Partial fraction expansion 1

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:02.910
Let's see if we can learn a
thing or two about partial

00:00:02.910 --> 00:00:06.210
fraction expansion, or
sometimes it's called partial

00:00:06.210 --> 00:00:07.560
fraction decomposition.

00:00:14.390 --> 00:00:17.800
The whole idea is to take
rational functions-- and a

00:00:17.800 --> 00:00:20.980
rational function is just a
function or expression where

00:00:20.980 --> 00:00:25.170
it's one expression divided by
another-- and to essentially

00:00:25.170 --> 00:00:28.560
expand them or decompose
them into simpler parts.

00:00:28.560 --> 00:00:31.020
And the first thing you've got
to do, before you can even

00:00:31.020 --> 00:00:34.850
start the actual partial
fraction expansion process, is

00:00:34.850 --> 00:00:37.760
to make sure that the
numerator has a lower degree

00:00:37.760 --> 00:00:38.650
than the denominator.

00:00:38.650 --> 00:00:42.440
In the situation, the problem,
that I've drawn right here,

00:00:42.440 --> 00:00:44.560
I've written right here,
that's not the case.

00:00:44.560 --> 00:00:47.190
The numerator has the same
degree as the denominator.

00:00:47.190 --> 00:00:50.430
So the first step we want to do
to simplify this and to get it

00:00:50.430 --> 00:00:52.850
to the point where the
numerator has a lower degree

00:00:52.850 --> 00:00:54.780
than the denominator is
to do a little bit of

00:00:54.780 --> 00:00:56.270
algebraic division.

00:00:56.270 --> 00:00:59.320
And I've done a video on this,
but it never hurts to get a

00:00:59.320 --> 00:01:02.530
review here, so to do that, we
divide the denominator into the

00:01:02.530 --> 00:01:05.810
numerator to figure out the
remainder, so we divide x

00:01:05.810 --> 00:01:18.590
squared minus 3x minus 40 into
x squared minus 2x minus 37.

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So how many times?

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You look at the highest degree
term, so x squared goes into x

00:01:22.620 --> 00:01:28.380
squared one time, one times
this whole thing is x squared

00:01:28.380 --> 00:01:34.690
minus 3x minus 40, and now you
want to subtract this from

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that to get the remainder.

00:01:37.470 --> 00:01:40.010
And see, if I'm subtracting, so
I'm going to subtract, and then

00:01:40.010 --> 00:01:43.410
minus minus is a plus, a plus,
and then you can add them.

00:01:43.410 --> 00:01:44.540
These cancel out.

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Minus 2x plus 3x, that's x.

00:01:47.320 --> 00:01:51.680
Minus 37 plus 40,
that's plus 3.

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So this expression up here can
be rewritten as-- let me scroll

00:01:57.410 --> 00:02:05.680
down a little bit-- as 1 plus x
plus 3 over x squared

00:02:05.680 --> 00:02:09.690
minus 3x minus 40.

00:02:09.690 --> 00:02:12.250
This might seem like some type
of magic thing I just did, but

00:02:12.250 --> 00:02:15.380
this is no different than what
you did in the fourth or fifth

00:02:15.380 --> 00:02:18.930
grade, where you learned how
to convert improper regular

00:02:18.930 --> 00:02:21.620
fractions into mixed numbers.

00:02:21.620 --> 00:02:23.180
Let me just do a little
side example here.

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If I had 13 over 2, and I want
to turn it into a mixed number,

00:02:27.520 --> 00:02:29.910
what you do-- you can probably
do this in your head now-- but

00:02:29.910 --> 00:02:32.360
what you did is, you divide the
denominator into the numerator,

00:02:32.360 --> 00:02:33.810
just like we did over here.

00:02:33.810 --> 00:02:34.755
2 goes into 13.

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We see 2 goes into 13 six
times, 6 times 2 is 12, you

00:02:39.490 --> 00:02:42.050
subtract that from that,
you get a remainder of 1.

00:02:42.050 --> 00:02:44.870
2 doesn't go into 1, so
that's just the remainder.

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So if you wanted to rewrite
this, it would be the number of

00:02:48.310 --> 00:02:52.820
times the denominator goes into
the numerator, that's 6, plus

00:02:52.820 --> 00:02:55.160
the remainder over
the denominator.

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Plus 6-- plus 1 over 2.

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And when you did it in
elementary school, you would

00:02:59.110 --> 00:03:03.130
just write 6 1/2, but 6 1/2 is
the same thing as 6 plus 1/2.

00:03:03.130 --> 00:03:05.480
That's exactly the same
thing we did here.

00:03:05.480 --> 00:03:08.710
The denominator went to the
numerator one time, and then

00:03:08.710 --> 00:03:12.245
there was a remainder of x plus
3 left over, so it's 1 plus x

00:03:12.245 --> 00:03:14.590
plus 3 over this expression.

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Now we see that that numerator
in this rational expression

00:03:22.260 --> 00:03:24.760
does have a lower degree
than the denominator.

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The highest degree here is 1,
the highest degree here is 2,

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so we're ready to commence our
partial fraction decomposition.

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And all that is, is taking this
expression up here and turning

00:03:37.420 --> 00:03:41.290
it into two simpler expressions
where the denominators are the

00:03:41.290 --> 00:03:43.310
factors of this lower term.

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So given that, let's
factor this lower term.

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So let's see.

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What two numbers add up to
minus 3, and when you multiply

00:03:51.650 --> 00:03:54.050
them, you get minus 40?

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So let's see.

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They have to be different
signs, because when you

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multiply them you get a
negative, so it has to

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be minus 8 and plus 5.

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So we can rewrite this up here
as-- I'll switch colors--

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1 plus x plus 3 over x
plus 5 times x minus 8.

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5 times 8 is minus 40-- 5 times
negative 8 is minus 40, plus 5

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minus 8 is minus 3,
so we're all set.

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Now I'll just focus on
this part right now.

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We can just remember
that that 1 is sitting

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out there out front.

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This is the expression we
want to decompose or expand.

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And we're going to expand it
into two simpler expressions

00:04:39.010 --> 00:04:42.810
where each of these are the
denominator-- and I will make

00:04:42.810 --> 00:04:47.010
the claim, and if the numbers
work out then the claim is

00:04:47.010 --> 00:04:50.600
true-- I'll make the claim that
I can expand this, or decompose

00:04:50.600 --> 00:04:53.560
this, into two fractions where
the first fraction is just some

00:04:53.560 --> 00:05:00.500
number a over the first factor,
over x plus 5, plus some

00:05:00.500 --> 00:05:05.890
number b over the second
factor, over x minus 8.

00:05:05.890 --> 00:05:09.200
That's my claim, and if I can
solve for a and b in a way that

00:05:09.200 --> 00:05:11.530
it actually does add up to
this, then I'm done and I will

00:05:11.530 --> 00:05:15.250
have fully decomposed
this fraction.

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I guess is the way-- I
don't know if that's the

00:05:16.960 --> 00:05:17.850
correct terminology.

00:05:17.850 --> 00:05:18.750
So let's try to do that.

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So if I were to add these
two terms, what do I get?

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When you add anything, you find
the common denominator, and the

00:05:24.320 --> 00:05:26.640
common denominator, the easiest
common denominator, is to

00:05:26.640 --> 00:05:30.220
multiply the two denominators,
so let me write this here.

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So a over x plus 5 plus b
over x minus 8 is equal to--

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well, let's get the common
denominator-- it's equal to

00:05:42.230 --> 00:05:47.580
x plus 5 times x minus 8.

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And then the a term, we would--
a over x plus 5 is the same

00:05:52.080 --> 00:05:59.010
thing as a times x minus
8 over this whole thing.

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I mean, if I just wrote this
right here, you would just

00:06:01.030 --> 00:06:04.720
cancel these two terms out and
you would get a over x plus 5.

00:06:04.720 --> 00:06:11.010
And then you could add that to
the common denominator, x plus

00:06:11.010 --> 00:06:19.450
5 times x minus 8, and it
would be b times x plus 5.

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Important to realize,
that, look.

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This term is the exact same
thing as this term if you just

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cancel the x minus 8 out, and
this term is the exact same

00:06:27.330 --> 00:06:30.320
thing as this term if you just
cancel the x plus 5 out.

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But now that we have an actual
common denominator, we can add

00:06:32.800 --> 00:06:35.820
them together, so we get-- let
me just write the left side

00:06:35.820 --> 00:06:41.000
here over-- a over x
plus 5-- I'm sorry.

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I want to write this over here.

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I want to write x plus 3 over
plus 5 times x minus 8 is equal

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to is equal to the sum of
these two things on top.

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a times x minus 8 plus b times
x plus 5, all of that over

00:07:07.440 --> 00:07:15.830
their common denominator, x
plus 5 times x minus eight.

00:07:15.830 --> 00:07:18.600
So the denominators are the
same, so we know that this,

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when you add this together,
you have to get this.

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So if we want to solve
for a and b, let's just

00:07:22.600 --> 00:07:23.370
set that equality.

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We can ignore the denominators.

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So we can say that x plus 3
is equal to a times x minus

00:07:33.110 --> 00:07:38.430
8 plus b times x plus 5.

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Now, there's two ways to
solve for a and b from

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this point going forward.

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One is the way that I was
actually taught in the seventh

00:07:45.950 --> 00:07:48.150
or eighth grade, which tends to
take a little longer, then

00:07:48.150 --> 00:07:50.340
there's a fast way to do it and
it never hurts to do

00:07:50.340 --> 00:07:51.890
the fast way first.

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If you want to solve for a,
let's pick an x that'll

00:07:55.290 --> 00:07:56.940
make this term disappear.

00:07:56.940 --> 00:08:00.000
So what x would make
this term disappear?

00:08:00.000 --> 00:08:03.190
Well, if I say x is minus 5,
then this becomes 0, and

00:08:03.190 --> 00:08:04.760
then the b disappears.

00:08:04.760 --> 00:08:09.660
So if we say x is minus 5-- I'm
just picking an arbitrary x to

00:08:09.660 --> 00:08:14.610
be able to solve for this--
then this would become minus 5

00:08:14.610 --> 00:08:19.070
plus 3-- let me just write it
out, minus 5 plus 3-- is equal

00:08:19.070 --> 00:08:23.520
to a times minus 5 minus 8--
let me just write it out,

00:08:23.520 --> 00:08:29.750
minus 5 minus 8-- plus
b times minus 5 plus 5.

00:08:29.750 --> 00:08:33.610
And I picked the minus 5 to
make this expression 0.

00:08:33.610 --> 00:08:37.090
So then you get-- pick a
brighter color-- minus 5 plus

00:08:37.090 --> 00:08:44.230
3 is minus 2, is equal to--
what is this?-- minus 13a

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plus-- this is 0, right?

00:08:45.990 --> 00:08:46.690
That's 0.

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Minus 5 plus 5 is 0, 0 times b
is 0, and then you divide both

00:08:50.130 --> 00:08:54.600
sides by minus 13, you get--
negatives cancel out-- you get

00:08:54.600 --> 00:08:59.940
2 over 13 is equal to a, and
now we can do the same thing up

00:08:59.940 --> 00:09:05.270
here and get rid of the a terms
by making x is equal to 8.

00:09:05.270 --> 00:09:11.150
If x is equal to 8, you get x
plus 3 is equal to 11, is equal

00:09:11.150 --> 00:09:19.660
to a times 0 plus b times--
what's 5-- 8 plus 5

00:09:19.660 --> 00:09:23.340
is-- plus b times 13.

00:09:23.340 --> 00:09:25.880
Their b looks a bit like a 13.

00:09:25.880 --> 00:09:32.430
And then you get 11 is equal to
13b, divide both sides by 13,

00:09:32.430 --> 00:09:36.130
you get b is equal
to 11 over 13.

00:09:36.130 --> 00:09:39.300
So we were able to solve
for our a's and our b's.

00:09:39.300 --> 00:09:42.370
And so we can go back to
our original equation

00:09:42.370 --> 00:09:43.060
and we could say, wow.

00:09:43.060 --> 00:09:47.960
This just has to be equal to
2 over 13, and this just has

00:09:47.960 --> 00:09:49.765
to be equal to 11 over 13.

00:09:52.310 --> 00:09:56.080
So our original, our very
original thing we wrote up

00:09:56.080 --> 00:10:02.750
here, can be decomposed into 1,
that's this 1 over here, plus

00:10:02.750 --> 00:10:08.150
this, which is 2 over 13-- I'll
just write it like this for

00:10:08.150 --> 00:10:11.850
now-- 2 over 13, over x plus 5.

00:10:11.850 --> 00:10:13.735
You could bring the 13 down
here if you want to write

00:10:13.735 --> 00:10:15.970
it so you don't have a
fraction over a fraction.

00:10:15.970 --> 00:10:27.115
Plus 11 over 13 times--
over x minus 8.

00:10:27.115 --> 00:10:29.190
And once again, you could bring
the 13 down so you don't have

00:10:29.190 --> 00:10:30.400
a fraction over a fraction.

00:10:30.400 --> 00:10:36.075
But we have just successfully
decomposed this pretty-- I

00:10:36.075 --> 00:10:37.600
don't want to say that we
necessarily simplified it,

00:10:37.600 --> 00:10:39.320
because you could say, oh, we
only have one expression here,

00:10:39.320 --> 00:10:41.690
now I have three-- but I've
reduced the degree of both the

00:10:41.690 --> 00:10:43.660
numerators and the
denominators.

00:10:43.660 --> 00:10:45.420
And you might say, well,
Sal, why would I ever

00:10:45.420 --> 00:10:46.250
have to do this?

00:10:46.250 --> 00:10:47.140
And you're right.

00:10:47.140 --> 00:10:48.680
In algebra you probably won't.

00:10:48.680 --> 00:10:51.800
But this is actually a really
useful technique later on when

00:10:51.800 --> 00:10:53.560
you get to calculus, and
actually, differential

00:10:53.560 --> 00:10:56.160
equations, because a lot of
times it's much easier-- and

00:10:56.160 --> 00:10:58.080
I'll throw out a word here that
you don't understand-- to take

00:10:58.080 --> 00:11:00.000
the integral or the
antiderivative of

00:11:00.000 --> 00:11:02.550
something like this, then
something like this.

00:11:02.550 --> 00:11:05.930
And later, when you do inverse
Laplace transforms and

00:11:05.930 --> 00:11:08.600
differential equations, it's
much easier take an inverse

00:11:08.600 --> 00:11:10.920
Laplace transform of
something like this than

00:11:10.920 --> 00:11:12.070
something like that.

00:11:12.070 --> 00:11:16.120
So anyway, hopefully I've given
you another tool kit in your--

00:11:16.120 --> 00:11:19.950
or another tool in your tool
kit, and I'll probably do a

00:11:19.950 --> 00:11:22.750
couple more videos because we
haven't exhausted all of the

00:11:22.750 --> 00:11:24.800
examples that we could we
could show for partial

00:11:24.800 --> 00:11:25.770
fraction decomposition.


Proof: Hyperbola foci

WEBVTT
Kind: captions
Language: en

00:00:01.270 --> 00:00:04.040
In the last video, I told you
that if I had a hyperbola with

00:00:04.040 --> 00:00:11.080
the equation x squared over a
squared minus y squared over b

00:00:11.080 --> 00:00:15.500
squared is equal to 1, that the
focal distance for this

00:00:15.500 --> 00:00:17.750
hyperbola is just equal to the
square root of the sum

00:00:17.750 --> 00:00:18.995
of these two numbers.

00:00:18.995 --> 00:00:22.510
The square root of a
squared plus b squared.

00:00:22.510 --> 00:00:24.910
In this video I really just
want to show you that.

00:00:24.910 --> 00:00:29.290
And actually, just so you know,
this equation right here, this

00:00:29.290 --> 00:00:32.090
is a particular hyperbola that
opens to the left

00:00:32.090 --> 00:00:32.650
and the right.

00:00:32.650 --> 00:00:34.900
And that's because those
are the asymptote points.

00:00:34.900 --> 00:00:36.970
Those would be the axis.

00:00:36.970 --> 00:00:39.350
And that's because the
x term is positive.

00:00:39.350 --> 00:00:42.690
If the y term was positive and
the x term had a negative sign,

00:00:42.690 --> 00:00:45.210
then the hyperbola would open
upwards and downwards,

00:00:45.210 --> 00:00:46.470
like that.

00:00:46.470 --> 00:00:48.890
And the proof that I'm showing
you in this video, it's just a

00:00:48.890 --> 00:00:52.110
bunch of algebra, really, is
identical in the y case, you'd

00:00:52.110 --> 00:00:53.710
just switch around
x's and the y's.

00:00:53.710 --> 00:00:56.130
But I just wanted to make
sure you realize that.

00:00:56.130 --> 00:00:58.050
That I'm just doing a
particular case of hyperbola

00:00:58.050 --> 00:00:59.370
that opens the left
and the right.

00:00:59.370 --> 00:01:01.830
I could call it a horizontal
hyperbola, instead of a

00:01:01.830 --> 00:01:03.840
vertical one, but I wanted
to make it clear there is

00:01:03.840 --> 00:01:05.730
another type of hyperbola.

00:01:05.730 --> 00:01:07.790
But anyway, let's draw a
graphical representation of all

00:01:07.790 --> 00:01:10.530
this just to make sure
we understand, or we

00:01:10.530 --> 00:01:13.640
re-understand, or better
understand what the foci points

00:01:13.640 --> 00:01:18.570
are and where they sit on the
hyperbola So those are my axis.

00:01:18.570 --> 00:01:20.910
The asymptotes of this
hyperbola are the lines

00:01:20.910 --> 00:01:24.090
y is equal to plus
or minus b over a.

00:01:24.090 --> 00:01:27.830
Oh woops, not using
my line tool.

00:01:27.830 --> 00:01:34.530
So, that's one and that's
the other asymptote.

00:01:34.530 --> 00:01:38.410
then the hyperbola will
look something like this.

00:01:38.410 --> 00:01:39.670
It looks something like that.

00:01:39.670 --> 00:01:42.940
It's going to intersect at
a comma 0, right there.

00:01:42.940 --> 00:01:44.700
This is going to be a comma 0.

00:01:44.700 --> 00:01:47.140
And the intersect at
minus a comma 0.

00:01:47.140 --> 00:01:48.460
We saw this in the
previous video.

00:01:48.460 --> 00:01:50.080
It looks something like that.

00:01:50.080 --> 00:01:52.370
And then the focus
points are going to sit

00:01:52.370 --> 00:01:54.070
out here someplace.

00:01:54.070 --> 00:01:54.980
There and there.

00:01:54.980 --> 00:01:58.090
And the focal length this
a squared plus b square.

00:01:58.090 --> 00:01:59.940
The square root of a
squared plus b squared.

00:01:59.940 --> 00:02:02.690
That's just this
distance right here.

00:02:02.690 --> 00:02:04.000
That distance is
the focal length.

00:02:04.000 --> 00:02:08.080
So this is going to be the
point f,0 and this is going

00:02:08.080 --> 00:02:12.330
to be the point minus f,0.

00:02:12.330 --> 00:02:15.510
Now we learned in the last
video that one of the

00:02:15.510 --> 00:02:19.070
definitions of a hyperbola is
the locus of all points, or the

00:02:19.070 --> 00:02:21.940
set of all points, where if I
take the difference of the

00:02:21.940 --> 00:02:25.700
distances to the two foci, that
difference will be

00:02:25.700 --> 00:02:27.340
a constant number.

00:02:27.340 --> 00:02:31.620
So if this is the point x comma
y, and it could be any point

00:02:31.620 --> 00:02:33.730
that satisfies this equation,
it's any point on the

00:02:33.730 --> 00:02:38.840
byperbola, we know, or we are
told, that if we take this

00:02:38.840 --> 00:02:45.120
distance right here-- --let's
call that d1 --and subtract

00:02:45.120 --> 00:02:53.370
from that the distance to the
high other foci-- call that d2

00:02:53.370 --> 00:02:56.190
--that that number is a
constant regardless of where we

00:02:56.190 --> 00:02:57.950
are on the hyperbola.

00:02:57.950 --> 00:03:01.530
In fact the locus of all points
are-- the hyperbole in fact

00:03:01.530 --> 00:03:04.820
is all of the points that
satisfy that condition.

00:03:04.820 --> 00:03:08.500
And we learned in last video
just by taking the difference

00:03:08.500 --> 00:03:10.390
of the distance, we picked this
point and we said, OK, what's

00:03:10.390 --> 00:03:12.070
that distance minus
that distance.

00:03:12.070 --> 00:03:14.070
And we figured out
that it's 2a.

00:03:14.070 --> 00:03:19.570
So d1 minus d2 is equal to--
I'm going off the video screen

00:03:19.570 --> 00:03:22.800
--d1 minus d2 is equal to 2a.

00:03:22.800 --> 00:03:27.130
So let's use this fact right
here that d1 minus d2 is equal

00:03:27.130 --> 00:03:32.190
to 2a, to try to prove this.

00:03:32.190 --> 00:03:34.160
Right there.

00:03:34.160 --> 00:03:38.890
So the first thing to do is
figure out what d1 and d2.

00:03:38.890 --> 00:03:40.140
Just using the
distance formula.

00:03:40.140 --> 00:03:41.930
So what's d1?

00:03:41.930 --> 00:03:46.670
d1 is the distance between this
point and this point minus f0.

00:03:46.670 --> 00:03:49.280
So what you do is we just
use the distance formula,

00:03:49.280 --> 00:03:50.980
which is really just
Pythagorean theorem.

00:03:50.980 --> 00:03:53.840
So it's the difference
of the x's.

00:03:53.840 --> 00:03:54.850
It's the x distance.

00:03:54.850 --> 00:04:01.890
So it's x minus f squared
plus the y distances.

00:04:01.890 --> 00:04:07.010
y minus 0, so that's
just y, squared.

00:04:07.010 --> 00:04:09.290
Take the square root of that.

00:04:09.290 --> 00:04:12.530
So that's d1, right there.

00:04:12.530 --> 00:04:14.140
d1.

00:04:14.140 --> 00:04:17.570
And we want to subtract
from that d2.

00:04:17.570 --> 00:04:20.090
the difference of the
distances, and in this case d1

00:04:20.090 --> 00:04:21.770
is definitely bigger than d2.

00:04:21.770 --> 00:04:23.540
Or you could take the absolute
values if you didn't want

00:04:23.540 --> 00:04:24.870
to worry about that.

00:04:24.870 --> 00:04:31.740
And so here, we get the square
root of x minus f, x minus

00:04:31.740 --> 00:04:36.720
f squared, plus y squared.

00:04:36.720 --> 00:04:37.790
What does that equal to?

00:04:37.790 --> 00:04:40.570
Well, we said that equals
to 2a, that equals this

00:04:40.570 --> 00:04:42.350
distance right here.

00:04:42.350 --> 00:04:44.255
So that is equal to 2a.

00:04:44.255 --> 00:04:47.180
Now let's see if we can
simplify this at all.

00:04:47.180 --> 00:04:50.430
Well an interesting thing to do
my just to be the put this on

00:04:50.430 --> 00:04:53.350
the other side of the equation.

00:04:53.350 --> 00:04:55.120
And this just can get hairy,
so I really hope I don't

00:04:55.120 --> 00:04:56.230
make any careless mistakes.

00:04:56.230 --> 00:05:00.850
So this becomes-- and I might
write small to save space

00:05:00.850 --> 00:05:06.220
--this becomes x plus f, right,
minus minus, squared plus y

00:05:06.220 --> 00:05:15.810
squared is equal to 2a plus the
square root of x minus f

00:05:15.810 --> 00:05:19.180
squared plus y squared.

00:05:19.180 --> 00:05:21.240
Now, to get rid of these
radicals, let's square both

00:05:21.240 --> 00:05:22.740
sides of this equation.

00:05:22.740 --> 00:05:26.990
The left hand side, if you were
to square it just becomes x

00:05:26.990 --> 00:05:32.980
plus f squared plus y squared.

00:05:32.980 --> 00:05:35.970
And then to square this we have
to square the first term,

00:05:35.970 --> 00:05:39.140
which is 4a squared.

00:05:39.140 --> 00:05:42.630
Then we multiply the two terms
and multiply that by 2, right?

00:05:42.630 --> 00:05:44.530
We're just taking this whole
thing and squaring it, so

00:05:44.530 --> 00:05:47.370
that's-- and this is just a
review of kind of binomial

00:05:47.370 --> 00:05:53.960
algebra --so this is equal to
plus 2a times this times 2 is

00:05:53.960 --> 00:06:01.980
4a times the square root of
x minus f squared plus y.

00:06:01.980 --> 00:06:04.430
y squared, I don't want to lose
that squared right there.

00:06:04.430 --> 00:06:05.780
And then we square this term.

00:06:05.780 --> 00:06:08.870
And this is just
multiplying a binomial.

00:06:08.870 --> 00:06:11.300
So that's equal to-- you just
get rid of the radical sign,

00:06:11.300 --> 00:06:13.760
and I'm just going to be
staying in that color for now

00:06:13.760 --> 00:06:20.270
--that's equal to x minus
f squared plus y squared.

00:06:20.270 --> 00:06:22.670
And already it looks like
there's some cancellation

00:06:22.670 --> 00:06:23.220
that we can do.

00:06:23.220 --> 00:06:26.430
We can cancel out-- there's a
y squared on both sides of

00:06:26.430 --> 00:06:28.270
this equation --so let's
just cancel those out.

00:06:28.270 --> 00:06:30.940
Subtract y squared from both
sides of the equation.

00:06:30.940 --> 00:06:33.070
And let's multiply
this term out.

00:06:33.070 --> 00:06:42.840
So this right here is x squared
plus 2xf plus f squared.

00:06:42.840 --> 00:06:49.940
And then that is equal to 4a
squared plus 4a times the

00:06:49.940 --> 00:06:58.896
square root of x minus f
squared plus y squared.

00:06:58.896 --> 00:07:00.750
And then multiply this out.

00:07:00.750 --> 00:07:07.230
Plus x squared minus
2xf plus f squared.

00:07:07.230 --> 00:07:08.860
And then let's see, what
can we cancel out.

00:07:08.860 --> 00:07:11.150
We have x squared on both
sides of this, we subtract

00:07:11.150 --> 00:07:13.400
x squared from both
sides of the equation.

00:07:13.400 --> 00:07:15.460
We have an f squared on both
sides of the equation so

00:07:15.460 --> 00:07:17.760
let's cancel that out.

00:07:17.760 --> 00:07:19.430
And let's see, what can
we do to simplify it.

00:07:19.430 --> 00:07:22.330
So we have a minus
2xf and a plus 2xf.

00:07:22.330 --> 00:07:25.080
Let's add 2xf to both sides
this equation or bring

00:07:25.080 --> 00:07:26.670
this term over here.

00:07:26.670 --> 00:07:30.610
So if you add 2xf to both sides
of this equation-- let's see,

00:07:30.610 --> 00:07:34.180
my phone is ringing, let me
just turn it off --if you add

00:07:34.180 --> 00:07:37.780
2xf to both sides of this
equation, what do you get?

00:07:37.780 --> 00:07:43.010
You get 4xf-- remember I just
brought this term over this

00:07:43.010 --> 00:07:52.710
left hand side --is equal to
4a squared plus 4a times a

00:07:52.710 --> 00:07:58.120
square root of x minus f
squared plus y squared.

00:07:58.120 --> 00:07:59.270
It's easy to get lost
in the algebra.

00:07:59.270 --> 00:08:02.690
Remember all we're doing, just
to kind of remind you of what

00:08:02.690 --> 00:08:05.410
this whole point was, we're
just simplifying the difference

00:08:05.410 --> 00:08:08.290
of the distances between these
two points, and then see how

00:08:08.290 --> 00:08:11.840
it relates to the equation
of the hyperbola itself.

00:08:11.840 --> 00:08:13.570
The a's and the b's.

00:08:13.570 --> 00:08:19.980
Let's take this 4a put it on
this side, so you get 4xf minus

00:08:19.980 --> 00:08:27.820
4a squared is equal to 4a times
the square root of-- well let's

00:08:27.820 --> 00:08:29.950
just multiply this out 'cause
we'll probably have to

00:08:29.950 --> 00:08:36.470
eventually --x squared minus
2xf plus f squared

00:08:36.470 --> 00:08:38.500
plus y squared.

00:08:38.500 --> 00:08:39.990
That's this just
multiplied out.

00:08:39.990 --> 00:08:42.000
That's the y squared
right there.

00:08:42.000 --> 00:08:44.160
We could divide both
sides of this by 4.

00:08:44.160 --> 00:08:46.390
All I'm trying to do is just
simplify this as much as

00:08:46.390 --> 00:08:54.180
possible, so then this becomes
xf minus a squared is equal to

00:08:54.180 --> 00:08:57.450
a times the square root
of this whole thing.

00:08:57.450 --> 00:09:05.970
x squared minus 2xf plus f
squared plus why squared.

00:09:05.970 --> 00:09:09.140
And now we could square
both sides of this

00:09:09.140 --> 00:09:10.440
equation right here.

00:09:10.440 --> 00:09:15.060
And then if you square both
sides, this side becomes x

00:09:15.060 --> 00:09:26.900
squared f squared minus 2a
squared xf plus a

00:09:26.900 --> 00:09:28.020
to the fourth.

00:09:28.020 --> 00:09:29.610
That's this side squared.

00:09:29.610 --> 00:09:33.600
And that's equal to, if you
square the right hand side, a

00:09:33.600 --> 00:09:37.490
squared times the square of
a square root is just that

00:09:37.490 --> 00:09:46.180
expression, x squared minus 2xf
plus f squared plus y squared.

00:09:46.180 --> 00:09:48.470
This really is quite hairy.

00:09:48.470 --> 00:09:50.350
And let's see what
we can do now.

00:09:50.350 --> 00:09:55.340
Let's divide both sides of this
equation by a squared, and then

00:09:55.340 --> 00:10:00.150
you get x squared-- I'm really
just trying to simplify this as

00:10:00.150 --> 00:10:04.000
much as possible --over a
squared minus-- so the a

00:10:04.000 --> 00:10:10.500
squareds cancel out --minus 2xf
plus a to the fourth divided a

00:10:10.500 --> 00:10:12.470
square well that's
just a squared.

00:10:12.470 --> 00:10:19.840
So a squared is equal to x
squared minus 2xf plus f

00:10:19.840 --> 00:10:23.900
squared plus y squared.

00:10:23.900 --> 00:10:24.410
Well good.

00:10:24.410 --> 00:10:25.360
There's something
to cancel out.

00:10:25.360 --> 00:10:28.280
There's a mine 2xf on both
sides of this equation so

00:10:28.280 --> 00:10:29.870
let's cancel that out.

00:10:29.870 --> 00:10:32.530
Simplify our situation
a little bit.

00:10:32.530 --> 00:10:34.210
And let's see, we have.

00:10:34.210 --> 00:10:37.580
so we could do is subtract
this x squared from this.

00:10:37.580 --> 00:10:43.300
So you get-- let me rewrite
it --so you get x squared

00:10:43.300 --> 00:10:48.100
f squared over a squared
minus x squared.

00:10:48.100 --> 00:10:52.330
And let's bring this y to this
side of the equation too.

00:10:52.330 --> 00:10:54.240
So minus y squared.

00:10:54.240 --> 00:10:56.900
That's all I did, I just
brought that to that side.

00:10:56.900 --> 00:10:59.540
And then let's bring-- and I'm
kind of skipping a couple of

00:10:59.540 --> 00:11:02.310
steps, but I don't want to take
too long --let's take this a

00:11:02.310 --> 00:11:04.580
and put it on that
side equation.

00:11:04.580 --> 00:11:09.430
So we took the x and the y,
we subtracted that from

00:11:09.430 --> 00:11:10.220
both sides of the equation.

00:11:10.220 --> 00:11:12.040
So they ended up on
the left hand side.

00:11:12.040 --> 00:11:14.600
And then if we subtract a
squared from both sides of this

00:11:14.600 --> 00:11:17.510
equation-- this is a fatiguing
problem --then you get f

00:11:17.510 --> 00:11:21.850
squared minus a squared.

00:11:21.850 --> 00:11:23.370
I think we're almost there.

00:11:23.370 --> 00:11:26.970
This can simplify to,
let's see we can factor

00:11:26.970 --> 00:11:28.340
out the x squared.

00:11:28.340 --> 00:11:33.640
This becomes f squared
over a squared minus

00:11:33.640 --> 00:11:35.790
1 times x squared.

00:11:35.790 --> 00:11:40.420
I just factor out the x squared
there, minus y squared is equal

00:11:40.420 --> 00:11:46.750
to f squared, the focal length
squared, minus a squared.

00:11:46.750 --> 00:11:49.310
And let's see, let's divide
both sides of the equation by

00:11:49.310 --> 00:11:53.230
this expression right there,
and we get-- and this should

00:11:53.230 --> 00:12:00.490
start to look familiar --we get
f squared over a squared minus

00:12:00.490 --> 00:12:09.770
1, x squared divided by f
squared minus a squared minus y

00:12:09.770 --> 00:12:15.450
squared over f squared mine
is a squared is equal 1.

00:12:15.450 --> 00:12:17.880
Right, I divided both sides
by this, so I just get a 1

00:12:17.880 --> 00:12:18.910
on this right hand side.

00:12:18.910 --> 00:12:20.930
Let's see if I can
simplify this.

00:12:20.930 --> 00:12:23.790
If I multiply the numerator
and the denominator

00:12:23.790 --> 00:12:25.850
by a squared, right?

00:12:25.850 --> 00:12:27.420
As long as I multiply the
numerator and the denominator

00:12:27.420 --> 00:12:29.430
by the same number I'm just
multiplying my 1, so I'm

00:12:29.430 --> 00:12:30.870
not changing anything.

00:12:30.870 --> 00:12:36.200
So if I do that, the numerator
becomes f-- if I multiply it,

00:12:36.200 --> 00:12:40.880
it becomes f squared
minus a squared.

00:12:40.880 --> 00:12:43.080
I'm just multiplying
that times a squared.

00:12:43.080 --> 00:12:46.750
And the denominator
becomes a squared times f

00:12:46.750 --> 00:12:49.240
squared minus a squared.

00:12:49.240 --> 00:12:51.680
And all that times x squared.

00:12:51.680 --> 00:12:55.100
Minus y squared minus
a squared mine is a

00:12:55.100 --> 00:12:57.160
squared is equal to 1.

00:12:57.160 --> 00:12:59.450
This cancels with this.

00:12:59.450 --> 00:13:01.630
And we get something to
starting to look like the

00:13:01.630 --> 00:13:03.180
equation of a hyperbola.

00:13:03.180 --> 00:13:05.080
My energy is coming back!

00:13:05.080 --> 00:13:07.510
It seems like I see the light
at the end of the tunnel.

00:13:07.510 --> 00:13:14.550
We get x squared over a squared
minus y squared over f minus

00:13:14.550 --> 00:13:17.260
a squared is equal to 1.

00:13:17.260 --> 00:13:19.820
Now this looks a lot like our
original equation of the

00:13:19.820 --> 00:13:25.940
hyperbola, which was x squared
over a squared minus y squared

00:13:25.940 --> 00:13:28.630
over b squared is equal to 1.

00:13:28.630 --> 00:13:30.790
In fact this is equation of the
hyperbola but instead set of

00:13:30.790 --> 00:13:34.580
writing b squared, since we
wrote it, we essentially said,

00:13:34.580 --> 00:13:37.030
what is the locus of all points
where the difference of the

00:13:37.030 --> 00:13:41.410
distances to those two
foci is equal to 2a?

00:13:41.410 --> 00:13:44.010
And we just played with
the algebra for while.

00:13:44.010 --> 00:13:46.410
It was pretty tiring, and I'm
impressed if you've gotten this

00:13:46.410 --> 00:13:48.860
far into the video, and we got
this equation, which should be

00:13:48.860 --> 00:13:50.890
the equation of the hyperbola,
and it is the equation

00:13:50.890 --> 00:13:51.520
of the hyperbole.

00:13:51.520 --> 00:13:53.110
It is this equation.

00:13:53.110 --> 00:13:56.690
So this is the same
thing is that.

00:13:56.690 --> 00:13:59.390
So f squared minus a square.

00:13:59.390 --> 00:14:02.460
Or the focal length squared
minus a squared is

00:14:02.460 --> 00:14:04.830
equal to b squared.

00:14:04.830 --> 00:14:08.170
You add a squared to both
sides, and you get f squared

00:14:08.170 --> 00:14:11.740
is equal to b squared plus
a squared or a squared

00:14:11.740 --> 00:14:13.190
plus b squared.

00:14:13.190 --> 00:14:16.310
Which tells us that the focal
length is equal to the

00:14:16.310 --> 00:14:18.960
square root of this.

00:14:18.960 --> 00:14:21.440
Of a squared plus b squared.

00:14:21.440 --> 00:14:25.440
And that's what we set out to
figure out in the beginning.

00:14:25.440 --> 00:14:28.020
So hopefully you're now
satisfied that the focal length

00:14:28.020 --> 00:14:31.480
of a hyperbola is the sum of
these two denominators.

00:14:31.480 --> 00:14:36.940
And it's also truth if it is an
upward or vertical hyperbola.

00:14:36.940 --> 00:14:39.220
And if we're dealing with an
ellipse, it's the difference of

00:14:39.220 --> 00:14:40.590
these two-- the square root
of the difference of

00:14:40.590 --> 00:14:41.730
these two numbers.

00:14:41.730 --> 00:14:43.240
Anyway, I'll leave you there.

00:14:43.240 --> 00:14:44.630
That was an exhausting problem.

00:14:44.630 --> 00:14:47.210
I have to go get a
glass of water now.


Foci of a hyperbola

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.040
In the last video, we learned
that an ellipse can be defined

00:00:05.040 --> 00:00:08.660
as the locus of all points
where the sum of the distances

00:00:08.660 --> 00:00:11.702
to two special points, called
foci-- and let me draw this all

00:00:11.702 --> 00:00:15.090
out, so that's my x-axis-- the
sum of the distance to these

00:00:15.090 --> 00:00:19.040
two special points, called
focuses or foci, is a constant.

00:00:19.040 --> 00:00:25.330
So if this is my ellipse-- I'll
draw it out, that looks about

00:00:25.330 --> 00:00:29.630
where I want it to be right
around, that looks about

00:00:29.630 --> 00:00:31.640
right-- it's centered at the
origin, doesn't have to be, but

00:00:31.640 --> 00:00:35.040
for our purposes, let's make
it centered on the origin.

00:00:35.040 --> 00:00:40.170
If this is one focus point
right here, and this is the

00:00:40.170 --> 00:00:45.180
other focus point, this ellipse
could be defined as the set of

00:00:45.180 --> 00:00:49.260
all points, or the locus of all
points, where if I take the

00:00:49.260 --> 00:00:52.920
distance of any one of these
points that exist on the

00:00:52.920 --> 00:00:56.300
ellipse and take the distance
to each of the locuses-- sorry,

00:00:56.300 --> 00:01:00.730
each of the focuses-- if I take
that-- nope, I don't want to do

00:01:00.730 --> 00:01:08.220
that-- if I take that distance
and add it to this distance, so

00:01:08.220 --> 00:01:19.770
let's call this d1-- nope, too
thick-- let's call that d1,

00:01:19.770 --> 00:01:23.100
this is d2-- that that's going
to be equal to a constant

00:01:23.100 --> 00:01:25.100
number along the whole ellipse.

00:01:25.100 --> 00:01:28.540
So if I take a random point
along the ellipse, say I take

00:01:28.540 --> 00:01:38.930
this point right here, and if I
were to sum this distance to

00:01:38.930 --> 00:01:48.160
that distance-- so let's call
this d3, this is d4-- the sums

00:01:48.160 --> 00:01:50.840
of these distances to the focus
along with this ellipse are

00:01:50.840 --> 00:01:52.100
going to be a constant.

00:01:52.100 --> 00:01:58.330
So in this case, d2 plus d1,
this plus that, is going

00:01:58.330 --> 00:02:01.120
to be equal to d3 plus d4.

00:02:03.840 --> 00:02:06.410
And this would be true wherever
you go along the whole ellipse,

00:02:06.410 --> 00:02:08.430
and we learned in the last
video that this quantity is

00:02:08.430 --> 00:02:12.715
actually going to be equal to
2a, where a is the distance

00:02:12.715 --> 00:02:14.800
of the semi-major radius.

00:02:14.800 --> 00:02:16.830
If this is the formula
for the ellipse, this is

00:02:16.830 --> 00:02:18.310
where the a comes from.

00:02:18.310 --> 00:02:21.330
x squared over a squared
plus y squared over b

00:02:21.330 --> 00:02:24.230
squared is equal to 1.

00:02:24.230 --> 00:02:27.750
And we learned that the focus,
the focal distance-- or the

00:02:27.750 --> 00:02:30.510
distance from the central of
the ellipse, which is this

00:02:30.510 --> 00:02:35.400
distance right here-- that
focal distance is just the

00:02:35.400 --> 00:02:39.640
square root of the difference
of these two numbers.

00:02:39.640 --> 00:02:44.040
If this is the focal distance
from here to here, it's just

00:02:44.040 --> 00:02:49.490
equal to the square root of--
if a is larger then it would be

00:02:49.490 --> 00:02:52.220
a squared minus b squared,
which is the case

00:02:52.220 --> 00:02:53.200
in this ellipse.

00:02:53.200 --> 00:02:55.430
If we have a vertical ellipse,
and I really didn't cover it in

00:02:55.430 --> 00:02:57.450
the last video, but let me just
show you what it

00:02:57.450 --> 00:02:58.690
would look like.

00:02:58.690 --> 00:03:01.650
Let's say that the ellipse
looks something like this.

00:03:05.400 --> 00:03:08.110
I'll use blue.

00:03:08.110 --> 00:03:12.560
Let's say the ellipse
looks like that.

00:03:12.560 --> 00:03:17.720
In this case, our semi-major
radius is now in the y

00:03:17.720 --> 00:03:24.280
direction, so this is b, this
is a, and in this case, b is

00:03:24.280 --> 00:03:31.200
greater than a, because the
ellipse is tall and skinny.

00:03:31.200 --> 00:03:34.460
In this case the, focuses
are always going to lie

00:03:34.460 --> 00:03:35.900
along the major axis.

00:03:35.900 --> 00:03:38.050
In this case the major axis is
the vertical axis, so the

00:03:38.050 --> 00:03:43.640
focuses are going to sit here
and here, and in this case, the

00:03:43.640 --> 00:03:48.520
focal lengths are going to be
vertical down from the origin

00:03:48.520 --> 00:03:52.550
and vertical up from the
origin, and we get, instead of

00:03:52.550 --> 00:03:55.770
it being a squared minus b
squared, now since b is larger

00:03:55.770 --> 00:04:00.940
than a, the focal length, which
is this, is going to be equal

00:04:00.940 --> 00:04:04.680
to b squared minus a squared.

00:04:04.680 --> 00:04:06.040
Fair enough.

00:04:06.040 --> 00:04:08.540
Now I did all of that to kind
of compare it to what we're

00:04:08.540 --> 00:04:11.940
going to cover in this video,
which is the focus points or

00:04:11.940 --> 00:04:14.360
the foci of a hyperbola.

00:04:14.360 --> 00:04:17.750
And a hyperbola, it's very
close to an ellipse, you could

00:04:17.750 --> 00:04:19.760
probably guess that, because if
this is the equation of

00:04:19.760 --> 00:04:23.000
an ellipse, this is the
equation of a hyperbola.

00:04:23.000 --> 00:04:28.340
x squared over a squared
minus y squared over b

00:04:28.340 --> 00:04:30.830
squared is equal to 1.

00:04:30.830 --> 00:04:32.517
Or we could switch these
around, where the minus

00:04:32.517 --> 00:04:34.850
is in front of the
x instead of the y.

00:04:34.850 --> 00:04:35.870
And we could cover
that in a second.

00:04:35.870 --> 00:04:39.810
But this hyperbola looks
something like this.

00:04:39.810 --> 00:04:43.000
Let me see if I can draw it.

00:04:43.000 --> 00:04:48.410
If I draw the axes, and then I
want to draw the asymptotes--

00:04:48.410 --> 00:04:50.250
you could prove it, you could
look at the some of the

00:04:50.250 --> 00:04:53.490
previous videos, but the
asymptotes for this hyperbola

00:04:53.490 --> 00:05:00.190
are going to be y is equal to
plus or minus b over ax, so

00:05:00.190 --> 00:05:04.430
it's going to look-- I'll just
draw them as kind of tilted

00:05:04.430 --> 00:05:10.430
lines, so it would look
something like that,

00:05:10.430 --> 00:05:12.810
something like-- nope.

00:05:12.810 --> 00:05:16.850
I want to make it
something like that.

00:05:16.850 --> 00:05:18.630
Those are the as-- but this
one's centered at the origin,

00:05:18.630 --> 00:05:22.210
because it hasn't been shifted,
and then that's those

00:05:22.210 --> 00:05:25.340
two lines right there.

00:05:25.340 --> 00:05:29.370
And then, this is what I call
kind of a horizontal hyperbola.

00:05:29.370 --> 00:05:33.180
The way you can think about
that is, well, if you solve for

00:05:33.180 --> 00:05:34.830
y, you'll see that you're
always going to be a little

00:05:34.830 --> 00:05:35.910
bit lower than the asymptote.

00:05:35.910 --> 00:05:39.010
The other option is you say,
OK, can x or y equal 0?

00:05:39.010 --> 00:05:44.130
Well if y is equal to 0, that
puts us along the x-axis, and

00:05:44.130 --> 00:05:54.380
you get x squared over a
squared is equal to 1, which

00:05:54.380 --> 00:05:59.090
means that x squared is equal
to a squared, which means that

00:05:59.090 --> 00:06:01.260
x is equal to plus or minus a.

00:06:01.260 --> 00:06:11.350
So the points a,0, and
the point minus a,0, are

00:06:11.350 --> 00:06:13.150
both on this hyperbola.

00:06:13.150 --> 00:06:15.120
And since they have to kind
of be contained by these

00:06:15.120 --> 00:06:17.080
asymptotes, never go through
it, you know that this is going

00:06:17.080 --> 00:06:20.795
to be a hyperbola that opens to
the left and the right, so

00:06:20.795 --> 00:06:23.714
it'll look something like this.

00:06:23.714 --> 00:06:26.022
Let me use its color.

00:06:26.022 --> 00:06:29.420
So it'll look something-- this
is the hard part-- it'll get

00:06:29.420 --> 00:06:32.890
closer and closer on that side,
and then you kind of view this

00:06:32.890 --> 00:06:37.480
as one of the vertexes or
vertices of the hyperbola,

00:06:37.480 --> 00:06:40.430
and it'll go like that.

00:06:40.430 --> 00:06:43.260
Where this distance-- and
notice the similarity here with

00:06:43.260 --> 00:06:45.900
the ellipse-- this distance
right here-- let me do it in a

00:06:45.900 --> 00:06:50.430
more vibrant color-- this
distance right here, between

00:06:50.430 --> 00:06:53.760
these two I guess you could
call them the elbows of the two

00:06:53.760 --> 00:06:55.380
ellipses-- that distance
right there, that is

00:06:55.380 --> 00:06:58.980
a, and this is also a.

00:06:58.980 --> 00:07:02.300
So you have a 2a distance,
which is very similar to this

00:07:02.300 --> 00:07:07.320
situation, where this distance
is a and this distance is a.

00:07:07.320 --> 00:07:10.430
So your distance between the
two left and right points in a

00:07:10.430 --> 00:07:13.440
horizontal ellipse is the same
as the distance between the two

00:07:13.440 --> 00:07:15.130
left and right points
on a hyperbola.

00:07:15.130 --> 00:07:17.440
It's just the hyperbola
opens outward while the

00:07:17.440 --> 00:07:19.020
ellipse opens inward.

00:07:19.020 --> 00:07:20.050
Fair enough.

00:07:20.050 --> 00:07:23.620
But the whole point of this
video is to discuss foci, and

00:07:23.620 --> 00:07:25.700
you might have guessed-- and I
did touch on it in the last

00:07:25.700 --> 00:07:29.580
video-- that hyperbola also
have foci, but they open.

00:07:29.580 --> 00:07:31.750
They're going to be to the
right and the left of these two

00:07:31.750 --> 00:07:36.190
points, so this is a-- let me
do them in a bright color,

00:07:36.190 --> 00:07:39.630
because I want you see them--
let's say that those are the

00:07:39.630 --> 00:07:42.810
two foci-- and a hyperbola,
this is-- and notice

00:07:42.810 --> 00:07:43.350
the difference.

00:07:43.350 --> 00:07:46.760
An ellipse, one of the
definitions of an ellipse, was

00:07:46.760 --> 00:07:49.690
the locus of all points or the
set of all points where the

00:07:49.690 --> 00:07:53.000
distance from each of those
points, the sum of the distance

00:07:53.000 --> 00:07:57.090
from each of those points to
the two foci is a constant.

00:07:57.090 --> 00:08:00.370
Now the definition of a
hyperbola, one of the

00:08:00.370 --> 00:08:04.470
definitions of a hyperbola, can
be the locus of all points

00:08:04.470 --> 00:08:07.750
where you take the difference--
not the sum, you take the

00:08:07.750 --> 00:08:13.840
difference of the distances
between the two foci, so if--

00:08:13.840 --> 00:08:14.800
let me write that down.

00:08:14.800 --> 00:08:24.270
So this is d1, and this is d2,
so you have a situation-- and

00:08:24.270 --> 00:08:26.310
we could take the absolute
value of the difference,

00:08:26.310 --> 00:08:29.210
because they might be, they
might at some points d1

00:08:29.210 --> 00:08:31.340
will be longer than d2
if you're on this curve.

00:08:31.340 --> 00:08:36.560
If you're on this curve, d1
will be shorter than d2-- So d1

00:08:36.560 --> 00:08:40.820
minus d2, the absolute value is
going to be equal to constant.

00:08:40.820 --> 00:08:44.560
In the ellipse situation,
d1 plus d2 was a constant.

00:08:44.560 --> 00:08:46.450
So they're very
closely related.

00:08:46.450 --> 00:08:49.635
In the ellipse, you're taking
the sum of the distances

00:08:49.635 --> 00:08:52.730
to the focus points and
saying it's a constant.

00:08:52.730 --> 00:08:56.200
In a hyperbola, you're taking
the difference of the distances

00:08:56.200 --> 00:08:58.070
to the focus points and
saying that's a constant.

00:08:58.070 --> 00:09:00.840
So this number right here is
going to be the exact same

00:09:00.840 --> 00:09:04.490
thing as if I took a point
right here-- and I'm picking

00:09:04.490 --> 00:09:06.650
these points arbitrarily, as
long as they're on the

00:09:06.650 --> 00:09:15.530
hyperbola, and if I called
these two points d3 and d4, the

00:09:15.530 --> 00:09:19.700
difference between d1 and d2 is
the same thing as the

00:09:19.700 --> 00:09:24.910
difference, between
d3 minus d4.

00:09:24.910 --> 00:09:27.370
This is going to be a
constant the entire way

00:09:27.370 --> 00:09:28.970
around the ellipse.

00:09:28.970 --> 00:09:31.400
And so the next question
is, what is this constant

00:09:31.400 --> 00:09:32.800
going to be equal to?

00:09:32.800 --> 00:09:37.180
And this is where it's useful
to find a point where you can

00:09:37.180 --> 00:09:40.270
kind of get the intuition, and
we did it with the ellipses,

00:09:40.270 --> 00:09:44.530
where we said if we take these
points, we used logic in the

00:09:44.530 --> 00:09:49.690
last video to say oh, the sum
of the distance between this

00:09:49.690 --> 00:09:55.170
and this, that sum, is going to
be equal to, we saw, is going

00:09:55.170 --> 00:09:57.830
to be equal to 2a, or
the distance of the

00:09:57.830 --> 00:09:59.680
semi-major axis.

00:09:59.680 --> 00:10:02.710
Because this distance was the
same as this distance, so this

00:10:02.710 --> 00:10:07.970
plus this is the same thing as
this plus that, which is 2a.

00:10:07.970 --> 00:10:11.540
So the entire time, the
contents, the sum of the

00:10:11.540 --> 00:10:15.130
distances to the two
foci was equal to 2a.

00:10:15.130 --> 00:10:17.490
Now in the hyperbola, what
is the difference of the

00:10:17.490 --> 00:10:19.830
distances to the two foci?

00:10:19.830 --> 00:10:23.650
So let's take this point right
here on the hyperbola, and

00:10:23.650 --> 00:10:28.960
we're saying so what is-- let
me do a good color-- what is

00:10:28.960 --> 00:10:33.750
the magenta distance-- that's
the distance to that foci--

00:10:33.750 --> 00:10:40.630
minus this-- let me pick
another color-- minus this

00:10:40.630 --> 00:10:42.190
light blue distance?

00:10:42.190 --> 00:10:45.500
This magenta distance minus
this light blue distance.

00:10:45.500 --> 00:10:47.780
So we can make a very similar
argument that we made in

00:10:47.780 --> 00:10:49.520
the ellipse situation.

00:10:49.520 --> 00:10:53.100
This light blue distance is the
same, the distance from this

00:10:53.100 --> 00:10:57.590
vertex or from this leftmost
point of this rightward opening

00:10:57.590 --> 00:11:02.170
hyperbola to this focus is
the same as this distance.

00:11:02.170 --> 00:11:04.460
Because a hyperbola is
symmetric around the origin or

00:11:04.460 --> 00:11:07.700
the focal length is the same on
either side of the center of

00:11:07.700 --> 00:11:09.490
the hyperbola depending on how
you may view it, but I think

00:11:09.490 --> 00:11:12.910
that's not too much of a
stretch of a statement for

00:11:12.910 --> 00:11:14.600
you to for you to accept.

00:11:14.600 --> 00:11:18.500
So if this distance is the same
as this distance, then the

00:11:18.500 --> 00:11:23.020
magenta distance minus this
blue distance is going to be

00:11:23.020 --> 00:11:25.410
equal to this green distance.

00:11:28.020 --> 00:11:29.690
And this green
distance is what?

00:11:29.690 --> 00:11:31.430
That's 2a.

00:11:31.430 --> 00:11:32.920
We saw that at the
beginning of this video.

00:11:32.920 --> 00:11:38.130
So this, once again,
is also equal to 2a.

00:11:38.130 --> 00:11:40.420
Anyway, I'll leave
you there right now.

00:11:40.420 --> 00:11:43.890
Actually, let's actually just
do one problem, just because

00:11:43.890 --> 00:11:45.150
I like to make one concrete.

00:11:45.150 --> 00:11:47.280
Because I told you at the
beginning that if you wanted to

00:11:47.280 --> 00:11:52.395
find the-- so if you have an
ellipse-- so if you have-- this

00:11:52.395 --> 00:11:56.880
is an ellipse, x squared over a
squared plus y squared over b

00:11:56.880 --> 00:12:00.470
squared is equal to 1, we
learned that the-- that's over

00:12:00.470 --> 00:12:03.290
b squared-- this is an ellipse.

00:12:03.290 --> 00:12:05.980
We learned that the focal
length is equal to the

00:12:05.980 --> 00:12:09.960
square root of a squared
minus b squared.

00:12:09.960 --> 00:12:12.530
Now for a hyperbola, you kind
of see that there's a very

00:12:12.530 --> 00:12:14.540
close relation between the
ellipse and the hyperbola,

00:12:14.540 --> 00:12:17.790
but it is kind of a fun
thing to ponder about.

00:12:17.790 --> 00:12:22.066
And a hyperbola's equation
looks like this.

00:12:22.066 --> 00:12:26.870
x squared over a squared minus
y squared over b squared, or it

00:12:26.870 --> 00:12:29.030
could be y squared over b
squared minus x squared over

00:12:29.030 --> 00:12:30.790
a square is equal to 1.

00:12:30.790 --> 00:12:33.980
It turns out, and I'll prove
this to you in the next video,

00:12:33.980 --> 00:12:37.460
it's a little bit of a hairy
math problem, that the focal

00:12:37.460 --> 00:12:42.490
length of a hyperbola is equal
to the square root of the sum

00:12:42.490 --> 00:12:45.940
of these two numbers, is equal
to the sum of a squared

00:12:45.940 --> 00:12:47.890
plus b squared.

00:12:47.890 --> 00:12:50.360
So if I were to give you--
so notice the difference.

00:12:50.360 --> 00:12:51.350
It's just a difference in sign.

00:12:51.350 --> 00:12:54.130
You're taking the difference of
those two denominators, and now

00:12:54.130 --> 00:12:56.330
you're taking the sum of
the two denominators.

00:12:56.330 --> 00:12:58.510
So if I were to give you
the following hyperbola.

00:12:58.510 --> 00:13:08.870
x squared over 9 plus y squared
over 16 is equal to 1.

00:13:08.870 --> 00:13:11.230
Well, the first thing you do
is-- though I've never-- well,

00:13:11.230 --> 00:13:12.870
we could just figure out
the focal length just by

00:13:12.870 --> 00:13:13.950
plugging into the formula.

00:13:13.950 --> 00:13:18.910
The focal length is equal
to the square root of a

00:13:18.910 --> 00:13:20.460
squared plus b squared.

00:13:20.460 --> 00:13:22.840
This is squared,
right? a is three.

00:13:22.840 --> 00:13:24.240
b is 4.

00:13:24.240 --> 00:13:28.490
So 9 plus 16 is 25,
which is equal to 5.

00:13:28.490 --> 00:13:35.430
And so if we were to graph
this-- that's my y-axis, that's

00:13:35.430 --> 00:13:38.970
my x-axis-- and the focal
length is the distance to, in

00:13:38.970 --> 00:13:40.640
this case, to the left and
the right of the origin.

00:13:40.640 --> 00:13:43.210
If it was kind of an up and
down opening hyperbola, it

00:13:43.210 --> 00:13:46.680
would be above and below the
origin, so this is a-- oh

00:13:46.680 --> 00:13:49.260
sorry, this should be a plus.

00:13:49.260 --> 00:13:51.660
We're doing with a hyperbola,
that should be a minus.

00:13:51.660 --> 00:13:52.680
Don't want to confuse you.

00:13:52.680 --> 00:13:54.360
What I had written before,
with a plus, that would

00:13:54.360 --> 00:13:55.320
have been an ellipse.

00:13:55.320 --> 00:13:57.450
A minus is the hyperbola.

00:13:57.450 --> 00:14:00.480
So the two asymptotes-- this is
centered at the origin, it

00:14:00.480 --> 00:14:03.790
hasn't been shifted-- are going
to be 16 over 9, so it's going

00:14:03.790 --> 00:14:06.050
to be fairly steep asymptotes,
it's going to look something

00:14:06.050 --> 00:14:14.620
like that and that, those
are the two asymptotes--

00:14:14.620 --> 00:14:19.920
The two vertex points are at 2
times a. a is three, right?

00:14:19.920 --> 00:14:24.930
a squared is equal to 9, b
squared is equal to 16, so this

00:14:24.930 --> 00:14:28.920
is the center, so the two
vertex points, this is 3 minus

00:14:28.920 --> 00:14:34.410
3, and then the focal points
are going to be at, from the

00:14:34.410 --> 00:14:36.860
center, 5 to the right, so
that's going to be right here,

00:14:36.860 --> 00:14:43.730
so that's 5,0, and minus 5,0.

00:14:43.730 --> 00:14:47.870
This is minus 3, and this is 3,
so if we were to graph it, it

00:14:47.870 --> 00:14:48.920
would look something like this.

00:14:51.900 --> 00:14:55.110
There you go.

00:14:55.110 --> 00:14:58.830
And if you were to take an
arbitrary point on that

00:14:58.830 --> 00:15:03.540
hyperbola and take this
distance and subtract that from

00:15:03.540 --> 00:15:06.420
that distance, that will be a
constant number that would be

00:15:06.420 --> 00:15:10.760
exactly equal to 2a or exactly
equal to 6, in this

00:15:10.760 --> 00:15:11.810
particular example.

00:15:11.810 --> 00:15:12.400
Anyway.

00:15:12.400 --> 00:15:14.397
Hopefully I didn't confuse you
too much with that little sign

00:15:14.397 --> 00:15:17.260
error near the end of the
video, but in the next video,

00:15:17.260 --> 00:15:20.620
I'll prove to you this formula,
which is a little bit of hairy

00:15:20.620 --> 00:15:22.980
algebra but it's fun
to do regardless.


Foci of an ellipse

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:05.650
Let's say we have an ellipse
formula, x squared over a

00:00:05.650 --> 00:00:10.950
squared plus y squared over
b squared is equal to 1.

00:00:10.950 --> 00:00:13.210
And for the sake of our
discussion, we'll assume

00:00:13.210 --> 00:00:15.030
that a is greater than b.

00:00:15.030 --> 00:00:17.420
And all that does for us is, it
lets us so this is going to

00:00:17.420 --> 00:00:19.110
be kind of a short
and fat ellipse.

00:00:19.110 --> 00:00:23.600
Or that the semi-major axis,
or, the major axis, is going

00:00:23.600 --> 00:00:24.730
to be along the horizontal.

00:00:24.730 --> 00:00:26.580
And the minor axis is
along the vertical.

00:00:26.580 --> 00:00:28.960
And let's draw that.

00:00:28.960 --> 00:00:31.330
Draw this ellipse.

00:00:31.330 --> 00:00:35.070
I want to draw a
thicker ellipse.

00:00:35.070 --> 00:00:40.120
Let's say, that's my ellipse,
and then let me draw my axes.

00:00:40.120 --> 00:00:45.490
OK, this is the
horizontal right there.

00:00:45.490 --> 00:00:48.120
And there we have the vertical.

00:00:48.120 --> 00:00:51.830
And we've studied an ellipse
in pretty good detail so far.

00:00:51.830 --> 00:00:58.160
We know how to figure out
semi-minor radius, which in

00:00:58.160 --> 00:01:01.970
this case we know is b.

00:01:01.970 --> 00:01:03.320
That's the same b right there.

00:01:03.320 --> 00:01:05.510
And that's only the
semi-minor radius.

00:01:05.510 --> 00:01:06.670
Because b is smaller than a.

00:01:06.670 --> 00:01:08.480
If b was greater, it would
be the major radius.

00:01:08.480 --> 00:01:11.170
And then, of course,
the major radius is a.

00:01:11.170 --> 00:01:15.350
And that distance is
this right here.

00:01:15.350 --> 00:01:18.540
Now, another super-interesting,
and perhaps the most

00:01:18.540 --> 00:01:21.750
interesting property of an
ellipse, is that if you take

00:01:21.750 --> 00:01:27.730
any point on the an ellipse,
and measure the distance from

00:01:27.730 --> 00:01:30.490
that point to two special
points which we, for the sake

00:01:30.490 --> 00:01:32.500
of this discussion, and not
just for the sake of this

00:01:32.500 --> 00:01:35.170
discussion, for pretty much
forever, we will call the

00:01:35.170 --> 00:01:38.120
focuses, or the foci,
of this ellipse.

00:01:38.120 --> 00:01:41.380
And these two points,
they always sit along

00:01:41.380 --> 00:01:42.280
the major axis.

00:01:42.280 --> 00:01:44.180
So, in this case, it's
the horizontal axis.

00:01:44.180 --> 00:01:47.090
And they're symmetric around
the center of the ellipse.

00:01:47.090 --> 00:01:49.230
So let's just call
these points, let me

00:01:49.230 --> 00:01:51.700
call this one f1.

00:01:51.700 --> 00:01:52.960
And this is f2.

00:01:52.960 --> 00:01:54.710
And it's for focus.

00:01:54.710 --> 00:01:55.920
Focuses.

00:01:55.920 --> 00:01:56.810
f2.

00:01:56.810 --> 00:02:00.220
So the super-interesting,
fascinating property

00:02:00.220 --> 00:02:01.100
of an ellipse.

00:02:01.100 --> 00:02:03.900
And it's often used as the
definition of an ellipse is, if

00:02:03.900 --> 00:02:07.490
you take any point on this
ellipse, and measure its

00:02:07.490 --> 00:02:09.360
distance to each of
these two points.

00:02:09.360 --> 00:02:14.160
So, let's say that I have
this distance right here.

00:02:14.160 --> 00:02:18.980
Let's call this distance d1.

00:02:18.980 --> 00:02:23.200
And then I have this distance
over here, so I'm taking any

00:02:23.200 --> 00:02:25.400
point on that ellipse, or this
particular point, and I'm

00:02:25.400 --> 00:02:29.980
measuring the distance to
each of these two foci.

00:02:29.980 --> 00:02:32.060
And this is d2.

00:02:32.060 --> 00:02:33.200
We'll do it in a
different color.

00:02:35.950 --> 00:02:36.730
So this is d2.

00:02:36.730 --> 00:02:39.030
This whole line right here.

00:02:39.030 --> 00:02:39.680
That's d2.

00:02:42.180 --> 00:02:47.100
So when you find these two
distances, you sum of them up.

00:02:47.100 --> 00:02:52.280
So this d2 plus d1, this
is going to be a constant

00:02:52.280 --> 00:02:54.980
that it actually turns
out is equal to 2a.

00:02:54.980 --> 00:02:58.000
But it turns out that it's true
anywhere you go on the ellipse.

00:02:58.000 --> 00:02:59.600
Let me make that point clear.

00:02:59.600 --> 00:03:01.620
And I'm actually going to prove
to you that this constant

00:03:01.620 --> 00:03:04.460
distance is actually 2a, where
this a is the same is

00:03:04.460 --> 00:03:05.280
that a right there.

00:03:05.280 --> 00:03:07.830
So, just to make sure you
understand what I'm saying.

00:03:07.830 --> 00:03:09.530
So let me take another
arbitrary point

00:03:09.530 --> 00:03:11.240
on this ellipse.

00:03:11.240 --> 00:03:12.690
Let's take it right there.

00:03:12.690 --> 00:03:15.570
And if I were to measure the
distance from this point to

00:03:15.570 --> 00:03:27.660
this focus, let's call that
point d3, and then measure the

00:03:27.660 --> 00:03:30.760
distance from this point to
that focus -- let's

00:03:30.760 --> 00:03:32.830
call that point d4.

00:03:35.680 --> 00:03:36.630
d4.

00:03:36.630 --> 00:03:39.490
If I were to sum up these two
points, it's still going

00:03:39.490 --> 00:03:41.030
to be equal to 2a.

00:03:41.030 --> 00:03:42.110
Let me write that down.

00:03:42.110 --> 00:03:49.260
d3 plus d4 is still going
to be equal to 2a.

00:03:49.260 --> 00:03:50.280
That's just neat.

00:03:50.280 --> 00:03:53.550
And, actually, this is often
used as the definition for an

00:03:53.550 --> 00:03:56.180
ellipse, where they say that
the ellipse is the set of all

00:03:56.180 --> 00:03:58.330
points, or sometimes they'll
use the word locus, which is

00:03:58.330 --> 00:04:01.090
kind of the graphical
representation of the set of

00:04:01.090 --> 00:04:06.630
all points, that where the sum
of the distances to each

00:04:06.630 --> 00:04:09.540
of these focuses is
equal to a constant.

00:04:09.540 --> 00:04:11.900
And we'll play with that a
little bit, and we'll figure

00:04:11.900 --> 00:04:13.930
out, how do you figure out
the focuses of an ellipse.

00:04:13.930 --> 00:04:17.110
But the first thing to do is
just to feel satisfied that the

00:04:17.110 --> 00:04:20.900
distance, if this is true,
that it is equal to 2a.

00:04:20.900 --> 00:04:25.060
And the easiest way to figure
that out is to pick these, I

00:04:25.060 --> 00:04:28.560
guess you could call them, the
extreme points along the

00:04:28.560 --> 00:04:30.570
x-axis here and here.

00:04:30.570 --> 00:04:33.630
We're already making the claim
that the distance from here to

00:04:33.630 --> 00:04:37.190
here, let me draw that
in another color.

00:04:37.190 --> 00:04:43.390
That this distance plus this
distance over here, is going

00:04:43.390 --> 00:04:45.450
to be equal to some
constant number.

00:04:45.450 --> 00:04:48.240
And using this extreme point,
I'm going to show you that that

00:04:48.240 --> 00:04:52.110
constant number is equal to 2a,
So let's figure out

00:04:52.110 --> 00:04:52.860
how to do that.

00:04:52.860 --> 00:04:55.150
So one thing to realize is that
these two focus points are

00:04:55.150 --> 00:04:57.270
symmetric around the origin.

00:04:57.270 --> 00:05:02.240
So, whatever distance this is,
right here, it's going to be

00:05:02.240 --> 00:05:04.510
the same as this distance.

00:05:04.510 --> 00:05:05.250
Right there.

00:05:05.250 --> 00:05:07.030
Because these two points are
symmetric around the origin.

00:05:07.030 --> 00:05:12.630
So this, right here, is the
same distance as that.

00:05:12.630 --> 00:05:15.920
And, of course, we have -- what
we want to do is figure out the

00:05:15.920 --> 00:05:20.891
sum of this distance and this
longer distance right there.

00:05:20.891 --> 00:05:24.740
Well, what's the sum of this
plus this green distance?

00:05:24.740 --> 00:05:27.190
Well, this right here
is the same as that.

00:05:27.190 --> 00:05:30.710
So this plus the green --
let me write that down.

00:05:30.710 --> 00:05:37.410
So let me write down these, let
me call this distance g, just

00:05:37.410 --> 00:05:40.890
to say, let's call that g,
and let's call this h.

00:05:40.890 --> 00:05:43.510
Now, if this is g and this is
h, we also know that this is g

00:05:43.510 --> 00:05:44.760
because everything's symmetric.

00:05:44.760 --> 00:05:47.010
So what's g plus h?

00:05:47.010 --> 00:05:49.130
Well, that's the same
thing as g plus h.

00:05:49.130 --> 00:05:53.750
Which is the entire major
diameter of this ellipse.

00:05:53.750 --> 00:05:54.970
Which is what?

00:05:54.970 --> 00:05:57.960
Well, we know the minor
radius is a, so this length

00:05:57.960 --> 00:05:59.960
right here is also a.

00:05:59.960 --> 00:06:03.970
So the distance, or the sum of
the distance from this point on

00:06:03.970 --> 00:06:07.630
the ellipse to this focus, plus
this point on the ellipse to

00:06:07.630 --> 00:06:12.740
that focus, is equal to g plus
h, or this big green part,

00:06:12.740 --> 00:06:16.320
which is the same thing as the
major diameter of this ellipse,

00:06:16.320 --> 00:06:18.430
which is the same thing as 2a.

00:06:18.430 --> 00:06:18.890
Fair enough.

00:06:18.890 --> 00:06:23.520
Hopefully that that is
good enough for you.

00:06:23.520 --> 00:06:26.090
Now, the next thing, now that
we've realized that, is how do

00:06:26.090 --> 00:06:28.150
we figure out where
these foci stand.

00:06:28.150 --> 00:06:30.770
Or, if we have this equation,
how can we figure out what

00:06:30.770 --> 00:06:32.410
these two points are?

00:06:32.410 --> 00:06:33.710
Let's figure that out.

00:06:33.710 --> 00:06:36.990
So, the first thing we realize,
all of a sudden is that no

00:06:36.990 --> 00:06:39.110
matter where we go, it was easy
to do it with these points.

00:06:39.110 --> 00:06:41.910
But even if we take this point
right here and we say, OK,

00:06:41.910 --> 00:06:46.040
what's this distance, and then
sum it to that distance, that

00:06:46.040 --> 00:06:47.560
should also be equal to 2a.

00:06:47.560 --> 00:06:50.290
And we could use that
information to actually figure

00:06:50.290 --> 00:06:53.210
out where the foci lie.

00:06:53.210 --> 00:06:57.905
So, let's say I have --
let me draw another one.

00:07:06.050 --> 00:07:07.990
So that's my ellipse.

00:07:07.990 --> 00:07:11.590
And then we want
to draw the axes.

00:07:11.590 --> 00:07:13.930
For clarity.

00:07:13.930 --> 00:07:15.480
Let me write down
the equation again.

00:07:15.480 --> 00:07:18.310
Just so we don't lose it.
x squared over a squared

00:07:18.310 --> 00:07:23.030
plus y squared over b
squared is equal to 1.

00:07:23.030 --> 00:07:25.120
Let's take this
point right here.

00:07:25.120 --> 00:07:28.310
These extreme points are always
useful when you're trying

00:07:28.310 --> 00:07:29.830
to prove something.

00:07:29.830 --> 00:07:32.720
Or they can be, I don't
want to say always.

00:07:32.720 --> 00:07:35.920
Now, we said that we have these
two foci that are symmetric

00:07:35.920 --> 00:07:38.240
around the center
of the ellipse.

00:07:38.240 --> 00:07:41.740
This is f1, this is f2.

00:07:41.740 --> 00:07:45.530
And we've already said that an
ellipse is the locus of all

00:07:45.530 --> 00:07:48.230
points, or the set of all
points, that if you take each

00:07:48.230 --> 00:07:52.780
of these points' distance from
each of the focuses, and add

00:07:52.780 --> 00:07:54.230
them up, you get a
constant number.

00:07:54.230 --> 00:07:56.120
And we've figured out that
that constant number is 2a.

00:07:56.120 --> 00:07:59.100
So we've figured out that if
you take this distance right

00:07:59.100 --> 00:08:03.320
here and add it to this
distance right here,

00:08:03.320 --> 00:08:06.240
it'll be equal to 2a.

00:08:06.240 --> 00:08:15.370
So we could say that if we
call this d, d1, this is d2.

00:08:15.370 --> 00:08:21.030
We know that d1 plus
d2 is equal to 2a.

00:08:21.030 --> 00:08:22.710
And an interesting thing
here is that this is

00:08:22.710 --> 00:08:24.080
all symmetric, right?

00:08:24.080 --> 00:08:27.020
This length is going to be the
same, d1 is is going to be the

00:08:27.020 --> 00:08:29.340
same, as d2, because everything
we're doing is symmetric.

00:08:29.340 --> 00:08:31.760
These two focal lengths
are symmetric.

00:08:31.760 --> 00:08:35.160
This distance is the
same distance as this

00:08:35.160 --> 00:08:36.710
distance right there.

00:08:36.710 --> 00:08:38.870
So, d1 and d2 have
to be the same.

00:08:38.870 --> 00:08:40.920
There's no way that you
could -- this is the exact

00:08:40.920 --> 00:08:43.050
center point the ellipse.

00:08:43.050 --> 00:08:47.090
The ellipse is symmetric
around the y-axis.

00:08:47.090 --> 00:08:50.990
So if d1 is equal to d2, and
that equals 2a, then we know

00:08:50.990 --> 00:08:54.240
that this has to be equal to a.

00:08:54.240 --> 00:08:56.740
And this has to be equal to a.

00:08:56.740 --> 00:08:58.690
I think we're making progress.

00:08:58.690 --> 00:09:00.760
And the other thing to think
about, and we already did that

00:09:00.760 --> 00:09:02.740
in the previous drawing of
the ellipse is, what

00:09:02.740 --> 00:09:04.880
is this distance?

00:09:04.880 --> 00:09:07.880
This distance is the
semi-minor radius.

00:09:07.880 --> 00:09:11.160
Which we already learned is b.

00:09:11.160 --> 00:09:14.250
And this of course is the
focal length that we're

00:09:14.250 --> 00:09:16.360
trying to figure out.

00:09:16.360 --> 00:09:19.530
This should already pop into
your brain as a Pythagorean

00:09:19.530 --> 00:09:20.830
theorem problem.

00:09:20.830 --> 00:09:24.460
So we have the focal length.

00:09:24.460 --> 00:09:26.120
And we could do it on this
triangle or this triangle.

00:09:26.120 --> 00:09:27.140
I'll do it on this
right one here.

00:09:27.140 --> 00:09:28.490
This focal length is f.

00:09:28.490 --> 00:09:30.020
Let's call that f.

00:09:30.020 --> 00:09:34.450
f squared plus b squared is
going to be equal to the

00:09:34.450 --> 00:09:38.060
hypotenuse squared, which
in this case is d2 or a.

00:09:38.060 --> 00:09:39.890
Which is equal to a squared.

00:09:39.890 --> 00:09:42.870
And now we have a nice
equation in terms of b and a.

00:09:42.870 --> 00:09:46.070
We know what b and a are,
from the equation we were

00:09:46.070 --> 00:09:47.160
given for this ellipse.

00:09:47.160 --> 00:09:48.820
So let's solve for
the focal length.

00:09:48.820 --> 00:09:52.760
The focal length, f squared,
is equal to a squared

00:09:52.760 --> 00:09:55.360
minus b squared.

00:09:55.360 --> 00:09:57.870
So, f, the focal length, is
going to be equal to the

00:09:57.870 --> 00:10:02.020
square root of a squared
minus b squared.

00:10:02.020 --> 00:10:04.490
Pretty neat and clean, and
a pretty intuitive way to

00:10:04.490 --> 00:10:05.250
think about something.

00:10:05.250 --> 00:10:08.520
So you just literally take
the difference of these two

00:10:08.520 --> 00:10:11.520
numbers, whichever is larger,
or whichever is smaller you

00:10:11.520 --> 00:10:12.340
subtract from the other one.

00:10:12.340 --> 00:10:15.140
You take the square root, and
that's the focal distance.

00:10:15.140 --> 00:10:17.840
Now, let's see if we can use
that to apply it to some some

00:10:17.840 --> 00:10:21.290
real problems where they might
ask you, hey, find

00:10:21.290 --> 00:10:21.930
the focal length.

00:10:21.930 --> 00:10:24.930
Or find the coordinates
of the focuses.

00:10:24.930 --> 00:10:34.820
So let's add the equation x
minus 1 squared over 9 plus

00:10:34.820 --> 00:10:42.530
y plus 2 squared over
4 is equal to 1.

00:10:42.530 --> 00:10:44.810
So let's just graph
this first of all.

00:10:44.810 --> 00:10:46.410
This could be interesting.

00:10:46.410 --> 00:10:48.770
So I'll draw the axes.

00:10:48.770 --> 00:10:50.320
That's the x-axis.

00:10:50.320 --> 00:10:52.790
This is the y-axis.

00:10:52.790 --> 00:10:54.890
And we immediately see,
what's the center of this?

00:10:54.890 --> 00:10:57.730
The center is going to be at
the point 1, negative 2.

00:10:57.730 --> 00:11:00.230
And if that's confusing, you
might want to review some

00:11:00.230 --> 00:11:01.740
of the previous videos.

00:11:01.740 --> 00:11:04.440
Center's at 1, x is equal to 1.

00:11:04.440 --> 00:11:07.560
y is equal to minus 2.

00:11:07.560 --> 00:11:08.970
That's the center.

00:11:08.970 --> 00:11:14.330
And then, the major axis is the
x-axis, because this is larger.

00:11:14.330 --> 00:11:18.050
And so, b squared is -- or
a squared, is equal to 9.

00:11:18.050 --> 00:11:20.870
And the semi-minor radius
is going to be equal to 3.

00:11:20.870 --> 00:11:23.230
So, if you go 1, 2, 3.

00:11:25.750 --> 00:11:26.270
Go there.

00:11:26.270 --> 00:11:29.940
Than you have 1, 2, 3.

00:11:29.940 --> 00:11:30.510
No.

00:11:30.510 --> 00:11:32.240
1, 2, 3.

00:11:32.240 --> 00:11:32.820
1, 2, 3.

00:11:32.820 --> 00:11:33.701
I think this -- let's see.

00:11:33.701 --> 00:11:35.530
1, 2, 3.

00:11:35.530 --> 00:11:38.130
You go there, roughly.

00:11:38.130 --> 00:11:41.010
And then in the y direction,
the semi-minor radius is

00:11:41.010 --> 00:11:42.110
going to be 2, right?

00:11:42.110 --> 00:11:43.430
The square root of that.

00:11:43.430 --> 00:11:44.670
So b is equal to 2.

00:11:44.670 --> 00:11:47.440
So you go up 2, then
you go down 2.

00:11:47.440 --> 00:11:50.110
And this ellipse is going
to look something like

00:11:50.110 --> 00:11:52.050
-- pick a good color.

00:11:52.050 --> 00:11:56.750
It's going to look
something like this.

00:11:56.750 --> 00:11:59.590
And what we want to do is,
we want to find out the

00:11:59.590 --> 00:12:01.830
coordinates of the
focal points.

00:12:01.830 --> 00:12:04.220
So, the focal points
are going to sit along

00:12:04.220 --> 00:12:06.970
the semi-major axis.

00:12:06.970 --> 00:12:09.920
And we need to figure out
these focal distances.

00:12:09.920 --> 00:12:11.690
And then we can essentially
just add and subtract

00:12:11.690 --> 00:12:12.860
them from the center.

00:12:12.860 --> 00:12:14.430
And then we'll have
the coordinates.

00:12:14.430 --> 00:12:17.260
What we just showed you, or
hopefully I showed you, that

00:12:17.260 --> 00:12:21.420
the the focal length or this
distance, f, the focal length

00:12:21.420 --> 00:12:24.670
is just equal to the square
root of the difference between

00:12:24.670 --> 00:12:26.750
these two numbers, right?

00:12:26.750 --> 00:12:30.660
It's just the square
root of 9 minus 4.

00:12:30.660 --> 00:12:35.660
So the focal length is equal
to the square root of 5.

00:12:35.660 --> 00:12:38.540
So, if this point right here is
the point, and we already

00:12:38.540 --> 00:12:42.920
showed that, this is the point
-- the center of the ellipse

00:12:42.920 --> 00:12:46.930
is the point 1, minus 2.

00:12:46.930 --> 00:12:51.470
The coordinate of this focus
right there is going to be 1

00:12:51.470 --> 00:12:55.970
plus the square root
of 5, minus 2.

00:12:55.970 --> 00:12:58.440
And the coordinate of this
focus right there is going

00:12:58.440 --> 00:13:02.630
to be 1 minus the square
root of 5, minus 2.

00:13:02.630 --> 00:13:04.650
And all I did is, I took the
focal length and I subtracted

00:13:04.650 --> 00:13:08.510
-- since we're along the major
axes, or the x axis, I just add

00:13:08.510 --> 00:13:11.420
and subtract this from the x
coordinate to get these two

00:13:11.420 --> 00:13:12.670
coordinates right there.

00:13:12.670 --> 00:13:15.840
So, anyway, this is the really
neat thing about conic

00:13:15.840 --> 00:13:18.240
sections, is they have these
interesting properties in

00:13:18.240 --> 00:13:22.230
relation to these foci or in
relation to these focus points.

00:13:22.230 --> 00:13:26.140
And in future videos I'll show
you the foci of a hyperbola or

00:13:26.140 --> 00:13:29.820
the the foci of a -- well, it
only has one focus

00:13:29.820 --> 00:13:31.440
of a parabola.

00:13:31.440 --> 00:13:33.340
But this is really starting
to get into what makes

00:13:33.340 --> 00:13:34.350
conic sections neat.

00:13:34.350 --> 00:13:36.740
Everything we've done up to
this point has been much more

00:13:36.740 --> 00:13:40.230
about the mechanics of graphing
and plotting and figuring out

00:13:40.230 --> 00:13:41.350
the centers of conic sections.

00:13:41.350 --> 00:13:43.700
But now we're getting into
a little bit of the the

00:13:43.700 --> 00:13:46.340
mathematical interesting
parts of conic sections.

00:13:46.340 --> 00:13:46.900
Anyway.

00:13:46.900 --> 00:13:48.770
See you in the next video.


Identifying circles and parabolas from equations

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.090
Let's see if we could do a
couple more of these conic

00:00:03.090 --> 00:00:05.420
section identification
problems.

00:00:05.420 --> 00:00:12.870
So I have this problem, x
squared plus y squared minus

00:00:12.870 --> 00:00:19.990
2x plus 4y is equal to 4.

00:00:19.990 --> 00:00:21.990
And so, the first thing I like
to do is just try to figure

00:00:21.990 --> 00:00:24.340
out what type of conic
section this will be.

00:00:24.340 --> 00:00:28.265
And we have our -- this
is my x squared term,

00:00:28.265 --> 00:00:29.200
my y squared term.

00:00:29.200 --> 00:00:31.440
They're on the same
side of the equation.

00:00:31.440 --> 00:00:33.590
And they both have
positive coefficients.

00:00:33.590 --> 00:00:35.280
So this tells me that
we're going to be

00:00:35.280 --> 00:00:36.055
dealing with an ellipse.

00:00:36.055 --> 00:00:38.320
And in this case in particular,
the coefficients are

00:00:38.320 --> 00:00:38.930
the same number.

00:00:38.930 --> 00:00:40.230
They're both positive 1.

00:00:40.230 --> 00:00:43.350
So this is going to tell
me that this is a circle.

00:00:43.350 --> 00:00:45.360
So let's get this in
the standard form.

00:00:45.360 --> 00:00:47.350
And try to graph the circle.

00:00:47.350 --> 00:00:49.100
So we're going to want
to complete the square.

00:00:49.100 --> 00:00:54.480
So let's take the x terms,
since we get x squared minus

00:00:54.480 --> 00:00:59.490
2x plus something to complete
the square later on.

00:00:59.490 --> 00:01:01.960
Plus, and now let's do
the y squared terms.

00:01:01.960 --> 00:01:11.760
y squared plus 4y plus
something is equal to 4.

00:01:11.760 --> 00:01:13.730
And now what are we at here?

00:01:13.730 --> 00:01:16.860
We take 1/2 of the
minus 2, minus 1.

00:01:16.860 --> 00:01:17.650
Square it.

00:01:17.650 --> 00:01:18.780
That becomes plus 1.

00:01:18.780 --> 00:01:19.730
Add a 1.

00:01:19.730 --> 00:01:22.260
We have nothing out here, so we
really just added a 1 to the

00:01:22.260 --> 00:01:23.580
left-hand side of
this equation.

00:01:23.580 --> 00:01:27.370
So we have to add 1 to
the right-hand side.

00:01:27.370 --> 00:01:29.310
And here we take 1/2 of 4.

00:01:29.310 --> 00:01:31.240
1/2 of 4 is 2.

00:01:31.240 --> 00:01:32.950
2 squared is 4.

00:01:32.950 --> 00:01:34.040
Put a 4 here.

00:01:34.040 --> 00:01:36.420
So you have to add a 4 to the
right-hand side as well.

00:01:36.420 --> 00:01:38.640
And we actually did add just a
4, because there was nothing

00:01:38.640 --> 00:01:40.790
multiplying the 4 out here.

00:01:40.790 --> 00:01:53.680
And so this becomes x minus 1
squared plus y plus 2 squared

00:01:53.680 --> 00:01:58.070
is equal to 4 plus 1
plus 4 is equal to 9.

00:01:58.070 --> 00:01:58.660
And there you have it.

00:01:58.660 --> 00:02:01.070
We have it in the standard
form of a circle.

00:02:01.070 --> 00:02:04.690
You remember that if a circle
is centered at 0, the standard

00:02:04.690 --> 00:02:08.830
form would be x squared plus y
squared is equal to r squared.

00:02:08.830 --> 00:02:11.860
So this is r squared, this
is the radius squared.

00:02:11.860 --> 00:02:14.510
So that tells us the radius
of the circle is 3.

00:02:14.510 --> 00:02:17.330
And it was just shifted so that
its origin, instead of being 0,

00:02:17.330 --> 00:02:21.860
0, is at the point 1, minus 2.

00:02:21.860 --> 00:02:23.440
And the reason why we got 1,
minus 2, we just have to

00:02:23.440 --> 00:02:27.410
think about what makes this
whole expression equal 0?

00:02:27.410 --> 00:02:28.960
In this case it was the origin.

00:02:28.960 --> 00:02:32.120
In this case, it's x equals 1.

00:02:32.120 --> 00:02:35.230
And what makes this whole
expression equal 0, in this

00:02:35.230 --> 00:02:36.620
case, it was y is equal to 0.

00:02:36.620 --> 00:02:38.950
In this case it's y
is equal to minus 2.

00:02:38.950 --> 00:02:41.380
So that's our center.

00:02:41.380 --> 00:02:46.700
That's our radius, and we're
ready to graph this circle.

00:02:46.700 --> 00:02:48.610
So it's at, let me see.

00:02:52.110 --> 00:02:54.340
I should graph the
circle first.

00:02:54.340 --> 00:02:55.080
That's fair enough.

00:02:58.870 --> 00:03:01.406
So it's going to
be at 1, minus 2.

00:03:01.406 --> 00:03:03.280
So, 1, minus 2.

00:03:03.280 --> 00:03:06.290
So it's going to be down here.

00:03:06.290 --> 00:03:10.945
So it's going to come out, this
circle's going to start there.

00:03:19.290 --> 00:03:24.290
It's going to be at 1,
and then you go 1, 2.

00:03:24.290 --> 00:03:25.970
So that's pretty
close to the center.

00:03:25.970 --> 00:03:26.995
Maybe I should do it at 1, 2.

00:03:26.995 --> 00:03:28.410
So that's at center.

00:03:28.410 --> 00:03:30.550
Right there, at 1, minus 2.

00:03:30.550 --> 00:03:31.860
And then its radius is 3.

00:03:31.860 --> 00:03:35.870
So this distance right here
is 3, in any direction.

00:03:35.870 --> 00:03:36.820
3.

00:03:36.820 --> 00:03:38.050
And that is 3.

00:03:38.050 --> 00:03:39.940
Fair enough, that was a pretty
straightforward problem.

00:03:39.940 --> 00:03:42.080
Circles in some ways
are the simplest.

00:03:42.080 --> 00:03:43.430
And remember I said it's
going to be an ellipse.

00:03:43.430 --> 00:03:44.870
And you say, oh, this isn't
the standard formula

00:03:44.870 --> 00:03:45.820
for an ellipse.

00:03:45.820 --> 00:03:47.630
So just as a refresher, if you
divide both sides of this

00:03:47.630 --> 00:03:49.470
equation by 9, what do you get?

00:03:49.470 --> 00:03:57.180
You get x minus 1 squared
over 9 plus y plus 2 squared

00:03:57.180 --> 00:03:59.430
over 9 is equal to 1.

00:03:59.430 --> 00:04:04.660
And then you see that the
horizontal axis, I guess.

00:04:04.660 --> 00:04:08.190
Or the horizontal diameter,
is going to be 3.

00:04:08.190 --> 00:04:09.810
Or the horizontal radius
is going to be 3.

00:04:09.810 --> 00:04:11.260
And the vertical radius
is also going to be 3.

00:04:11.260 --> 00:04:13.730
Because the radius never
changes in this ellipse.

00:04:13.730 --> 00:04:15.450
Which is really a circle.

00:04:15.450 --> 00:04:16.790
Let's do one more.

00:04:16.790 --> 00:04:20.600
Just to make sure you
know this stuff cold.

00:04:20.600 --> 00:04:33.660
So I have 2x squared plus y
plus 12x plus 16 is equal to 0.

00:04:33.660 --> 00:04:36.080
Let's look at the x squared
and the y squared terms.

00:04:36.080 --> 00:04:37.380
0's an squared term.

00:04:37.380 --> 00:04:39.340
But I don't see a
y squared term.

00:04:39.340 --> 00:04:41.170
So this is a bit
of a conundrum.

00:04:41.170 --> 00:04:45.440
And this will lead us to the
fourth of our conic sections,

00:04:45.440 --> 00:04:47.150
which I talk about in the
first video but we haven't

00:04:47.150 --> 00:04:47.950
really touched on yet.

00:04:47.950 --> 00:04:49.350
And that's the parabola.

00:04:49.350 --> 00:04:51.280
And how do I know
it's a parabola?

00:04:51.280 --> 00:04:54.210
You're familiar, and I'll go
more in future videos on all

00:04:54.210 --> 00:04:56.190
the different ways that
a parabola comes about.

00:04:56.190 --> 00:05:00.090
And how all the points are
equidistant between one point

00:05:00.090 --> 00:05:01.330
and a line and all of that.

00:05:01.330 --> 00:05:04.550
But just in very simple ways,
you recognize the most

00:05:04.550 --> 00:05:07.500
simple parabola is y
is equal to x squared.

00:05:07.500 --> 00:05:10.990
That parabola looks
something like this.

00:05:10.990 --> 00:05:15.940
Where its minimum point, or
its vertex, is at the origin.

00:05:15.940 --> 00:05:19.950
Or if you have a parabola like,
x is equal to y squared.

00:05:19.950 --> 00:05:22.660
That looks something like this.

00:05:22.660 --> 00:05:26.410
Where it's a sideways
version of that one.

00:05:26.410 --> 00:05:29.000
Where, once again, its
vertex is at the origin.

00:05:29.000 --> 00:05:36.660
And just out of interest, so we
know that this is a parabola,

00:05:36.660 --> 00:05:38.790
because we have a y and we
have an x squared, right?

00:05:38.790 --> 00:05:39.610
They're different degrees.

00:05:39.610 --> 00:05:42.870
There's no second
degree term of the y.

00:05:42.870 --> 00:05:45.485
And just to put this in a form
that's familiar to you, let's

00:05:45.485 --> 00:05:50.540
just subtract everything but
the y from the left-hand side.

00:05:50.540 --> 00:05:58.760
So you get y is equal to minus
2x squared minus 12x minus 16.

00:05:58.760 --> 00:06:01.890
And this is kind of the
traditional form that

00:06:01.890 --> 00:06:02.660
you're familiar with.

00:06:02.660 --> 00:06:05.360
You're probably even used to
finding the 0'z of this

00:06:05.360 --> 00:06:06.840
parabola, and we could
do that right now.

00:06:06.840 --> 00:06:09.720
We could say, OK, when
does this equation

00:06:09.720 --> 00:06:11.480
intersect the x axis.

00:06:11.480 --> 00:06:13.850
The x axis is when
y is equal to 0.

00:06:13.850 --> 00:06:14.930
So that equals 0.

00:06:14.930 --> 00:06:19.490
You get minus 2x squared
minus 12x minus 16.

00:06:19.490 --> 00:06:20.820
And this is different to
what we normally do.

00:06:20.820 --> 00:06:22.460
Normally I would immediately
break into completing

00:06:22.460 --> 00:06:22.730
the square.

00:06:22.730 --> 00:06:25.740
But I just want to figure out
the 0's of this parabola first.

00:06:25.740 --> 00:06:29.720
So this 0's equal to minus 2
times factoring out minus 2,

00:06:29.720 --> 00:06:34.110
you get x plus 6x plus 8.

00:06:34.110 --> 00:06:39.470
So the 0 is equal to minus 2
times x plus 2 times x plus

00:06:39.470 --> 00:06:41.420
4, I just factored that.

00:06:41.420 --> 00:06:43.770
And so for this whole thing
to be 0, either this

00:06:43.770 --> 00:06:46.260
is 0 or that is 0.

00:06:46.260 --> 00:06:50.710
And so, either x plus
2 is equal to 0, or x

00:06:50.710 --> 00:06:52.620
plus 4 is equal to 0.

00:06:52.620 --> 00:06:56.180
So x is equal to minus 2,
and x is equal to minus 4.

00:06:56.180 --> 00:06:58.530
That's the two 0's
of this parabola.

00:06:58.530 --> 00:07:02.180
So we immediately know one
thing about this parabola, and

00:07:02.180 --> 00:07:04.130
you've probably already done
this in your algebra classes.

00:07:04.130 --> 00:07:14.350
If we were to draw the x axis,
it intersects the x axis at 1

00:07:14.350 --> 00:07:19.260
and minus 2, and 3 and minus 4.

00:07:19.260 --> 00:07:21.170
That's all we know
about this right now.

00:07:21.170 --> 00:07:23.510
So let's see if we can use some
of our completing the square

00:07:23.510 --> 00:07:26.710
skills with the conic sections
we've done so far, to to come

00:07:26.710 --> 00:07:28.180
up with a little bit more
information about

00:07:28.180 --> 00:07:29.220
this parabola.

00:07:29.220 --> 00:07:30.850
So let's try to complete
the square with it.

00:07:30.850 --> 00:07:31.920
I'll rewrite it down here.

00:07:31.920 --> 00:07:36.330
So it's y is equal to, this
is the one I'm dealing with.

00:07:36.330 --> 00:07:40.680
Now, let me just take the x
terms by themselves, and

00:07:40.680 --> 00:07:42.430
factor out the minus 2.

00:07:42.430 --> 00:07:46.970
Minus 2 times x
squared plus 6x.

00:07:46.970 --> 00:07:49.240
And I'm going to add
something else.

00:07:49.240 --> 00:07:53.130
And then I have a
minus 16 over there.

00:07:53.130 --> 00:07:56.730
To make this a perfect square,
I have to take 1/2 of

00:07:56.730 --> 00:07:58.000
this, 1/2 of the 6.

00:07:58.000 --> 00:07:59.920
1/2 of the 6 is 3.

00:07:59.920 --> 00:08:02.090
3 squared is 9.

00:08:02.090 --> 00:08:05.400
If I add nine to the right-hand
side of the equation, remember,

00:08:05.400 --> 00:08:06.350
I didn't just add 9.

00:08:06.350 --> 00:08:08.830
This is 9 times
minus 2 I'm adding.

00:08:08.830 --> 00:08:12.280
So if I subtract, that's minus
18, if I subtract 18 from the

00:08:12.280 --> 00:08:14.825
right-hand side, I also have to
do it from the left-hand side.

00:08:14.825 --> 00:08:16.720
So, subtract 18 there.

00:08:16.720 --> 00:08:22.650
And now my equation becomes y
minus 18 is equal to minus

00:08:22.650 --> 00:08:24.500
2 times, what is this?

00:08:24.500 --> 00:08:30.740
This is x plus 3
squared minus 16.

00:08:30.740 --> 00:08:34.230
And let's just get it in a form
that we might start recognizing

00:08:34.230 --> 00:08:35.990
from our conic sections.

00:08:35.990 --> 00:08:38.260
Let's add 16 to both sides.

00:08:38.260 --> 00:08:42.070
If we add 16 to both sides,
y minus 18, plus 16.

00:08:42.070 --> 00:08:45.310
It's going to be y minus
2, I'll put parentheses

00:08:45.310 --> 00:08:45.810
around that.

00:08:45.810 --> 00:08:51.470
Is equal to minus 2
times x plus 3 squared.

00:08:51.470 --> 00:08:53.860
And you might wonder why
I put it in this form.

00:08:53.860 --> 00:08:57.850
And I did because this'll help
us, this is kind of the same

00:08:57.850 --> 00:08:59.830
pattern that you see with all
of the other conic sections.

00:08:59.830 --> 00:09:03.180
Like, if I were to tell you to
graph y is equal to x squared,

00:09:03.180 --> 00:09:06.110
y is equal to x squared.

00:09:06.110 --> 00:09:10.630
It would look something like,
let me draw the - whoops -

00:09:10.630 --> 00:09:12.790
Draw some axes here.

00:09:12.790 --> 00:09:16.170
y equals x squared looks
something like this.

00:09:16.170 --> 00:09:17.370
It looks like a parabola.

00:09:17.370 --> 00:09:19.540
I mean, it is a parabola,
with the vertex at 0,

00:09:19.540 --> 00:09:20.630
or its minimum point.

00:09:20.630 --> 00:09:23.800
And that's what the vertex is,
is the minimum point of, or the

00:09:23.800 --> 00:09:25.310
maximum point of the parabola.

00:09:25.310 --> 00:09:27.020
We'll talk more about that,
you'll learn a lot more about

00:09:27.020 --> 00:09:28.570
that when we go into calculus.

00:09:28.570 --> 00:09:31.090
But I think you can recognize
it, the bottom of the

00:09:31.090 --> 00:09:32.860
U or the top of the U.

00:09:32.860 --> 00:09:39.540
If I were to draw y is equal
to minus x squared, you

00:09:39.540 --> 00:09:40.490
could plot some points.

00:09:40.490 --> 00:09:45.160
But it looks
something like that.

00:09:45.160 --> 00:09:51.120
If I were to try to graph y is
equal to 2x squared, it would

00:09:51.120 --> 00:09:53.220
just be like the y is equal to
x squared, but it would

00:09:53.220 --> 00:09:54.370
go up twice as fast.

00:09:54.370 --> 00:09:57.010
It would look
something like that.

00:09:57.010 --> 00:10:00.330
So the vertex at the origin.

00:10:00.330 --> 00:10:04.480
And finally, if I were to
graph y is equal to minus

00:10:04.480 --> 00:10:07.150
2x squared, it would look
something like this.

00:10:07.150 --> 00:10:11.120
It would open downward and it
would go down twice as fast.

00:10:11.120 --> 00:10:14.590
Now, this one, this equation
that we ended up with, right

00:10:14.590 --> 00:10:18.640
here, this is the same thing as
y is equal to minus 2x squared.

00:10:18.640 --> 00:10:20.200
It has the same general shape.

00:10:20.200 --> 00:10:23.130
But instead of its vertex, or
its center or its starting

00:10:23.130 --> 00:10:24.780
point, or whatever you want
to call it, to be at the

00:10:24.780 --> 00:10:26.810
origin, it's shifted now.

00:10:26.810 --> 00:10:29.675
You say what y value makes
this term zero, well,

00:10:29.675 --> 00:10:31.470
y is equal to 2.

00:10:31.470 --> 00:10:34.890
Just why, in this case,
what y value makes this 0?

00:10:34.890 --> 00:10:37.000
Well, it's 0 here because
we were at the origin.

00:10:37.000 --> 00:10:38.690
And here, what x
value makes this 0?

00:10:38.690 --> 00:10:40.920
It's x is equal to minus 3.

00:10:40.920 --> 00:10:44.800
So this gives us information
about where the vertex is.

00:10:44.800 --> 00:10:49.620
It's at x is equal to
minus 3, y is equal to 2.

00:10:49.620 --> 00:10:55.190
So it's at x is equal to 1,
2, 3. y is equal to 1, 2.

00:10:55.190 --> 00:10:57.080
Right there.

00:10:57.080 --> 00:10:58.650
We already know those two
points, because we've

00:10:58.650 --> 00:10:59.480
figure out it's 0.

00:10:59.480 --> 00:11:01.810
But even if we didn't know
those two points -- we know

00:11:01.810 --> 00:11:03.920
that this has the same
general shape as y is

00:11:03.920 --> 00:11:05.770
equal to minus 2x squared.

00:11:05.770 --> 00:11:07.620
So it's going to open downward.

00:11:07.620 --> 00:11:08.330
Like this.

00:11:08.330 --> 00:11:10.660
And a little bit faster than y
is equal to minus x squared.

00:11:10.660 --> 00:11:14.020
So it's going to
look like this.

00:11:14.020 --> 00:11:16.150
And we know it goes through
that point and we know it

00:11:16.150 --> 00:11:17.590
goes through that point.

00:11:17.590 --> 00:11:18.730
There you go.

00:11:18.730 --> 00:11:21.970
So we've touched on
every conic section.

00:11:21.970 --> 00:11:23.720
And in the next two videos,
I'll go into a little bit more

00:11:23.720 --> 00:11:26.830
depth of the theory behind
conic sections and how they

00:11:26.830 --> 00:11:28.140
came about and all of that.

00:11:28.140 --> 00:11:29.990
But I think now you're ready
to tackle a lot of what you

00:11:29.990 --> 00:11:32.500
might actually see on
your algebra test.

00:11:32.500 --> 00:11:34.002
See you soon.