared plus y
squared minus 8x equals 0,

00:00:04.210 --> 00:00:06.720
and the hyperbola x squared
over 9 minus y squared

00:00:06.720 --> 00:00:09.400
over 4 equal 1
intersect at the points

00:00:09.400 --> 00:00:17.820
A and B. Equation of a common
tangent with positive slope

00:00:17.820 --> 00:00:21.650
to the circle as well so
the hyperbola is-- so let's

00:00:21.650 --> 00:00:23.550
just visualize what
they're asking first.

00:00:23.550 --> 00:00:25.970
And this is going to
take us multiple videos,

00:00:25.970 --> 00:00:26.730
I have a feeling.

00:00:26.730 --> 00:00:28.080
But let's just
visualize it just so we

00:00:28.080 --> 00:00:29.690
can get our head
around the problem.

00:00:29.690 --> 00:00:32.900
So this circle, let me complete
the square in terms of x.

00:00:32.900 --> 00:00:36.970
So this circle, as they wrote
it, is x squared minus 8x.

00:00:36.970 --> 00:00:39.370
And then we have
a plus y squared--

00:00:39.370 --> 00:00:41.210
I left some space here
so we can complete

00:00:41.210 --> 00:00:44.360
the square-- is equal to 0.

00:00:44.360 --> 00:00:48.860
And then let me add half of this
8 term squared to both sides.

00:00:48.860 --> 00:00:50.930
So half of negative
8 is negative 4,

00:00:50.930 --> 00:00:52.980
and negative 4 squared is 16.

00:00:52.980 --> 00:00:55.670
So add 16 to both sides.

00:00:55.670 --> 00:01:00.950
And that allowed me to turn the
x-term into a perfect square.

00:01:00.950 --> 00:01:05.800
This is the same thing
as x minus 4 squared.

00:01:05.800 --> 00:01:08.540
And then we have plus y squared.

00:01:08.540 --> 00:01:12.700
Plus y squared is equal to 16.

00:01:12.700 --> 00:01:14.140
So this is a circle.

00:01:14.140 --> 00:01:21.320
This right here is a circle
with center at x equals 4.

00:01:21.320 --> 00:01:24.290
At x equals 4, y is equal to 0.

00:01:24.290 --> 00:01:29.830
And it has a radius
of 4 as well.

00:01:29.830 --> 00:01:34.190
So let me graph
this circle here.

00:01:34.190 --> 00:01:38.330
So let me draw the
horizontal axis, my x-axis.

00:01:38.330 --> 00:01:41.500
Let me draw the y-axis.

00:01:41.500 --> 00:01:44.220
That is my y-axis over here.

00:01:44.220 --> 00:01:49.051
And let me draw its
center, so 1, 2, 3, 4.

00:01:49.051 --> 00:01:50.246
4 comma 0.

00:01:50.246 --> 00:01:52.120
That's it's center, and
it has a radius of 4.

00:01:52.120 --> 00:01:53.578
So it's going to
come out, and it's

00:01:53.578 --> 00:01:57.832
going to look something--
I could draw a better

00:01:57.832 --> 00:01:58.540
circle than that.

00:01:58.540 --> 00:02:02.210
It's going to look something
like-- that's the top half,

00:02:02.210 --> 00:02:03.860
and then the bottom
half is going

00:02:03.860 --> 00:02:05.680
to look something like that.

00:02:05.680 --> 00:02:06.980
So that's our circle.

00:02:06.980 --> 00:02:10.460
Now let's think
about the hyperbola.

00:02:10.460 --> 00:02:13.671
So if we just look at it, the
x squared term is positive,

00:02:13.671 --> 00:02:15.170
so it's going to
be a hyperbola that

00:02:15.170 --> 00:02:16.503
opens to the right and the left.

00:02:16.503 --> 00:02:18.550
We do this a bunch in
the conic sections videos

00:02:18.550 --> 00:02:20.090
if you want to review of that.

00:02:20.090 --> 00:02:24.910
And we could just figure out
where it intersects the x-axis.

00:02:24.910 --> 00:02:28.320
So then when y is equal to
0, we have x squared over 9

00:02:28.320 --> 00:02:29.620
must be equal to 1.

00:02:29.620 --> 00:02:31.600
Or x would be plus or minus 3.

00:02:31.600 --> 00:02:34.630
So the hyperbola is
going to look like this.

00:02:34.630 --> 00:02:37.360
So this is at plus 3 comma 0.

00:02:37.360 --> 00:02:41.150
The hyperbola will
open up like that.

00:02:41.150 --> 00:02:45.840
And then at 1, 2, 3,
negative 3 comma 0,

00:02:45.840 --> 00:02:49.680
the hyperbola is going
to open up to the left.

00:02:49.680 --> 00:02:52.180
And so in the problem when they
describe the points A and B,

00:02:52.180 --> 00:02:56.450
they're probably talking about
that point A and that point B.

00:02:56.450 --> 00:02:59.130
Now, let's think about what
this question is asking us.

00:02:59.130 --> 00:03:02.290
Equation of a common tangent
with positive slope--

00:03:02.290 --> 00:03:04.600
so it has to have a positive
slope-- to the circle

00:03:04.600 --> 00:03:08.380
as well as to the
hyperbola-- a common tangent.

00:03:08.380 --> 00:03:10.490
So let's just think
about this a little bit.

00:03:10.490 --> 00:03:14.110
So it's going to have
a positive slope,

00:03:14.110 --> 00:03:16.072
so it won't be
tangent to the circle

00:03:16.072 --> 00:03:18.030
anywhere where the circle
has a negative slope.

00:03:18.030 --> 00:03:22.170
So it can't be
tangent over here.

00:03:22.170 --> 00:03:24.884
It can't be tangent over there.

00:03:24.884 --> 00:03:26.300
And then we could
say, well, if it

00:03:26.300 --> 00:03:28.591
was tangent to the circle
over here, what would happen?

00:03:28.591 --> 00:03:32.880
Well , it wouldn't be able to
be tangent to the hyperbola.

00:03:32.880 --> 00:03:35.430
So it has to be
tangent to the circle

00:03:35.430 --> 00:03:40.260
someplace in this blue
region right over here.

00:03:40.260 --> 00:03:42.300
And then how can it be
tangent to the hyperbola?

00:03:42.300 --> 00:03:44.130
It might be tempting
to say that it would

00:03:44.130 --> 00:03:47.980
be tangent to the hyperbola
in this way somehow,

00:03:47.980 --> 00:03:50.360
but what you need to
realize is the hyperbola

00:03:50.360 --> 00:03:51.920
is asymptoting
towards some line.

00:03:51.920 --> 00:03:55.320
And we could figure
out what that line is.

00:03:55.320 --> 00:03:57.590
Its asymptoting
towards some line.

00:03:57.590 --> 00:03:58.785
So let me draw that line.

00:04:03.810 --> 00:04:05.950
The hyperbola is always
going to have a higher

00:04:05.950 --> 00:04:09.330
slope than that line, a
very slightly higher slope.

00:04:09.330 --> 00:04:11.980
It's slowly
approaching that line.

00:04:11.980 --> 00:04:13.930
So if you go out
here, the hyperbola

00:04:13.930 --> 00:04:18.490
is going to have a higher
slope than the asymptote line.

00:04:18.490 --> 00:04:20.529
And so if you had
to be tangent to it,

00:04:20.529 --> 00:04:23.932
you would have to
have a higher slope.

00:04:23.932 --> 00:04:25.640
And anything that's
coming from this part

00:04:25.640 --> 00:04:28.520
of the circle towards anything
out here on the hyperbola

00:04:28.520 --> 00:04:31.181
is going to have to have a lower
slope than the tangent line,

00:04:31.181 --> 00:04:31.680
right?

00:04:31.680 --> 00:04:33.040
Because it's going
to have to catch up.

00:04:33.040 --> 00:04:34.415
The tangent line
is going to have

00:04:34.415 --> 00:04:36.340
to catch up to
whatever-- let me draw it

00:04:36.340 --> 00:04:38.420
again-- to whatever
we draw over here.

00:04:38.420 --> 00:04:41.370
I want to make this clear.

00:04:41.370 --> 00:04:42.940
The hyperbola, as
you go out here,

00:04:42.940 --> 00:04:46.880
actually this whole period, this
whole part of the hyperbola,

00:04:46.880 --> 00:04:49.620
is going to have a
higher slope than what

00:04:49.620 --> 00:04:50.900
it is asymptoting towards.

00:04:50.900 --> 00:04:54.870
That's what allows it to get
closer and closer to that line.

00:04:54.870 --> 00:04:57.440
So any tangent is going
to have to have a higher

00:04:57.440 --> 00:04:59.280
slope out here.

00:04:59.280 --> 00:05:00.725
Anything tangent
to the hyperbola

00:05:00.725 --> 00:05:02.100
is going to have
to have a higher

00:05:02.100 --> 00:05:04.340
slope than this actual line.

00:05:04.340 --> 00:05:07.230
It's going to have to have
a slightly higher slope.

00:05:07.230 --> 00:05:08.920
So if we take something
out here and we

00:05:08.920 --> 00:05:12.200
try to draw a tangent from this
part of the circle out there,

00:05:12.200 --> 00:05:13.210
it wouldn't work.

00:05:13.210 --> 00:05:16.970
Because this tangent,
by definition,

00:05:16.970 --> 00:05:19.050
in order to meet
the hyperbola, is

00:05:19.050 --> 00:05:22.430
going to have to have a lower
slope than this asymptote.

00:05:22.430 --> 00:05:25.050
So this can't be tangent to
that part of the hyperbola.

00:05:25.050 --> 00:05:26.540
So what else can we do?

00:05:26.540 --> 00:05:28.389
Well, the only other
part of the hyperbola

00:05:28.389 --> 00:05:30.180
that we might be able
to work something out

00:05:30.180 --> 00:05:33.890
is this part of the
hyperbola right over here.

00:05:33.890 --> 00:05:40.740
So if we find a line that's
tangent maybe there and there

00:05:40.740 --> 00:05:44.870
on the hyperbola, then we might
have found our common tangent

00:05:44.870 --> 00:05:46.060
with positive slope.

00:05:46.060 --> 00:05:47.550
So let me draw that.

00:05:47.550 --> 00:05:50.790
So our common tangent
with positive slope,

00:05:50.790 --> 00:05:52.090
I'll do that in pink.

00:05:52.090 --> 00:05:57.650
Our common tangent with positive
slope could look like that.

00:05:57.650 --> 00:05:59.680
So now that we have
the visualization down,

00:05:59.680 --> 00:06:03.120
in the next video
let's try to figure out

00:06:03.120 --> 00:06:09.290
what that line might look like,
especially when we constrain it

00:06:09.290 --> 00:06:11.470
to having to be
tangent to the circle

00:06:11.470 --> 00:06:15.430
and having to be tangent
to this hyperbola.