Identifying a hyperbola from an equation

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.330
Let's do another conic section
identification problem.

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So, I have 4y squared
minus 50x is equal to 25x

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squared plus 16y plus 109.

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So, the first thing I like to
do is to group all of the x and

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y terms onto one side of the
equation and leave all the

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constants on the other side.

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So let's do that.

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So, on the left-hand side
I'll put the 4y squared.

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4yy squared.

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And, actually, I'm also going
to group all the x and

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y terms in this step.

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So the 4y squared.

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Let's move this 16y onto
the left-hand side.

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So if I subtract 16y from both
sides of this equation, I get

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minus 16y, minus 16y on the
left-hand side and of course

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it will disappear on
the right-hand side.

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And then I want to subtract
the 25x squared from both

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sides of this equation.

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So I get minus 25x
squared minus 50x.

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That's that right there.

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And then I'll leave this 109
on the right-hand side.

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It's equal to 109.

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And now that we have the x's
and the y's on the same side of

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the equation, we know what type
of -- we know the general

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direction we're going to go in.

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Because they're on
the same side.

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They have different
coefficients.

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And one is positive
and one is negative.

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So that lets us know that we're
dealing with a hyperbola.

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So let's complete the
square and get it into

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the standard form.

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So, the easiest way to complete
the square is if you have a 1

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coefficient on the y squared
and the x squared terms.

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So let's factor out
a 4, in this case.

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So you get 4 times y
squared minus 4y.

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I'm going to add
something later, when

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I complete the square.

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Minus 25 times x squared,
plus, let's see, minus 50

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divided by minus 25 is 2.

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Plus 2x, I'm going to
add something later.

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Is equal to 109.

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And the things we're going
to add, those are what

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complete the square.

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Make these things
a perfect square.

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So, if I take this, you
have a minus 4 here.

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I take half of that number.

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This is just completing the
square, I encourage you to

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watch the video on completing
the square where I

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explain why this works.

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But I think I have a minus 4.

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I take half of that,
it's minus 2.

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And then minus 2
squared is plus 4.

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Now, I can't do one thing to
one side of the equation

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without doing it to the other.

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And I didn't add a 4 to the
left-hand side of the equation.

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I actually added a
4 times 4, right?

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Because you have this 4
multiplying it out front.

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So I added at 16 to the left
side of the equation, so I

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have to also add it to
the right-hand side

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of the equation.

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Right?

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This is equivalent to also
having a plus 16 here.

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That might make a little
bit clearer, right?

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When you factor it out,
and it becomes a 4.

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And we would have added
a 16 up here as well.

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Likewise, if we take half
of this number here.

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Half of 2 is 1.

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1 squared is 1.

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We didn't add a 1 to the
left-hand side of the equation,

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we added a 1 times minus 25.

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So we want to put
a minus 25 here.

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And, likewise, this would
have been the same thing as

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adding a minus 25 up here.

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And you do a minus
25 over here.

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And now, what does this become?

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The y terms become 4
times y minus 2 squared.

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y minus 2 squared.

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Might want to review factoring
a polynomial, if you found that

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a little confusing, that step.

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Minus 25 times x
plus 1 squared.

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That's that, right there.

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x plus 1 squared, is equal to,
let's see, 109 plus 16 is

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25 minus 25, it equals 100.

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We're almost there.

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So we want a 1 here, so
let's divide both sides

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of this equation by 100.

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So, you will get y
minus 2 squared.

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4 divided by 100 is the
same thing as 1/25, so

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this becomes over 25.

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Minus, let's see, 25/100 is the
same thing as 1/4, so this

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becomes x plus 1 squared
over 4 is equal to 1.

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And there you have it.

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We have it in standard
form and, yes, indeed,

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we do have a hyperbola.

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Now, let's graph
this hyperbola.

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So the first thing we
know is where the center

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of this hyperbola is.

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Is the center of this hyperbola
is at the point x is

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equal to minus one.

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So it's an x is equal to
minus 1. y is equal to 2.

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And let's figure out the
asymptotes of this hyperbola.

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So if this was -- this is the
way I always do it, because I

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always forget the
actual formula.

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If this was centered at 0 and
it looked something like this.

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y squared over 25 minus x
squared over 4 is equal to 1.

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I do this to figure out what
the asymptotes would have been

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if we were centered at 0.

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Because it's a lot easier to
deal with these equations

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than to deal with these.

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So we could solve -- we
multiply both sides by 100.

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We're kind of unwinding
what we just did.

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So if you -- actually, let's
multiply both sides by 25.

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So then you get y squared
minus 2 over 4x squared

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is equal to 25.

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And then, I'll just
go right here.

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And then if I add 25 over 4x
squared to both sides, I get

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y squared is equal to 25
over 4x squared plus 25.

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And so, y is equal to the plus
or minus square root of 25

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over 4x squared plus 25.

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And like always, the
asymptotes, the hyperbola will

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never equal the asymptotes or
intersect the asymptotes, but

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it's what the graph approaches
as x approaches positive

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and negative infinity.

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So, as x approaches positive
and negative infinity, and

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you'll learn the concept
of limits later on.

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But I think you get
it at this point.

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Because that's what the idea
of an asymptote even is.

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Is that as x gets really
larger, approaching this line

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-- so as x approaches positive
or negative infinity, as we've

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done in the previous videos,
this term starts to

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matter a lot less.

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Because this term is huge.

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So then y is approximately
equal to plus or minus the

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square root of just this term.

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Now, the square root of
just this term is 5/2 x.

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So those would be our
asymptotes if we

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were centered at 0.

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But, of course, we're
centered at negative 1, 2.

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So let's graph that.

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And then we could figure out
if it's a upward-opening

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or downward-opening graph.

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We're centered at
negative 1, 2.

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So I want to be in the,
that's my y axis.

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This is my x axis.

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And we're centered
at minus 1, 1, 2.

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That's the center.

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And this would've been the
two lines of the asymptotes

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if we were centered at 0.

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But now this tells us the
slope of the two asymptotes.

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So the asymptotes are going to
intersect at the center of

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our hyperbola, so to speak.

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So these are the slopes
of the two asymptotes.

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And one is positive 5/2.

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So, positive 5/2 means
if we go over 2, so 1,

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2, in x, we go up 5.

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So, 1, 2, 3, 4, 5.

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So we'll end up
right over there.

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So I can draw that line, I just
need two points for a line.

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So that line would
look like that.

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And the other asymptote
is minus 5/2.

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So for every 2 we go over to
the right, we go down 5.

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So, 1, 2.

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1, 2, 3 4, 5.

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So we end up right about there.

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And so that line would
look like that.

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Good enough.

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So those are the two
asymptotes, and they go on

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forever in those directions.

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And now we can think
of it two ways.

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We could either say, OK, if
we look at -- actually,

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look at this one.

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If it was centered at
0, could x equals 0?

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Well, sure x could equal 0.

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If x is 0, then y squared
over 25 equals 1. y

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squared equals 25.

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y would be plus or minus 5.

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So, in this case, this
term could be equal to 0.

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So we could say that x
could equal negative 1.

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If x is equal to negative 1,
then y minus 2 squared over 25

00:09:06.050 --> 00:09:07.610
will equal -- let's do that.

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Let's set.

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If x is equal to negative 1,
x is equal to negative 1,

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then what does this
expression become?

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I don't want to lose it, so
I'll write it right there.

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So then you get y minus
2 squared over 25.

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This becomes 0 minus
0 is equal to 1.

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So you get y minus 2 squared
over 25 is equal to 1.

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y minus 2 squared
is equal to 25.

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Just multiplied
both sides by 25.

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y minus 2 is equal to plus or
minus, I'm just taking the

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square root of both sides, 5.

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So y minus 2 is equal to
positive 5 or y minus

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2 is equal to minus 5.

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Add 2 to both sides of this,
you get y is equal to 7.

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Add 2 to both sides of this,
you get y is equal to minus 3.

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So we know that the points
minus 1, 7 and minus 1, minus

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3, are both on this graph.

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So, minus 1 is here.

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1, 2, 3, 4, 5, 6, 7, minus
1, 7, and minus 1, 1, 2,

00:10:26.410 --> 00:10:28.660
3, are both on this graph.

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So that lets us know, since
we're inside here, this tells

00:10:31.610 --> 00:10:33.830
us that this is kind of a
vertical asymptote, and another

00:10:33.830 --> 00:10:37.580
way to guess it is, if you
see that y squared

00:10:37.580 --> 00:10:39.770
term is positive.

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Or the other way to think about
it is, when you take the

00:10:42.200 --> 00:10:46.460
positive square root, when you
take the positive square root,

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you're always going to be a
little bit above the asymptote.

00:10:51.210 --> 00:10:52.730
That's the other way
to think about this.

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That we're always going to be
a little bit -- and this is

00:10:56.470 --> 00:10:57.780
the positive square root.

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The positive square
root is the top line.

00:10:58.860 --> 00:11:02.480
So we're always going to be a
little bit above the asymptote.

00:11:02.480 --> 00:11:03.150
This is the asymptote.

00:11:03.150 --> 00:11:04.870
But we're always a
little bit above it.

00:11:04.870 --> 00:11:07.370
And obviously as this number
gets larger, this starts to

00:11:07.370 --> 00:11:09.080
matter a lot less, so the
graph is going to look

00:11:09.080 --> 00:11:10.740
something like this.

00:11:10.740 --> 00:11:13.870
It's going to come down and
then go off, and never quite

00:11:13.870 --> 00:11:15.220
touch the asymptote,
but approach it.

00:11:15.220 --> 00:11:17.770
So it's going to get really
close to the asymptote, and

00:11:17.770 --> 00:11:21.030
then go off, and go off
in that direction.

00:11:21.030 --> 00:11:22.970
Anyway, hope you
found that helpful.

00:11:22.970 --> 00:11:25.480
This was a slightly hairier
problem, so it should

00:11:25.480 --> 00:11:27.050
be instructive.


Identifying an ellipse from equation

WEBVTT
Kind: captions
Language: en

00:00:01.180 --> 00:00:03.560
The standard question you often
get in your algebra class is

00:00:03.560 --> 00:00:06.300
they will give you this
equation and it'll say identify

00:00:06.300 --> 00:00:09.050
the conic section and
graph it if you can.

00:00:09.050 --> 00:00:11.140
And the equation they give you
won't be in the standard form,

00:00:11.140 --> 00:00:13.845
because if it was you could
just kind of pattern match with

00:00:13.845 --> 00:00:15.470
what I showed in some of the
previous videos and you'd

00:00:15.470 --> 00:00:16.080
be able to get it.

00:00:16.080 --> 00:00:17.600
So let's do a question
like and let's see if

00:00:17.600 --> 00:00:19.230
we can figure it out.

00:00:19.230 --> 00:00:28.990
So what I have here is 9x
squared plus 4y squared

00:00:28.990 --> 00:00:41.400
plus 54x minus 8y plus
49 is equal to 0.

00:00:41.400 --> 00:00:44.050
And once again, I mean who
knows what this is it's just

00:00:44.050 --> 00:00:45.000
not in the standard form.

00:00:45.000 --> 00:00:48.500
And actually one quick clue to
tell you what this is you look

00:00:48.500 --> 00:00:51.980
at the x squared and the y
squared terms if there are.

00:00:51.980 --> 00:00:55.130
If there's only an x squared
term and then there's just a y

00:00:55.130 --> 00:00:56.990
and not a y squared term, then
you're probably dealing with a

00:00:56.990 --> 00:00:58.970
parabola, and we'll go
into that more later.

00:00:58.970 --> 00:01:00.770
Or if it's the other way
around, if it's just an x

00:01:00.770 --> 00:01:03.330
term and a y squared term,
it's probably a parabola.

00:01:03.330 --> 00:01:06.555
But assuming that we're dealing
with a circle, an ellipse, or a

00:01:06.555 --> 00:01:09.450
hyperbola, there will be an x
squared term and a

00:01:09.450 --> 00:01:10.970
y squared term.

00:01:10.970 --> 00:01:13.970
If they both kind of have the
same number in front of them,

00:01:13.970 --> 00:01:15.580
that's a pretty good clue
that we're going to be

00:01:15.580 --> 00:01:16.710
dealing with a circle.

00:01:16.710 --> 00:01:19.550
If they both have different
numbers, but they're both

00:01:19.550 --> 00:01:21.550
positive in front of them,
that's a pretty good clue

00:01:21.550 --> 00:01:25.500
we're probably going to be
dealing with an ellipse.

00:01:25.500 --> 00:01:27.906
If one of them has a negative
number in front of them and

00:01:27.906 --> 00:01:30.420
the other one has a positive
number, that tells you that

00:01:30.420 --> 00:01:32.690
we're probably going to be
dealing with a hyperbola.

00:01:32.690 --> 00:01:35.290
But with that said, I mean that
might help you identify things

00:01:35.290 --> 00:01:37.440
very quickly at this level, but
it doesn't help you graph it or

00:01:37.440 --> 00:01:38.500
get into the standard form.

00:01:38.500 --> 00:01:40.370
So let's get it in
the standard form.

00:01:40.370 --> 00:01:42.720
And the key to getting it in
the standard form is really

00:01:42.720 --> 00:01:43.790
just completing the square.

00:01:43.790 --> 00:01:45.710
And I encourage you to re-watch
the completing the square

00:01:45.710 --> 00:01:48.245
video, because that's all we're
going to do right here to get

00:01:48.245 --> 00:01:49.740
it into the standard form.

00:01:49.740 --> 00:01:51.490
So the first thing I like to do
to complete the square, and

00:01:51.490 --> 00:01:53.590
you're going to have to do it
for the x variables and for

00:01:53.590 --> 00:01:56.190
the y terms, is group
the x and y terms.

00:01:56.190 --> 00:01:56.590
Let's see.

00:01:56.590 --> 00:02:02.796
The x terms are 9x
squared plus 54x.

00:02:09.370 --> 00:02:12.740
And let's do the y
terms in magenta.

00:02:12.740 --> 00:02:22.130
So then you have plus 4y
squared minus 8y and then you

00:02:22.130 --> 00:02:24.500
have-- let me do this in a
different color-- plus

00:02:24.500 --> 00:02:30.290
49 is equal to 0.

00:02:30.290 --> 00:02:33.060
And so the easy thing to do
when you complete the square,

00:02:33.060 --> 00:02:36.320
the thing I like to do is, it's
very clear we can factor out a

00:02:36.320 --> 00:02:38.390
9 out of both of these numbers,
and we can factor out a

00:02:38.390 --> 00:02:39.460
4 out of both of those.

00:02:39.460 --> 00:02:40.790
Let's do that, because
that will help us

00:02:40.790 --> 00:02:42.260
complete the square.

00:02:42.260 --> 00:02:48.630
So this is the same thing
is 9 times x squared plus

00:02:48.630 --> 00:02:51.790
9 times 6 is 54, 6x.

00:02:51.790 --> 00:02:53.490
I'm going to add something
else here, but I'll

00:02:53.490 --> 00:02:56.480
leave it blank for now.

00:02:56.480 --> 00:03:04.250
Plus 4 times y squared minus
2y I'm probably going to add

00:03:04.250 --> 00:03:08.520
something here too, so I'll
leave it blank for now.

00:03:08.520 --> 00:03:13.870
Plus 49 is equal to 0.

00:03:13.870 --> 00:03:14.880
So what are we
going to add here?

00:03:14.880 --> 00:03:15.980
We're going to
complete the square.

00:03:15.980 --> 00:03:21.340
We want to add some number here
so that this whole three term

00:03:21.340 --> 00:03:23.780
expression becomes
a perfect square.

00:03:23.780 --> 00:03:25.990
Likewise, we're going to add
some number here, so this

00:03:25.990 --> 00:03:28.750
three term number expression
becomes a perfect square.

00:03:28.750 --> 00:03:31.240
And of course whatever we add
on the side, we're going

00:03:31.240 --> 00:03:32.670
to have to multiply it by
9, because we're really

00:03:32.670 --> 00:03:34.170
adding nine times that.

00:03:34.170 --> 00:03:35.740
And add it on to that side.

00:03:35.740 --> 00:03:37.135
Whatever we add here, we're
going to have to multiply

00:03:37.135 --> 00:03:39.160
it times 4 and add
it on that side.

00:03:39.160 --> 00:03:42.180
If I put a 1 here, it's really
like as if I had a 4 here,

00:03:42.180 --> 00:03:45.720
because 1 times 4 is 4 and if I
had a 1 here it's 1 times 9.

00:03:45.720 --> 00:03:46.930
So 9 there.

00:03:46.930 --> 00:03:48.220
Let's do that.

00:03:48.220 --> 00:03:49.990
When we complete the square,
we just take half of

00:03:49.990 --> 00:03:51.130
this coefficient.

00:03:51.130 --> 00:03:54.100
This coefficient is 6, we
take half of it is 3, we

00:03:54.100 --> 00:03:55.290
square it, we get a 9.

00:03:58.420 --> 00:04:00.850
Remember it's an equation, so
what you do to one side, you

00:04:00.850 --> 00:04:01.140
have to do to the other.

00:04:01.140 --> 00:04:06.320
So if we added a 9 here, we're
actually adding 9 times 9 to

00:04:06.320 --> 00:04:10.030
the left-hand side of the
equation, so we have to add 81

00:04:10.030 --> 00:04:13.070
to the right-hand side to make
the equation still hold.

00:04:13.070 --> 00:04:15.550
And you could kind of view
it if we go back up here.

00:04:15.550 --> 00:04:17.490
This is the same thing, just
to make that clear as if I

00:04:17.490 --> 00:04:20.200
added plus 81 right here.

00:04:20.200 --> 00:04:23.640
Of course I would have had
to add plus 81 up here.

00:04:23.640 --> 00:04:25.210
Now let's go to the y terms.

00:04:25.210 --> 00:04:27.980
You take half of this
coefficient is minus 2,

00:04:27.980 --> 00:04:29.750
half of that is minus 1.

00:04:29.750 --> 00:04:32.660
You square it, you get plus 1.

00:04:32.660 --> 00:04:35.350
1 times 4, so we're really
adding 4 to the left-hand

00:04:35.350 --> 00:04:38.170
side of the equation.

00:04:38.170 --> 00:04:41.210
And just so you understand
what I did here.

00:04:41.210 --> 00:04:44.030
This is equivalent as if I just
added a 4 here, and then I

00:04:44.030 --> 00:04:45.520
later just factored out this 4.

00:04:48.350 --> 00:04:49.930
And so what does this become?

00:04:49.930 --> 00:04:54.630
This expression
is 9 times what?

00:04:54.630 --> 00:04:57.820
This is the square of-- you
could factor this, but we did

00:04:57.820 --> 00:05:05.940
it on purpose-- it's x plus 3
squared and then we have plus

00:05:05.940 --> 00:05:08.910
4 times-- What is
this right here?

00:05:08.910 --> 00:05:13.430
That's y minus 1 squared.

00:05:13.430 --> 00:05:15.560
You might want to review
factoring of polynomial or

00:05:15.560 --> 00:05:18.280
completing the square if
you found that step

00:05:18.280 --> 00:05:19.290
a little daunting.

00:05:19.290 --> 00:05:26.330
And then we have plus 49 is
equal to 0 plus 81 plus

00:05:26.330 --> 00:05:29.550
84 is equal to 85.

00:05:29.550 --> 00:05:38.080
All right, so now we have 9
times plus 3 squared plus 4

00:05:38.080 --> 00:05:41.180
times y minus 1 squared.

00:05:41.180 --> 00:05:44.360
And let's subtract
49 from both sides.

00:05:44.360 --> 00:05:51.380
That is equal to-- let's see if
I subtract 50 from 85 I get 35,

00:05:51.380 --> 00:05:53.360
so if I subtract 49, I get 36.

00:05:56.560 --> 00:06:00.300
And now we are getting close to
the standard form of something,

00:06:00.300 --> 00:06:02.130
but remember all the standard
forms we did except for the

00:06:02.130 --> 00:06:04.000
circle-- we had a y-- and we
know this isn't a circle,

00:06:04.000 --> 00:06:05.780
because we have these weird
coefficients, well not weird

00:06:05.780 --> 00:06:08.520
but different coefficients
in front of these terms.

00:06:08.520 --> 00:06:10.040
So to get the 1 on the
right-hand side let's

00:06:10.040 --> 00:06:12.140
divide everything by 36.

00:06:12.140 --> 00:06:20.770
If you divide everything by 36,
this term becomes x plus 3

00:06:20.770 --> 00:06:28.030
squared over see 9 over 36 is
the same thing as 1 over 4, and

00:06:28.030 --> 00:06:34.780
then you have plus y minus 1
squared 4 over 36 is the same

00:06:34.780 --> 00:06:39.140
thing as 1 over 9 and all
of that is equal to 1.

00:06:39.140 --> 00:06:39.740
And there you go.

00:06:39.740 --> 00:06:42.580
We have it in the standard
form, and you can see our

00:06:42.580 --> 00:06:44.990
intuition at the beginning
the problem was correct.

00:06:44.990 --> 00:06:50.010
This is indeed an ellipse, and
now we can actually graph it.

00:06:50.010 --> 00:06:51.860
So first of all, actually
good place to start, where's

00:06:51.860 --> 00:06:54.030
the center of the this
ellipse going to be?

00:06:54.030 --> 00:06:56.640
It's going to be x is
equal to negative 3.

00:06:56.640 --> 00:06:59.820
What x value makes
this whole terms 0?

00:06:59.820 --> 00:07:03.980
So it's going to be x is
equal to minus 3, and y is

00:07:03.980 --> 00:07:04.940
going to be equal to 1.

00:07:04.940 --> 00:07:08.910
What y value makes this
term 0? y is equal to 1.

00:07:08.910 --> 00:07:10.310
That's our center.

00:07:10.310 --> 00:07:15.510
So let's graph that, and then
we can draw the ellipse.

00:07:15.510 --> 00:07:17.475
It's going to be in the
negative quadrant.

00:07:17.475 --> 00:07:22.680
This is our x-axis and
this is our y-axis.

00:07:22.680 --> 00:07:29.660
And then the center of our
ellipse is at minus 3

00:07:29.660 --> 00:07:34.010
and positive 1, so
that's the center.

00:07:34.010 --> 00:07:37.350
And then, what is the
radius in the x direction?

00:07:37.350 --> 00:07:40.640
We just take the square
root of this, so it's 2.

00:07:40.640 --> 00:07:45.200
So in the x direction we
go two to the right.

00:07:45.200 --> 00:07:47.800
We go two to the left.

00:07:47.800 --> 00:07:49.340
And in the y direction,
what do we do?

00:07:49.340 --> 00:07:51.120
Well we go up three
and down three.

00:07:51.120 --> 00:07:52.890
The square root of this.

00:07:52.890 --> 00:07:53.610
Let me do that.

00:08:06.150 --> 00:08:10.430
Remember you have to take the
square root of both of those.

00:08:10.430 --> 00:08:14.080
The vertical axis is actually
the major radius or the

00:08:14.080 --> 00:08:18.430
semi-major axis is 3, because
that's the longer one.

00:08:18.430 --> 00:08:21.960
And then the 2 is the minor
radius, because that's

00:08:21.960 --> 00:08:23.100
the shorter one.

00:08:23.100 --> 00:08:26.780
And now we're ready to
draw this ellipse.

00:08:26.780 --> 00:08:28.310
I'll draw it in brown.

00:08:28.310 --> 00:08:30.410
Let me see if I can
do this properly.

00:08:30.410 --> 00:08:32.820
I have a shaky hand.

00:08:32.820 --> 00:08:35.940
All right, it looks
something like that.

00:08:35.940 --> 00:08:36.950
And there you go.

00:08:36.950 --> 00:08:40.730
We took this kind of crazy
looking thing, and all we did

00:08:40.730 --> 00:08:42.470
is algebraically manipulate it.

00:08:42.470 --> 00:08:46.980
We just completed the squares
with the x's and the y terms.

00:08:46.980 --> 00:08:49.180
And then we divided both sides
by this number right here and

00:08:49.180 --> 00:08:50.465
we got it into the
standard form.

00:08:50.465 --> 00:08:51.970
We said oh this is an ellipse.

00:08:51.970 --> 00:08:54.800
We have both of these terms,
they're both positive, we're

00:08:54.800 --> 00:08:57.090
adding we're not subtracting,
they have different

00:08:57.090 --> 00:08:58.930
coefficients underneath here.

00:08:58.930 --> 00:09:01.510
So we're ready to go over the
ellipse, and we realized that

00:09:01.510 --> 00:09:04.160
the center was at minus 3,1,
and then we just drew the

00:09:04.160 --> 00:09:07.990
major radius, or the major
axis and the minor axis.

00:09:07.990 --> 00:09:08.490
See you in the next video.


Conic sections: Hyperbolas 3

WEBVTT
Kind: captions
Language: en

00:00:01.020 --> 00:00:04.000
Let's see if we can tackle
a slightly more difficult

00:00:04.000 --> 00:00:05.810
hyperbola graphing problem.

00:00:05.810 --> 00:00:08.150
Let's add the hyperbola.

00:00:08.150 --> 00:00:21.180
Make this up on the fly x minus
1 squared over 16 minus y plus

00:00:21.180 --> 00:00:29.660
1 squared over 4 is equal to 1.

00:00:29.660 --> 00:00:32.480
So the first thing to recognize
is that this is a hyperbola and

00:00:32.480 --> 00:00:35.410
we'll in a few videos, do a
bunch of problems where the

00:00:35.410 --> 00:00:39.025
first point is just to identify
what type of conic section we

00:00:39.025 --> 00:00:41.480
have and then the second
step is actually graph

00:00:41.480 --> 00:00:43.440
the conic section.

00:00:43.440 --> 00:00:45.400
I already told you that we're
going to be doing a hyperbola

00:00:45.400 --> 00:00:46.550
problem, so you know
it's a hyperbola.

00:00:46.550 --> 00:00:49.500
But the way to recognize that
is that you have this minus of

00:00:49.500 --> 00:00:52.670
the y squared term and then
we actually have it shifted.

00:00:52.670 --> 00:00:56.640
The classic or the standard
non-shifted form of a hyperbola

00:00:56.640 --> 00:00:59.940
or a hyperbola centered at 0
would look something like this.

00:00:59.940 --> 00:01:04.950
Especially if it has the same
asymptotes just shifted, but

00:01:04.950 --> 00:01:10.600
centered at 0 it would look
like this: x squared over 16

00:01:10.600 --> 00:01:15.820
minus y squared over
4 is equal to 1.

00:01:15.820 --> 00:01:18.830
And the difference between this
hyperbola and this hyperbola

00:01:18.830 --> 00:01:28.670
the center of this hyperbola is
at the point x is equal to

00:01:28.670 --> 00:01:32.250
1 y is equal to minus 1.

00:01:32.250 --> 00:01:35.670
And the way to think about it
is x equals 1 makes this whole

00:01:35.670 --> 00:01:39.570
term 0, and so that's
why it's the center.

00:01:39.570 --> 00:01:42.720
And y equal to minus 1
makes this whole term 0.

00:01:42.720 --> 00:01:45.750
And on here, of course,
the center is the origin.

00:01:45.750 --> 00:01:49.030
Center is 0, 0.

00:01:49.030 --> 00:01:53.430
So the easy way to graph this
is to really graph this one,

00:01:53.430 --> 00:01:58.140
but you shift it so you use the
center being 1 minus 1 instead

00:01:58.140 --> 00:01:59.770
of the center being 0, 0.

00:01:59.770 --> 00:02:00.440
So let's do that.

00:02:00.440 --> 00:02:06.990
So let's figure out the slope
of the two asymptotes here and

00:02:06.990 --> 00:02:10.270
then we can shift those two
slopes so that it's appropriate

00:02:10.270 --> 00:02:12.220
for this hyperbola right here.

00:02:12.220 --> 00:02:15.140
So if we go with this one,
let's just solve for y.

00:02:15.140 --> 00:02:19.480
That's what I always
like to do whenever I'm

00:02:19.480 --> 00:02:21.270
graphing a hyperbola.

00:02:21.270 --> 00:02:26.100
So we get minus y
squared over 4.

00:02:26.100 --> 00:02:29.470
Subtracting x squared over
16 from both sides minus

00:02:29.470 --> 00:02:34.730
x squared over 16 plus 1.

00:02:34.730 --> 00:02:37.190
I'm working on this hyperbola
right here, not this one,

00:02:37.190 --> 00:02:38.880
and then I'm going to
just shift it later.

00:02:38.880 --> 00:02:41.810
And then let's say multiply
both sides by minus 4 and you

00:02:41.810 --> 00:02:47.100
get y squared is equal to-- see
the minus cancels out with that

00:02:47.100 --> 00:02:59.020
and then 4 over 16 is x squared
over 4 minus 4 and so y is

00:02:59.020 --> 00:03:05.900
equal to plus or minus square
root of x squared

00:03:05.900 --> 00:03:08.480
over 4 minus 4.

00:03:08.480 --> 00:03:10.340
And to figure out the
asymptotes you just have to

00:03:10.340 --> 00:03:13.640
think about well what happens
as x approaches positive

00:03:13.640 --> 00:03:14.220
or negative infinity.

00:03:14.220 --> 00:03:17.270
As x gets really positive
or x gets really negative.

00:03:17.270 --> 00:03:19.040
And we've done this a
bunch of times already.

00:03:19.040 --> 00:03:19.900
I think this is important.

00:03:19.900 --> 00:03:21.560
This is more important than
just memorizing the formula,

00:03:21.560 --> 00:03:23.640
because it gives you an
intuition of where those

00:03:23.640 --> 00:03:26.750
equations for the lines of the
asymptote actually come from.

00:03:26.750 --> 00:03:30.710
Because these are what this
graph or this equation or this

00:03:30.710 --> 00:03:33.330
function approaches as x
approaches positive or

00:03:33.330 --> 00:03:34.600
negative infinity.

00:03:34.600 --> 00:03:47.090
As x approaches positive or
negative infinity, what is y

00:03:47.090 --> 00:03:49.680
approximately equal
to, in this case?

00:03:49.680 --> 00:03:52.860
Well once again, this term
is going to dominate.

00:03:52.860 --> 00:03:54.110
This is just a 4 right here.

00:03:54.110 --> 00:03:57.410
You could imagine when x is
like a trillion or a negative

00:03:57.410 --> 00:03:59.650
trillion, this is going to be
huge number and this is going

00:03:59.650 --> 00:04:01.230
to be just like you know
you almost view it like

00:04:01.230 --> 00:04:01.920
the ground off area.

00:04:01.920 --> 00:04:03.630
You take the square root
of that and so this

00:04:03.630 --> 00:04:04.520
is going to dominate.

00:04:04.520 --> 00:04:07.910
So as you approach positive or
negative infinity, y is going

00:04:07.910 --> 00:04:10.570
to be approximately equal to
the square root, the positive

00:04:10.570 --> 00:04:15.510
and negative square root,
of x squared over 4.

00:04:15.510 --> 00:04:18.950
So y would be approximately
equal to positive or

00:04:18.950 --> 00:04:25.340
negative x over 2, or 1/2x.

00:04:25.340 --> 00:04:26.560
Let's do that.

00:04:26.560 --> 00:04:28.525
Let's draw our asymptotes.

00:04:28.525 --> 00:04:31.820
And remember, these are the
asymptotes for this situation.

00:04:31.820 --> 00:04:34.070
But now of course, we're
centered at 1 negative 1.

00:04:34.070 --> 00:04:38.330
So I'm going to draw two lines
with these slopes, with

00:04:38.330 --> 00:04:41.840
positive 1/2 and negative 1/2
slopes, but they're going to be

00:04:41.840 --> 00:04:44.450
centered at this point.

00:04:44.450 --> 00:04:47.090
I just got rid of the shift
just so I could figure out the

00:04:47.090 --> 00:04:49.120
asymptotes but of course this
is the real thing that we're

00:04:49.120 --> 00:04:53.480
trying to graph, so
let me do that.

00:04:53.480 --> 00:05:02.990
This is my y-axis this is my
x-axis and the center of

00:05:02.990 --> 00:05:05.340
this is at 1 negative 1.

00:05:08.190 --> 00:05:14.360
So x is equal to 1 y
is equal to minus 1.

00:05:14.360 --> 00:05:17.530
And then the slopes of the
asymptotes were positive

00:05:17.530 --> 00:05:18.420
and negative 1/2.

00:05:22.060 --> 00:05:24.040
So let's do the positive 1/2.

00:05:24.040 --> 00:05:28.330
So that means for every 2 you
run over, so if you go positive

00:05:28.330 --> 00:05:31.770
in the positive x direction
2, you move up 1.

00:05:31.770 --> 00:05:34.710
So you go to the
right 2 and up 1.

00:05:34.710 --> 00:05:35.880
So that's the first one.

00:05:35.880 --> 00:05:37.620
Let me draw that asymptote.

00:05:41.320 --> 00:05:43.970
Looks something like that, and
then we draw it from this

00:05:43.970 --> 00:05:45.610
point to that point.

00:05:45.610 --> 00:05:46.790
Got to have a steady hand.

00:05:51.520 --> 00:05:53.040
And then the other asymptote
is going to have a

00:05:53.040 --> 00:05:56.850
minus 1/2 slope.

00:05:56.850 --> 00:05:59.350
Remember this is our
center 1 minus 1, so if

00:05:59.350 --> 00:06:04.350
I go down 1 and over.

00:06:04.350 --> 00:06:07.580
So when I go over 2, I go down
1, so that will be right there,

00:06:07.580 --> 00:06:10.850
Let me draw that asymptote.

00:06:14.790 --> 00:06:18.320
And then just to continue it
in the other direction I want

00:06:18.320 --> 00:06:20.280
to make the lines overlap.

00:06:20.280 --> 00:06:22.230
It's going to look
something like that.

00:06:22.230 --> 00:06:24.690
So we've drawn our asymptotes
for this function, and now we

00:06:24.690 --> 00:06:28.830
have to figure out if it's
going to be a vertical

00:06:28.830 --> 00:06:31.600
hyperbola or a
horizontal hyperbola.

00:06:31.600 --> 00:06:34.860
And the easy way to think
about it is to try and make--

00:06:34.860 --> 00:06:36.230
and we can do it two ways.

00:06:36.230 --> 00:06:39.650
I mean if you just look at
this equation right here.

00:06:39.650 --> 00:06:42.560
When you're taking the positive
square root, we're always going

00:06:42.560 --> 00:06:45.720
to be slightly below
the asymptote.

00:06:45.720 --> 00:06:47.890
The asymptote is this thing,
but we're always going

00:06:47.890 --> 00:06:48.810
to be slightly below it.

00:06:48.810 --> 00:06:50.550
So that tells us that were
always going to be slightly

00:06:50.550 --> 00:06:53.810
below the asymptote on the
positive square root, and we're

00:06:53.810 --> 00:06:55.460
always going to be slightly
above the asymptote on the

00:06:55.460 --> 00:06:56.100
negative square root.

00:06:56.100 --> 00:06:58.000
Because it's going to be little
less, and it's negative.

00:06:58.000 --> 00:07:00.850
But I'll let you
think about that.

00:07:00.850 --> 00:07:02.890
My intuition is that it's
going to be there and there.

00:07:02.890 --> 00:07:03.600
It's more than intuition.

00:07:03.600 --> 00:07:05.770
I know that we're going to be
a little bit less than the

00:07:05.770 --> 00:07:07.920
negative square root, but
I'll do it the other way.

00:07:07.920 --> 00:07:09.410
I'll do it the way I
did in the last video.

00:07:09.410 --> 00:07:12.430
So the other way to think
about it is what happens

00:07:12.430 --> 00:07:13.850
when this term is 0?

00:07:13.850 --> 00:07:18.330
For this term to be 0, x
has to be equal to 1.

00:07:18.330 --> 00:07:20.160
And does that ever happen?

00:07:20.160 --> 00:07:21.680
Can x be equal to 1?

00:07:21.680 --> 00:07:25.700
If x is equal to 1
here this term is 0.

00:07:25.700 --> 00:07:33.810
And then you have a situation
where-- and then you have a

00:07:33.810 --> 00:07:39.530
minus y squared over 4 would
have to equal 1, or this would

00:07:39.530 --> 00:07:40.975
have to be a negative number.

00:07:40.975 --> 00:07:43.610
So x could not be equal to 1.

00:07:43.610 --> 00:07:46.720
So y could be equal
to negative 1.

00:07:46.720 --> 00:07:47.770
Let's try that out.

00:07:47.770 --> 00:07:54.120
If y is equal to negative
1, this term right

00:07:54.120 --> 00:07:54.680
here disappears.

00:08:01.300 --> 00:08:07.800
So when y is equal to negative
1, you're just left with-- x

00:08:07.800 --> 00:08:13.450
minus 1 squared over
16 is equal to 1.

00:08:13.450 --> 00:08:15.270
I just canceled out this term,
because I'm saying what happens

00:08:15.270 --> 00:08:17.400
when y is equal to negative 1.

00:08:17.400 --> 00:08:19.970
You multiply both sides by 16.

00:08:19.970 --> 00:08:21.330
Let me do it over here.

00:08:21.330 --> 00:08:26.280
These get messy. x minus 1
squared is equal to 16.

00:08:26.280 --> 00:08:27.860
Take the square root
of both sides.

00:08:27.860 --> 00:08:34.710
x minus 1 is equal to
positive or negative 4.

00:08:34.710 --> 00:08:38.440
And so if x is equal to
positive 4, if you add 1 to

00:08:38.440 --> 00:08:41.700
that x would be equal to 5.

00:08:41.700 --> 00:08:46.660
And then if x minus 1 would be
minus 4 and you add 1 to that

00:08:46.660 --> 00:08:49.080
you will have x is equal to 3.

00:08:49.080 --> 00:08:54.260
So our 2 points or our 2 points
closest to our center are the

00:08:54.260 --> 00:08:59.830
points 5 comma negative 1
and 3 comma negative 1.

00:08:59.830 --> 00:09:00.770
Let's plot those 2.

00:09:00.770 --> 00:09:09.110
So 5, 1 2 3 4 5, negative
1 and 3, negative 1.

00:09:09.110 --> 00:09:11.310
Is that right?

00:09:11.310 --> 00:09:14.070
No, minus 3, because x
minus 1 could be minus 4.

00:09:14.070 --> 00:09:15.210
That's what happens
when you skip steps.

00:09:24.650 --> 00:09:26.770
If you have the minus 4
situation, then x is

00:09:26.770 --> 00:09:28.570
equal to minus 3.

00:09:31.160 --> 00:09:35.910
You go 1 2 3 minus 3, minus 1.

00:09:35.910 --> 00:09:38.690
So those are both points
on this hyperbola.

00:09:38.690 --> 00:09:40.980
And then our intuition was
correct, or it was what I said.

00:09:40.980 --> 00:09:47.110
That-- the positive square root
is always going to be slightly

00:09:47.110 --> 00:09:49.490
below the asymptote,
so we get our curve.

00:09:49.490 --> 00:09:52.500
It's going to look
something like this.

00:09:52.500 --> 00:09:54.590
It's going to get closer and
closer, and then here it's

00:09:54.590 --> 00:09:56.590
going to get closer and
closer in that direction.

00:09:56.590 --> 00:09:59.070
It keeps getting closer and
closer to that asymptote.

00:09:59.070 --> 00:10:01.690
And here, it's going to keep
getting closer and closer to

00:10:01.690 --> 00:10:06.080
the asymptote on that side
and then on that side.

00:10:06.080 --> 00:10:07.760
And of course these
asymptotes keep going

00:10:07.760 --> 00:10:09.600
on forever and forever.

00:10:09.600 --> 00:10:11.370
If you want you could try
out some other points

00:10:11.370 --> 00:10:12.100
just to confirm.

00:10:12.100 --> 00:10:14.690
You could plot that point
there, or that point there

00:10:14.690 --> 00:10:16.220
just to confirm that
that's the case.

00:10:16.220 --> 00:10:18.720
The hard part really is just to
identify the asymptotes and

00:10:18.720 --> 00:10:22.620
just to figure out do we sit on
the left and the right, or do

00:10:22.620 --> 00:10:23.950
we sit on the top
and the bottom.

00:10:23.950 --> 00:10:24.570
And then you're done.

00:10:24.570 --> 00:10:27.200
You can graph your hyperbola.

00:10:27.200 --> 00:10:27.690
See you in the next video.


Conic sections: Hyperbolas 2

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.010
In the last hyperbola video
I didn't get a chance to

00:00:03.010 --> 00:00:04.470
do some concrete examples.

00:00:04.470 --> 00:00:06.390
So I'll do that right now.

00:00:06.390 --> 00:00:13.050
So, let's say I had the
hyperbola y squared over

00:00:13.050 --> 00:00:19.770
4 minus x squared over,
I don't know, let me

00:00:19.770 --> 00:00:21.150
think of a good number.

00:00:21.150 --> 00:00:27.310
Let's say, x squared
over 9 is equal to 1.

00:00:27.310 --> 00:00:31.040
So the first thing to figure
out about this hyperbola is,

00:00:31.040 --> 00:00:32.880
what are its asymptotes?

00:00:32.880 --> 00:00:34.970
And, once again, I always
forget the formulas.

00:00:34.970 --> 00:00:38.110
And I just try to solve for y
and see what happens when x

00:00:38.110 --> 00:00:40.510
approaches positive or
negative infinity.

00:00:40.510 --> 00:00:42.680
So if you solve for y,
you can add x squared

00:00:42.680 --> 00:00:43.980
over 9 to both sides.

00:00:43.980 --> 00:00:50.500
And you get y squared
over 4 is equal to x

00:00:50.500 --> 00:00:55.140
squared over 9 plus 1.

00:00:55.140 --> 00:00:57.190
Now, I can multiply
4 times both sides.

00:00:57.190 --> 00:01:02.920
And you get y squared is
equal to 4 over 9 times

00:01:02.920 --> 00:01:05.790
x squared plus 4.

00:01:05.790 --> 00:01:08.790
I distribute the 4, take
the positive and negative

00:01:08.790 --> 00:01:10.550
square root both sides.

00:01:10.550 --> 00:01:17.270
y is equal to the plus or
minus square root of 4

00:01:17.270 --> 00:01:21.140
over 9x squared plus 4.

00:01:21.140 --> 00:01:22.880
And you can't really
simplify this anymore.

00:01:22.880 --> 00:01:26.000
But we can think about, what
does this approach as x

00:01:26.000 --> 00:01:28.640
approaches positive or
negative infinity.

00:01:28.640 --> 00:01:36.530
So, as x approaches plus
or minus infinity, what

00:01:36.530 --> 00:01:38.380
does this roughly equal?

00:01:38.380 --> 00:01:39.350
What does this approximate?

00:01:39.350 --> 00:01:41.450
What does the graph
get a lot closer to?

00:01:41.450 --> 00:01:46.115
Well, then, y is approximately
equal to just the square

00:01:46.115 --> 00:01:46.810
root of this term.

00:01:46.810 --> 00:01:51.256
Because this becomes super huge
and relative to this term, this

00:01:51.256 --> 00:01:52.690
starts to matter less
and less and less.

00:01:52.690 --> 00:01:55.480
And that's why we get closer
and closer to the asymptotes.

00:01:55.480 --> 00:01:59.410
Because when this number
is, like, a trillion, or a

00:01:59.410 --> 00:02:02.100
googol, then this number
is almost insignificant.

00:02:02.100 --> 00:02:04.000
You take the square root,
you're pretty much taking the

00:02:04.000 --> 00:02:05.720
square root of this, and
you'll just be a little

00:02:05.720 --> 00:02:06.900
bit above the graph.

00:02:06.900 --> 00:02:09.260
Because you have this
extra plus-4 there.

00:02:09.260 --> 00:02:12.660
So as you approach positive
or negative infinity, this

00:02:12.660 --> 00:02:17.260
equation is approximately equal
to the plus or minus square

00:02:17.260 --> 00:02:23.800
root of 4 over 9x squared.

00:02:23.800 --> 00:02:27.170
And so, that is -- so y would
be approximately equal

00:02:27.170 --> 00:02:27.950
to the plus or minus.

00:02:27.950 --> 00:02:30.010
We can take the
square root of this.

00:02:30.010 --> 00:02:33.770
Plus or minus the square
root of 4/9 is 2/3, right?

00:02:33.770 --> 00:02:37.880
Square root of 4 over
square root of 9, times x.

00:02:37.880 --> 00:02:39.020
So, these are the asymptotes.

00:02:39.020 --> 00:02:40.160
There's two lines here.

00:02:40.160 --> 00:02:42.940
There's y is equal to 2/3 x.

00:02:42.940 --> 00:02:45.860
And then there's y is
equal to minus 2/3 x.

00:02:45.860 --> 00:02:48.660
So let's draw those two lines.

00:02:48.660 --> 00:02:50.080
Let me draw my axes.

00:02:53.870 --> 00:02:56.100
Let's make that my y axis.

00:02:56.100 --> 00:02:57.470
Make that the x axis.

00:02:57.470 --> 00:03:00.100
Let me switch some colors, just
to make things interesting.

00:03:00.100 --> 00:03:01.620
So let me draw the first one.

00:03:01.620 --> 00:03:04.760
See, y is equal to 2/3 x.

00:03:04.760 --> 00:03:08.240
So, you rise 2 for
every 3 that you run.

00:03:08.240 --> 00:03:09.970
So let me draw that.

00:03:09.970 --> 00:03:17.640
So if this is 1, 2, 3, 1, 2.

00:03:17.640 --> 00:03:20.540
So that would be a
point on the line.

00:03:20.540 --> 00:03:21.990
Let me draw the line now.

00:03:21.990 --> 00:03:25.380
Actually go to the origin.

00:03:25.380 --> 00:03:26.750
No, that's not it.

00:03:26.750 --> 00:03:28.170
Let me draw it like this.

00:03:28.170 --> 00:03:29.930
This way I can make sure
it goes to the origin.

00:03:29.930 --> 00:03:31.940
This is going to go
through like that.

00:03:31.940 --> 00:03:33.850
Then I can go from here.

00:03:33.850 --> 00:03:35.670
And then go like that.

00:03:35.670 --> 00:03:37.280
So that's one asymptote.

00:03:37.280 --> 00:03:39.140
And the other asymptote is
y is going to be equal

00:03:39.140 --> 00:03:40.690
to minus 2/3 x, right?

00:03:40.690 --> 00:03:43.450
Because plus or minus 2/3 x.

00:03:43.450 --> 00:03:49.110
So, minus 2/3 x, you go down 2
for every 3 that you go out.

00:03:49.110 --> 00:03:54.160
So that point will show
up, see if I do, 1, 2.

00:03:54.160 --> 00:03:56.320
So you go down 2 for
every 3 that you go out.

00:03:56.320 --> 00:03:57.110
So it'll go 3.

00:03:57.110 --> 00:03:59.610
So if I draw that asymptote,
it'll look something

00:03:59.610 --> 00:04:03.531
like there.

00:04:03.531 --> 00:04:05.120
Go out there.

00:04:05.120 --> 00:04:08.180
And then go from here.

00:04:08.180 --> 00:04:09.590
Go out there.

00:04:09.590 --> 00:04:10.620
And we've drawn our asymptotes.

00:04:10.620 --> 00:04:13.630
Now the question is, is it
going to open up to the left

00:04:13.630 --> 00:04:15.030
or the right, or up and down?

00:04:15.030 --> 00:04:16.430
There's two ways we
can think about it.

00:04:16.430 --> 00:04:19.010
And I'll do it the way that
might be more intuitive for

00:04:19.010 --> 00:04:23.910
you is, can x -- what happens
when x is equal to 0?

00:04:23.910 --> 00:04:27.520
Well, when x is equal to
0, when x is equal to

00:04:27.520 --> 00:04:28.860
0, this disappears.

00:04:28.860 --> 00:04:32.880
And we're just left with,
I'll do it here, y squared

00:04:32.880 --> 00:04:36.480
over 4 is equal to 1.

00:04:36.480 --> 00:04:39.150
Or, y squared is equal to 4.

00:04:39.150 --> 00:04:42.290
Or, y is equal to
plus or minus 2.

00:04:42.290 --> 00:04:46.720
So, we know that the point
0, the points, 0 plus or

00:04:46.720 --> 00:04:48.710
minus 2, is on this graph.

00:04:48.710 --> 00:04:52.060
So x can be equal to 0
so 0 plus or minus 2.

00:04:52.060 --> 00:04:55.140
So 0 plus 2 is this
point right here.

00:04:55.140 --> 00:04:58.310
And 0 minus 2 is this
point right there.

00:04:58.310 --> 00:05:00.600
So that, by itself, actually,
is enough of a clue to know

00:05:00.600 --> 00:05:02.600
that it opens down here.

00:05:02.600 --> 00:05:03.160
And up here.

00:05:03.160 --> 00:05:04.800
Because it will never,
a hyperbola will never

00:05:04.800 --> 00:05:06.190
cross the asymptotes.

00:05:06.190 --> 00:05:09.610
It's not like it can go out
here and across this asymptote.

00:05:09.610 --> 00:05:09.830
So?

00:05:09.830 --> 00:05:13.430
We already know that the graph
of this parabola -- and you can

00:05:13.430 --> 00:05:15.340
try other points, if you
want, just to verify.

00:05:15.340 --> 00:05:16.550
It's going to look
something like this.

00:05:16.550 --> 00:05:18.880
It's going to go and then
-- nope, I want to make

00:05:18.880 --> 00:05:20.330
it so it never touches.

00:05:20.330 --> 00:05:24.470
It's going to get really
close, but no, I touched it.

00:05:24.470 --> 00:05:28.180
It's going to get really
close but never touch.

00:05:28.180 --> 00:05:29.970
And then on this side
it's going to get really

00:05:29.970 --> 00:05:32.100
close, but never touch.

00:05:32.100 --> 00:05:33.310
And I don't want to touch it.

00:05:33.310 --> 00:05:35.424
And then on the top side it's
going to do the same thing,

00:05:35.424 --> 00:05:37.243
it's going to get really close,
and as you approach infinity

00:05:37.243 --> 00:05:38.510
it's never going to touch it.

00:05:38.510 --> 00:05:40.970
And as you get reall close,
it'll get infiniitely

00:05:40.970 --> 00:05:41.850
close but never touch it.

00:05:41.850 --> 00:05:44.030
So that's what this parabola
-- this hyperbola --

00:05:44.030 --> 00:05:45.100
is going to look like.

00:05:45.100 --> 00:05:47.930
And I did it by just trying to
see if x could be equal to 0.

00:05:47.930 --> 00:05:50.370
And I encourage you to try
what happens when y equals 0.

00:05:50.370 --> 00:05:52.540
And you'll get no solution.

00:05:52.540 --> 00:05:55.390
And that makes sense because
this hyperbola never

00:05:55.390 --> 00:05:57.180
crosses y equals 0, right?

00:05:57.180 --> 00:06:00.330
It never crosses the x axis.

00:06:00.330 --> 00:06:02.630
And this should also be
intuitive, because if we

00:06:02.630 --> 00:06:05.250
saw here when we did ths
approximation, as x approaches

00:06:05.250 --> 00:06:09.820
positive or negative infinity,
we saw that we always did have

00:06:09.820 --> 00:06:12.270
this plus 4 sitting here.

00:06:12.270 --> 00:06:17.160
We said, oh, well, as x gets
super large or super negative,

00:06:17.160 --> 00:06:18.470
this starts to matter
less and less.

00:06:18.470 --> 00:06:22.190
But we will always be slightly
larger than this number.

00:06:22.190 --> 00:06:24.810
Especially in the positive
quadrant, right?

00:06:24.810 --> 00:06:26.605
We're always going to be --
so the positive quadrant is

00:06:26.605 --> 00:06:29.250
always going to be slightly
larger than the asymptote.

00:06:29.250 --> 00:06:33.210
And even when we take the
positive square root, I guess,

00:06:33.210 --> 00:06:33.990
is the best way to say it.

00:06:33.990 --> 00:06:36.600
When we take the positive
square root, we'll always

00:06:36.600 --> 00:06:39.140
be larger than either
of the asymptotes.

00:06:39.140 --> 00:06:42.100
And, likewise, when you take
the negative square root,

00:06:42.100 --> 00:06:44.010
you're always going to be
a little bit smaller than

00:06:44.010 --> 00:06:44.910
either of the asymptotes.

00:06:44.910 --> 00:06:46.690
Because this number is
going to be a little bit

00:06:46.690 --> 00:06:48.540
bigger than this number.

00:06:48.540 --> 00:06:49.740
Then we take the negative
square root, you're going to

00:06:49.740 --> 00:06:51.150
be a little bit smaller,
and that's why we're

00:06:51.150 --> 00:06:52.280
a little bit below.

00:06:52.280 --> 00:06:54.260
I don't know which one's
more intutive for you.

00:06:54.260 --> 00:06:57.590
Maybe just the -- trying it
when x is equal to 0 and when y

00:06:57.590 --> 00:06:59.740
equals 0, and see what points
you get and say, oh, then I'm

00:06:59.740 --> 00:07:04.990
in kind of a vertical hyperbola
as opposed to a horizontal one.

00:07:04.990 --> 00:07:07.960
So let's see if I have time
for -- I'll leave that

00:07:07.960 --> 00:07:08.770
video right there.

00:07:08.770 --> 00:07:10.090
And then I'll do another
video where I actually

00:07:10.090 --> 00:07:11.990
shift the hyperbola.

00:07:11.990 --> 00:07:15.350
And shifting it is actually
no different than shifting

00:07:15.350 --> 00:07:16.470
an ellipse a circle.

00:07:16.470 --> 00:07:19.200
You just have, you know, y
minus something squared, and x

00:07:19.200 --> 00:07:21.370
plus something, or x minus
something squared and that just

00:07:21.370 --> 00:07:23.890
tells you where you
shift the origin.

00:07:23.890 --> 00:07:26.610
This hyperbola, of course, is
just centered at the origin.

00:07:26.610 --> 00:07:28.890
Anyway, see you in
the next video.


Conic sections: Intro to hyperbolas

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:02.500
Let's see if we can learn
a thing or two about

00:00:02.500 --> 00:00:03.160
the hyperbola.

00:00:08.320 --> 00:00:10.680
And out of all the conic
sections, this is probably the

00:00:10.680 --> 00:00:14.340
one that confuses people the
most, because it's not quite

00:00:14.340 --> 00:00:17.820
as easy to draw as the
circle and the ellipse.

00:00:17.820 --> 00:00:19.160
You have to do a little
bit more algebra.

00:00:19.160 --> 00:00:20.930
But hopefully over the course
of this video you'll get pretty

00:00:20.930 --> 00:00:23.540
comfortable with that, and
you'll see that hyperbolas in

00:00:23.540 --> 00:00:27.660
some way are more fun than any
of the other conic sections.

00:00:27.660 --> 00:00:30.420
So just as a review, I want to
do this just so you see the

00:00:30.420 --> 00:00:33.750
similarity in the formulas or
the standard form of the

00:00:33.750 --> 00:00:34.740
different conic sections.

00:00:34.740 --> 00:00:40.330
If you have a circle centered
at 0, its equation is x

00:00:40.330 --> 00:00:44.730
squared plus y squared
is equal to r squared.

00:00:44.730 --> 00:00:47.720
And we saw that this could also
be written as-- and I'm doing

00:00:47.720 --> 00:00:50.150
this because I want to show
that this is really just the

00:00:50.150 --> 00:00:52.890
same thing as the standard
equation for an ellipse.

00:00:52.890 --> 00:00:56.870
If you divide both sides of
this by r squared, you get x

00:00:56.870 --> 00:01:03.760
squared over r squared plus y
squared over r squared

00:01:03.760 --> 00:01:04.775
is equal to 1.

00:01:04.775 --> 00:01:08.080
And so this is a circle.

00:01:08.080 --> 00:01:11.000
And once again, just as review,
a circle, all of the points on

00:01:11.000 --> 00:01:13.370
the circle are equidistant
from the center.

00:01:13.370 --> 00:01:16.650
Or in this case, you can kind
of say that the major axis and

00:01:16.650 --> 00:01:19.630
the minor axis are the same
distance, that there isn't any

00:01:19.630 --> 00:01:20.470
distinction between the two.

00:01:20.470 --> 00:01:23.280
You're always an equal distance
away from the center.

00:01:23.280 --> 00:01:24.780
So that was a circle.

00:01:24.780 --> 00:01:28.460
An ellipse was pretty much
this, but these two numbers

00:01:28.460 --> 00:01:29.600
could be different.

00:01:29.600 --> 00:01:32.310
Because your distance from
the center could change.

00:01:32.310 --> 00:01:38.850
So it's x squared over a
squared plus y squared over

00:01:38.850 --> 00:01:41.590
b squared is equal to 1.

00:01:41.590 --> 00:01:44.670
That's an ellipse.

00:01:44.670 --> 00:01:48.620
And now, I'll skip parabola for
now, because parabola's kind

00:01:48.620 --> 00:01:50.770
of an interesting case, and
you've already touched on it.

00:01:50.770 --> 00:01:53.050
So I'll go into more depth
in that in a future video.

00:01:53.050 --> 00:01:57.820
But a hyperbola is very
close in formula to this.

00:01:57.820 --> 00:02:00.630
And so there's two ways that a
hyperbola could be written.

00:02:00.630 --> 00:02:01.770
And I'll do those two ways.

00:02:01.770 --> 00:02:06.750
So it could either be written
as x squared over a squared

00:02:06.750 --> 00:02:11.815
minus y squared over b
squared is equal to 1.

00:02:11.815 --> 00:02:15.060
And notice the only difference
between this equation and this

00:02:15.060 --> 00:02:16.880
one is that instead of a
plus y squared, we have

00:02:16.880 --> 00:02:18.630
a minus y squared here.

00:02:18.630 --> 00:02:21.870
So that would be one hyperbola.

00:02:21.870 --> 00:02:24.400
The other one would be
if the minus sign was

00:02:24.400 --> 00:02:25.730
the other way around.

00:02:25.730 --> 00:02:33.970
If it was y squared over b
squared minus x squared over

00:02:33.970 --> 00:02:36.330
a squared is equal to 1.

00:02:36.330 --> 00:02:38.850
So now the minus is in front
of the x squared term instead

00:02:38.850 --> 00:02:40.820
of the y squared term.

00:02:40.820 --> 00:02:43.650
And what I want to do now is
try to figure out, how do

00:02:43.650 --> 00:02:46.970
we graph either of
these parabolas?

00:02:46.970 --> 00:02:48.170
Maybe we'll do both cases.

00:02:48.170 --> 00:02:51.450
And in a lot of text books, or
even if you look it up over the

00:02:51.450 --> 00:02:52.920
web, they'll give you formulas.

00:02:52.920 --> 00:02:53.990
But I don't like
those formulas.

00:02:53.990 --> 00:02:55.205
One, because I'll
always forget it.

00:02:55.205 --> 00:02:56.530
And you'll forget it
immediately after

00:02:56.530 --> 00:02:57.070
taking the test.

00:02:57.070 --> 00:02:58.790
You might want to memorize
it if you just want to

00:02:58.790 --> 00:03:00.450
be able to do the test
a little bit faster.

00:03:00.450 --> 00:03:01.500
But you'll forget it.

00:03:01.500 --> 00:03:03.260
And the second thing is, not
only will you forget it, but

00:03:03.260 --> 00:03:04.240
you'll probably get confused.

00:03:04.240 --> 00:03:07.440
Because sometimes they always
use the a under the x and the b

00:03:07.440 --> 00:03:09.640
under the y, or sometimes they
always use the a under the

00:03:09.640 --> 00:03:13.070
positive term and to b
under the negative term.

00:03:13.070 --> 00:03:16.110
So if you just memorize, oh, a
divided by b, that's the slope

00:03:16.110 --> 00:03:18.150
of the asymptote and all of
that, you might be using

00:03:18.150 --> 00:03:19.040
the wrong a and b.

00:03:19.040 --> 00:03:22.490
So I encourage you to always
re-prove it to yourself.

00:03:22.490 --> 00:03:23.770
And that's what we're
going to do right here.

00:03:23.770 --> 00:03:25.450
It actually doesn't
take too long.

00:03:25.450 --> 00:03:28.440
So these are both hyperbolas.

00:03:28.440 --> 00:03:30.040
And what I like to do
whenever I have a

00:03:30.040 --> 00:03:33.070
hyperbola is solve for y.

00:03:33.070 --> 00:03:37.480
So in this case, if I subtract
x squared over a squared from

00:03:37.480 --> 00:03:41.970
both sides, I get-- let me
change the color-- I get minus

00:03:41.970 --> 00:03:45.730
y squared over b squared.

00:03:45.730 --> 00:03:46.420
That stays there.

00:03:46.420 --> 00:03:53.680
Is equal to 1 minus x
squared over a squared.

00:03:53.680 --> 00:03:55.960
And then, let's see, I want to
get rid of this minus, and I

00:03:55.960 --> 00:03:57.360
want to get rid of
this b squared.

00:03:57.360 --> 00:03:59.840
So let's multiply both sides
of this equation times

00:03:59.840 --> 00:04:03.070
minus b squared.

00:04:03.070 --> 00:04:05.900
If you multiply the left hand
side times minus b squared, the

00:04:05.900 --> 00:04:08.550
minus and the b squared go
away, and you're just left

00:04:08.550 --> 00:04:14.920
with y squared is equal
to minus b squared.

00:04:14.920 --> 00:04:17.440
And then minus b squared
times a plus, it becomes

00:04:17.440 --> 00:04:24.990
a plus b squared over
a squared x squared.

00:04:24.990 --> 00:04:26.220
We're almost there.

00:04:26.220 --> 00:04:29.890
And then you get y is equal
to-- and I'm doing this on

00:04:29.890 --> 00:04:32.720
purpose-- the plus or minus
square root, because it can be

00:04:32.720 --> 00:04:35.010
the plus or minus square root.

00:04:35.010 --> 00:04:38.680
Of-- and let's switch these
around, just so I have

00:04:38.680 --> 00:04:40.430
the positive term first.

00:04:40.430 --> 00:04:49.270
b squared over a squared x
squared minus b squared.

00:04:49.270 --> 00:04:52.060
Now you said, Sal, you
said this was simple.

00:04:52.060 --> 00:04:52.890
I'm solving this.

00:04:52.890 --> 00:04:54.430
This looks like a really
complicated thing.

00:04:54.430 --> 00:04:57.380
But remember, we're doing this
to figure out asymptotes of the

00:04:57.380 --> 00:04:59.740
hyperbola, just to kind of
give you a sense of

00:04:59.740 --> 00:05:01.090
where we're going.

00:05:01.090 --> 00:05:03.120
Let me do it here--
actually, I want to do

00:05:03.120 --> 00:05:04.390
that other hyperbola.

00:05:04.390 --> 00:05:07.630
So a hyperbola, if that's
the x, that's the y-axis,

00:05:07.630 --> 00:05:09.340
it has two asymptotes.

00:05:09.340 --> 00:05:13.410
And the asymptotes, they're
these lines that the

00:05:13.410 --> 00:05:15.420
hyperbola will approach.

00:05:15.420 --> 00:05:17.520
So if those are the two
asymptotes-- and they're always

00:05:17.520 --> 00:05:21.440
the negative slope of each
other-- we know that this

00:05:21.440 --> 00:05:24.110
hyperbola's is either, and
we'll show in a second which

00:05:24.110 --> 00:05:28.090
one it is, it's either going to
look something like this, where

00:05:28.090 --> 00:05:30.650
as we approach infinity we get
closer and closer this line and

00:05:30.650 --> 00:05:32.150
closer and closer to that line.

00:05:32.150 --> 00:05:34.395
And here it's either going to
look like that-- I didn't draw

00:05:34.395 --> 00:05:36.095
it perfectly; it never
touches the asymptote.

00:05:36.095 --> 00:05:39.000
It just gets closer and closer
and closer, arbitrarily

00:05:39.000 --> 00:05:39.750
close to the asymptote.

00:05:39.750 --> 00:05:42.830
It's either going to look
like that, where it opens

00:05:42.830 --> 00:05:44.230
up to the right and left.

00:05:44.230 --> 00:05:51.610
Or our hyperbola's going
to open up and down.

00:05:51.610 --> 00:05:53.860
And once again, as you go
further and further, and

00:05:53.860 --> 00:05:56.430
asymptote means it's just going
to get closer and closer to one

00:05:56.430 --> 00:05:58.260
of these lines without
ever touching it.

00:05:58.260 --> 00:06:01.030
It will get infinitely close as
you get infinitely far away,

00:06:01.030 --> 00:06:03.010
as x gets infinitely large.

00:06:03.010 --> 00:06:04.930
So in order to figure out which
one of these this is, let's

00:06:04.930 --> 00:06:10.420
just think about what happens
as x becomes infinitely large.

00:06:10.420 --> 00:06:14.240
So as x approaches infinity.

00:06:14.240 --> 00:06:17.250
So as x approaches infinity, or
x approaches negative infinity.

00:06:17.250 --> 00:06:19.540
So I'll say plus or
minus infinity, right?

00:06:19.540 --> 00:06:21.740
It doesn't matter, because
when you take a negative,

00:06:21.740 --> 00:06:22.430
this gets squared.

00:06:22.430 --> 00:06:26.100
So this number becomes really
huge as you approach positive

00:06:26.100 --> 00:06:27.460
or negative infinity.

00:06:27.460 --> 00:06:29.160
And you'll learn more about
this when we actually do

00:06:29.160 --> 00:06:30.490
limits, but I think
that's intuitive.

00:06:30.490 --> 00:06:31.650
That this number becomes huge.

00:06:31.650 --> 00:06:33.020
This number's just a constant.

00:06:33.020 --> 00:06:34.810
It just stays the same.

00:06:34.810 --> 00:06:38.970
So as x approaches positive or
negative infinity, as it gets

00:06:38.970 --> 00:06:44.050
really, really large, y is
going to be approximately equal

00:06:44.050 --> 00:06:46.900
to-- actually, I think
that's congruent.

00:06:50.570 --> 00:06:52.250
I always forget notation.

00:06:52.250 --> 00:06:53.420
Approximately.

00:06:53.420 --> 00:06:56.820
This just means not exactly
but approximately equal to.

00:06:56.820 --> 00:06:59.450
When x approaches infinity,
it's going to be approximately

00:06:59.450 --> 00:07:03.420
equal to the plus or minus
square root of b squared

00:07:03.420 --> 00:07:07.070
over a squared x squared.

00:07:07.070 --> 00:07:10.540
And that is equal to-- now you
can take the square root.

00:07:10.540 --> 00:07:12.520
You couldn't take the square
root of this algebraically,

00:07:12.520 --> 00:07:13.680
but this you can.

00:07:13.680 --> 00:07:19.280
This is equal to plus
or minus b over a x.

00:07:19.280 --> 00:07:22.080
So that tells us, essentially,
what the two asymptotes are.

00:07:22.080 --> 00:07:26.040
Where the slope of one
asymptote will be b over a x.

00:07:26.040 --> 00:07:29.260
This could give you positive b
over a x, and the other one

00:07:29.260 --> 00:07:30.650
would be minus b over a x.

00:07:30.650 --> 00:07:32.930
And I'll do this with
some example so it makes

00:07:32.930 --> 00:07:34.670
it a little clearer.

00:07:34.670 --> 00:07:36.990
But we still know what the
asymptotes look like.

00:07:36.990 --> 00:07:38.320
It's these two lines.

00:07:38.320 --> 00:07:42.340
Because it's plus b a x is one
line, y equals plus b a x.

00:07:42.340 --> 00:07:44.860
Let's say it's this one.

00:07:44.860 --> 00:07:51.410
This asymptote right here is y
is equal to plus b over a x.

00:07:51.410 --> 00:07:52.510
I know you can't read that.

00:07:52.510 --> 00:07:56.920
And then the downward sloping
asymptote we could say is y is

00:07:56.920 --> 00:07:58.730
equal to minus b over a x.

00:07:58.730 --> 00:08:00.200
So those are two asymptotes.

00:08:00.200 --> 00:08:03.580
But we still have to figure out
whether the hyperbola opens up

00:08:03.580 --> 00:08:06.930
to the left and right, or
does it open up and down?

00:08:06.930 --> 00:08:10.350
And there, there's
two ways to do this.

00:08:10.350 --> 00:08:12.440
One, you say, well this
is an approximation.

00:08:12.440 --> 00:08:14.865
This is what you approach
as x approaches infinity.

00:08:14.865 --> 00:08:17.980
But we see here that even when
x approaches infinity, we're

00:08:17.980 --> 00:08:20.060
always going to be a little
bit smaller than that number.

00:08:20.060 --> 00:08:23.380
Because we're subtracting a
positive number from this.

00:08:23.380 --> 00:08:24.950
We're subtracting a positive
number, and then we're taking

00:08:24.950 --> 00:08:25.660
the square root of
the whole thing.

00:08:25.660 --> 00:08:28.060
So we're always going to be a
little bit lower than the

00:08:28.060 --> 00:08:32.110
asymptote, especially when
we're in the positive quadrant.

00:08:32.110 --> 00:08:32.720
Right?

00:08:32.720 --> 00:08:34.940
So to me, that's how
I like to do it.

00:08:34.940 --> 00:08:37.185
I think, we're always-- at
least in the positive quadrant;

00:08:37.185 --> 00:08:38.800
it gets a little more confusing
when you go to the other

00:08:38.800 --> 00:08:41.340
quadrants-- we're always going
to be a little bit lower

00:08:41.340 --> 00:08:42.260
than the asymptote.

00:08:42.260 --> 00:08:44.460
So we're going to approach
from the bottom there.

00:08:44.460 --> 00:08:46.050
And since you know you're
there, you know it's going

00:08:46.050 --> 00:08:47.790
to be like this and
approach this asymptote.

00:08:47.790 --> 00:08:49.960
And then since it's opening
to the right here, it's also

00:08:49.960 --> 00:08:53.330
going to open to the left.

00:08:53.330 --> 00:08:55.900
The other way to test it, and
maybe this is more intuitive

00:08:55.900 --> 00:08:58.840
for you, is to figure out,
in the original equation

00:08:58.840 --> 00:09:01.130
could x or y equal to 0?

00:09:01.130 --> 00:09:03.440
Because when you open to the
right and left, notice you

00:09:03.440 --> 00:09:05.110
never get to x equal to 0.

00:09:05.110 --> 00:09:08.680
You get to y equal 0,
right here and here.

00:09:08.680 --> 00:09:10.940
But you never get
to x equals 0.

00:09:10.940 --> 00:09:12.620
And actually your teacher
might want you to plot these

00:09:12.620 --> 00:09:14.730
points, and there you just
substitute y equals 0.

00:09:14.730 --> 00:09:18.300
And you can just look at
the original equation.

00:09:18.300 --> 00:09:22.240
Actually, you could even look
at this equation right here.

00:09:22.240 --> 00:09:24.600
Can x ever equal 0?

00:09:24.600 --> 00:09:27.890
If you look at this equation,
if x is equal to 0, this whole

00:09:27.890 --> 00:09:30.370
term right here would cancel
out, and you'd just be left

00:09:30.370 --> 00:09:31.730
with a minus b squared.

00:09:31.730 --> 00:09:34.330
Which is, you're taking b
squared, and you put a

00:09:34.330 --> 00:09:35.270
negative sign in front of it.

00:09:35.270 --> 00:09:36.560
So that's a negative number.

00:09:36.560 --> 00:09:38.780
And then you're taking a square
root of a negative number.

00:09:38.780 --> 00:09:41.420
So we're not dealing with
imaginaries right now.

00:09:41.420 --> 00:09:45.440
So you can never
have x equal to 0.

00:09:45.440 --> 00:09:46.930
But y could be
equal to 0, right?

00:09:46.930 --> 00:09:49.360
You can set y equal to 0 and
then you could solve for it.

00:09:49.360 --> 00:09:51.680
So in this case,
actually let's do that.

00:09:51.680 --> 00:09:55.050
If y is equal to 0, you get 0
is equal to the square root of

00:09:55.050 --> 00:09:59.300
b squared over a squared x
squared minus b squared.

00:09:59.300 --> 00:10:03.140
If you square both sides,
you get b squared over a

00:10:03.140 --> 00:10:08.280
squared x squared minus
b squared is equal to 0.

00:10:08.280 --> 00:10:09.740
I know this is messy.

00:10:09.740 --> 00:10:13.960
So then you get b squared
over a squared x squared

00:10:13.960 --> 00:10:17.010
is equal to b squared.

00:10:17.010 --> 00:10:19.535
You could divide both sides
by b squared, I guess.

00:10:19.535 --> 00:10:21.600
You get a 1 and a 1.

00:10:21.600 --> 00:10:24.250
And then you could multiply
both sides by a squared.

00:10:24.250 --> 00:10:29.875
You get x squared is equal to
a squared, and then you get

00:10:29.875 --> 00:10:33.270
x is equal to the plus or
minus square root of a.

00:10:33.270 --> 00:10:37.310
So this point right here is the
point a comma 0, and this point

00:10:37.310 --> 00:10:39.860
right here is the point
minus a comma 0.

00:10:39.860 --> 00:10:42.936
Now let's go back to
the other problem.

00:10:42.936 --> 00:10:45.600
I have a feeling I might
be running out of time.

00:10:45.600 --> 00:10:52.920
So notice that when the x term
was positive, our hyperbola

00:10:52.920 --> 00:10:55.250
opened to the right
and the left.

00:10:55.250 --> 00:10:57.660
And you could probably get from
detective reasoning that when

00:10:57.660 --> 00:11:00.850
the y term is positive, which
is the case in this one, we're

00:11:00.850 --> 00:11:02.810
probably going to
open up and down.

00:11:02.810 --> 00:11:04.780
And let's just prove
that to ourselves.

00:11:04.780 --> 00:11:06.240
So let's solve for y.

00:11:06.240 --> 00:11:08.730
You get y squared
over b squared.

00:11:08.730 --> 00:11:10.750
We're going to add x squared
over a squared to both sides.

00:11:10.750 --> 00:11:16.910
So you get equals x squared
over a squared plus 1.

00:11:16.910 --> 00:11:18.960
Multiply both sides
by b squared.

00:11:18.960 --> 00:11:23.310
y squared is equal to b
squared over a squared x

00:11:23.310 --> 00:11:25.815
squared plus b squared.

00:11:25.815 --> 00:11:27.970
You have to distribute
the b squared.

00:11:27.970 --> 00:11:29.500
Now take the square root.

00:11:29.500 --> 00:11:31.020
I'll switch colors for that.

00:11:31.020 --> 00:11:36.260
So y is equal to the plus
or minus square root of b

00:11:36.260 --> 00:11:42.150
squared over a squared x
squared plus b squared.

00:11:42.150 --> 00:11:45.170
And once again-- I've run out
of space-- we can make that

00:11:45.170 --> 00:11:50.640
same argument that as x
approaches positive or negative

00:11:50.640 --> 00:11:58.140
infinity, this equation, this
b, this little constant term

00:11:58.140 --> 00:11:59.630
right here isn't going
to matter as much.

00:11:59.630 --> 00:12:00.800
You're just going to
take the square root of

00:12:00.800 --> 00:12:02.300
this term right here.

00:12:02.300 --> 00:12:08.300
Which essentially b over a x,
plus or minus b over a x.

00:12:08.300 --> 00:12:11.790
And once again, those are the
same two asymptotes, which

00:12:11.790 --> 00:12:17.170
I'll redraw here, that
line and that line.

00:12:17.170 --> 00:12:20.610
But in this case, we're
always a little bit larger

00:12:20.610 --> 00:12:21.390
than the asymptotes.

00:12:24.640 --> 00:12:26.810
So in the positive quadrant,
that tells us we're going to

00:12:26.810 --> 00:12:31.330
be up here and down there.

00:12:31.330 --> 00:12:33.430
Another way to think about it,
in this case, when the

00:12:33.430 --> 00:12:38.510
hyperbola is a vertical
hyperbola, where it opens up

00:12:38.510 --> 00:12:42.730
and down, you notice x could be
equal to 0, but y could

00:12:42.730 --> 00:12:44.280
never be equal to 0.

00:12:44.280 --> 00:12:47.250
And that makes sense, too.

00:12:47.250 --> 00:12:50.735
Because if you look at our
original formula right here,

00:12:50.735 --> 00:12:52.060
x could be equal to 0.

00:12:52.060 --> 00:12:53.940
If x was 0, this would
cancel out and you

00:12:53.940 --> 00:12:55.250
could just solve for y.

00:12:55.250 --> 00:13:00.670
But if y were equal to 0, you'd
have minus x squared over a

00:13:00.670 --> 00:13:03.700
squared is equal to 1, and then
you would have, if you solved

00:13:03.700 --> 00:13:06.950
this, you'd get x squared is
equal to minus a squared.

00:13:06.950 --> 00:13:09.340
And we're not dealing with
imaginary numbers, so you can't

00:13:09.340 --> 00:13:11.740
square something, you can't
get a negative number.

00:13:11.740 --> 00:13:14.110
So once again, this
would be impossible.

00:13:14.110 --> 00:13:17.490
So that's this other clue that
tells you it opens up and down.

00:13:17.490 --> 00:13:20.600
Because in this case y
could never equal 0.

00:13:20.600 --> 00:13:22.690
Anyway, you might be a little
confused because I stayed

00:13:22.690 --> 00:13:24.560
abstract with the
b's and the a's.

00:13:24.560 --> 00:13:26.770
In the next couple of videos
I'll do a bunch of problems

00:13:26.770 --> 00:13:29.490
where we draw a bunch of
hyperbolas, ellipses, and

00:13:29.490 --> 00:13:31.840
circles with actual numbers.

00:13:31.840 --> 00:13:33.342
See you soon.


Conic sections: Intro to ellipse

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.330
In the last video, we learned a
little bit about the circle.

00:00:03.330 --> 00:00:07.350
And the circle is really just
a special case of an ellipse.

00:00:07.350 --> 00:00:13.220
It's a special case because in
a circle you're always an equal

00:00:13.220 --> 00:00:15.420
distance away from the center
of the circle, while in an

00:00:15.420 --> 00:00:18.730
ellipse, the distance from the
center of the circle

00:00:18.730 --> 00:00:20.110
is always changing.

00:00:20.110 --> 00:00:21.820
You know what an
ellipse looks like.

00:00:21.820 --> 00:00:23.300
Well, I showed you that
in the first video.

00:00:23.300 --> 00:00:26.020
It looks something like that.

00:00:26.020 --> 00:00:28.140
What I mean is that the radius
or the distance from the

00:00:28.140 --> 00:00:29.360
center is always changing.

00:00:29.360 --> 00:00:33.630
Let me say this is
centered at the origin.

00:00:33.630 --> 00:00:36.190
So that's the origin
right there.

00:00:36.190 --> 00:00:40.440
You see here, we're really, if
we're on this point on the

00:00:40.440 --> 00:00:43.010
ellipse, we're really
close to the origin.

00:00:43.010 --> 00:00:45.110
This is actually the closest
we'll ever get, just as close

00:00:45.110 --> 00:00:46.520
as well get down here.

00:00:46.520 --> 00:00:49.540
And when we're out here we're
really far away from the origin

00:00:49.540 --> 00:00:54.650
and that's about as far as
we're going to get right there.

00:00:54.650 --> 00:00:58.400
So a circle is a special case
of this, because in a circle's

00:00:58.400 --> 00:01:01.770
case, the furthest we get from
the origin is the same distance

00:01:01.770 --> 00:01:04.670
as the closest we get, or, in
other words, we are always

00:01:04.670 --> 00:01:07.190
the exact same distance
away from the origin.

00:01:07.190 --> 00:01:09.240
Well, with that said, let's
actually go a little

00:01:09.240 --> 00:01:10.070
bit into the math.

00:01:10.070 --> 00:01:14.640
So the general or the standard
form for an ellipse centered at

00:01:14.640 --> 00:01:23.120
the origin is x squared over a
squared plus y squared over

00:01:23.120 --> 00:01:25.170
b squared is equal to 1.

00:01:25.170 --> 00:01:29.040
Where a and b are just
any two numbers.

00:01:29.040 --> 00:01:31.170
I could have written this as
c squared and d squared.

00:01:31.170 --> 00:01:32.920
I mean, they're just
place holders.

00:01:32.920 --> 00:01:35.430
Just to give you an idea of
what this means, if this was

00:01:35.430 --> 00:01:41.180
our ellipse in question right
now, a is the length of the

00:01:41.180 --> 00:01:43.330
radius in the x-direction.

00:01:43.330 --> 00:01:45.030
Remember, we're going to
have a squared down here.

00:01:45.030 --> 00:01:46.760
So if you took the square
root of whatever is in the

00:01:46.760 --> 00:01:49.130
denominator, a is the x-radius.

00:01:49.130 --> 00:01:53.260
So this distance in our little
chart right here, in our little

00:01:53.260 --> 00:01:56.700
graph here, that distance is a,
or that this point right here,

00:01:56.700 --> 00:01:59.200
since we're centered at the
origin, will be the point x is

00:01:59.200 --> 00:02:02.160
equal to a y is equal to 0.

00:02:02.160 --> 00:02:06.050
And of course this point right
here this will be a, so this

00:02:06.050 --> 00:02:09.620
would be the point
minus a comma 0.

00:02:09.620 --> 00:02:14.320
And then the radius in the
y-direction would be this

00:02:14.320 --> 00:02:16.860
radius right here and is b.

00:02:16.860 --> 00:02:21.430
So this point would be x is
equal to 0, y is equal to b.

00:02:21.430 --> 00:02:24.720
Likewise this point right here
would be x is equal to 0,

00:02:24.720 --> 00:02:27.010
y is equal to minus b.

00:02:27.010 --> 00:02:29.730
And the way I drew this, we
have kind of a short and fat

00:02:29.730 --> 00:02:33.090
ellipse you can also have kind
of a tall and skinny ellipse.

00:02:33.090 --> 00:02:35.600
But in the short and fat
ellipse, the direction that

00:02:35.600 --> 00:02:39.710
you're short in that's
called your minor axis.

00:02:39.710 --> 00:02:44.010
And so b, I always forget the
exact terminology, but b you

00:02:44.010 --> 00:02:48.125
can call it your semi or the
length of your semi-minor axis.

00:02:56.520 --> 00:02:58.120
And where did that
word come from?

00:02:58.120 --> 00:03:00.660
Well if this whole thing is
your minor axis or maybe you

00:03:00.660 --> 00:03:03.820
could call your minor diameter
if this whole thing is your

00:03:03.820 --> 00:03:05.850
minor diameter it's called
minor, because it's the

00:03:05.850 --> 00:03:09.470
shortest of all of the
diameters of this ellipse.

00:03:09.470 --> 00:03:12.210
And then the semi
is half of that.

00:03:15.720 --> 00:03:17.840
b is the length of
the semi-minor axis.

00:03:21.180 --> 00:03:24.650
That's b in this example, just
because as I drew this ellipse

00:03:24.650 --> 00:03:27.450
it just happens to be that
b is smaller then a.

00:03:27.450 --> 00:03:31.860
If b was larger than a, I would
have a tall and skinny ellipse.

00:03:31.860 --> 00:03:32.830
Let me actually draw one.

00:03:32.830 --> 00:03:34.100
It could have been like this.

00:03:34.100 --> 00:03:36.550
I could have an ellipse that
looks something like that.

00:03:36.550 --> 00:03:41.190
In which case, all of a sudden
b would be the semi-major axis,

00:03:41.190 --> 00:03:43.710
because b would be
greater than a.

00:03:43.710 --> 00:03:46.440
That this would be
taller than it is wide.

00:03:46.440 --> 00:03:49.640
But let me not confuse
the graph too much.

00:03:49.640 --> 00:03:54.860
And in this case, a is the
length of-- I think you've

00:03:54.860 --> 00:04:03.520
guessed it-- a is the length of
the semi-major axis or you can

00:04:03.520 --> 00:04:06.210
even call it the length
of the major radius.

00:04:06.210 --> 00:04:07.460
I think that makes more sense.

00:04:12.330 --> 00:04:14.220
And you can call this
the minor radius.

00:04:22.390 --> 00:04:23.360
So let's just do an example.

00:04:23.360 --> 00:04:26.120
And I think when I've done an
example with actual numbers,

00:04:26.120 --> 00:04:28.250
it'll make it all a
little bit clearer.

00:04:28.250 --> 00:04:31.510
So let's say I were to show up
at your door with the

00:04:31.510 --> 00:04:39.580
following: If I were to say x
squared over 9 plus y squared

00:04:39.580 --> 00:04:44.800
over 25 is equal to 1.

00:04:44.800 --> 00:04:48.350
So what is your radius
in the x-direction?

00:04:48.350 --> 00:04:51.050
This is your radius in
the x-direction squared.

00:04:51.050 --> 00:04:53.002
So your radius in the
x-direction if we just map

00:04:53.002 --> 00:04:56.030
it, we would say that
a is equal to 3.

00:04:56.030 --> 00:04:57.690
Because this is a squared.

00:04:57.690 --> 00:05:00.260
And if we were just map it we'd
say this is b squared than this

00:05:00.260 --> 00:05:04.930
tells us that b is equal to 5.

00:05:04.930 --> 00:05:06.830
So if we wanted to graph
this, and once again this

00:05:06.830 --> 00:05:08.500
is centered at the origin.

00:05:08.500 --> 00:05:09.840
Let me draw the ellipse first.

00:05:09.840 --> 00:05:13.810
So, first of all, we have our
radius in the y-direction is

00:05:13.810 --> 00:05:17.140
larger than our radius
in the x-direction.

00:05:17.140 --> 00:05:20.320
The ellipse is going to
be taller and skinnier.

00:05:20.320 --> 00:05:23.170
It's going to look
something like that.

00:05:23.170 --> 00:05:28.060
Draw some axes, so that could
be your x-axis, your y-axis.

00:05:32.140 --> 00:05:35.260
This is your radius
in the y-direction.

00:05:35.260 --> 00:05:39.060
So this distance right here
is going to be 5, and

00:05:39.060 --> 00:05:42.030
so will this distance.

00:05:42.030 --> 00:05:43.960
And this is your radius
in the x-direction.

00:05:43.960 --> 00:05:48.310
So this will be 3,
and this will be 3.

00:05:48.310 --> 00:05:48.990
That's it.

00:05:48.990 --> 00:05:52.170
You have now plotted
this ellipse.

00:05:52.170 --> 00:05:53.530
Nothing too fancy about it.

00:05:53.530 --> 00:05:55.720
And actually just to kind of
hit the point home that the

00:05:55.720 --> 00:05:59.720
circle is a special
case of an ellipse.

00:05:59.720 --> 00:06:03.720
We learned in the last video
that the equation of a circle

00:06:03.720 --> 00:06:08.900
is x squared and a circle
centered at the origin.

00:06:08.900 --> 00:06:12.320
x squared plus y squared
is equal to r squared.

00:06:12.320 --> 00:06:16.350
So if we were to divide both
sides of this by r squared, we

00:06:16.350 --> 00:06:19.290
would get-- and this is just
little algebraic manipulation--

00:06:19.290 --> 00:06:26.510
x squared over r squared plus y
squared over r squared

00:06:26.510 --> 00:06:28.430
is equal to 1.

00:06:28.430 --> 00:06:33.940
Now in this case, your a
is r and so is your b.

00:06:33.940 --> 00:06:38.860
So your semi-minor axis
is r and so is your

00:06:38.860 --> 00:06:40.600
semi-major axis of r.

00:06:40.600 --> 00:06:42.500
Or, in other words, this
distance is the same as that

00:06:42.500 --> 00:06:45.200
distance, and so it will
neither be short and fat

00:06:45.200 --> 00:06:46.350
nor tall and skinny.

00:06:46.350 --> 00:06:47.710
It'll be perfectly round.

00:06:47.710 --> 00:06:51.750
And so that's why the circle is
a special case of an ellipse.

00:06:51.750 --> 00:06:54.740
So let me give you a slightly--
It'll look a lot more

00:06:54.740 --> 00:06:56.770
complicated, and this is
something you might

00:06:56.770 --> 00:06:57.600
see on exam.

00:06:57.600 --> 00:06:59.980
But I just want to show you
that this is just a shifting.

00:06:59.980 --> 00:07:06.960
Let's say we wanted to
shift this ellipse.

00:07:06.960 --> 00:07:11.990
Let's say we wanted to shift
it to the right by 5.

00:07:15.810 --> 00:07:19.040
So instead of the origin being
at x is equal to 0, the origin

00:07:19.040 --> 00:07:22.550
will now be at x is equal 5.

00:07:22.550 --> 00:07:27.620
So a way to think about that is
what does this term have to be

00:07:27.620 --> 00:07:32.340
so that at 5 this term
ends up being 0.

00:07:32.340 --> 00:07:33.780
Well I'll actually draw it
for you, because I think

00:07:33.780 --> 00:07:34.920
that might be confusing.

00:07:34.920 --> 00:07:39.170
So if we shift that over the
right by 5, the new equation of

00:07:39.170 --> 00:07:49.950
this ellipse will be x minus 5
squared over 9 plus y squared

00:07:49.950 --> 00:07:52.600
over 25 is equal to 1.

00:07:52.600 --> 00:07:55.460
So if I were to just draw
this ellipse right now,

00:07:55.460 --> 00:07:57.850
it would look like this.

00:07:57.850 --> 00:07:59.290
I want to make it look
fairly similar to the

00:07:59.290 --> 00:08:01.500
ellipse I had before.

00:08:01.500 --> 00:08:03.530
It would look just like that.

00:08:03.530 --> 00:08:07.180
Just shifted it over by five.

00:08:07.180 --> 00:08:11.170
And the intuition we learned a
little bit in the circle video

00:08:11.170 --> 00:08:13.240
where I said, oh well, you
know, if you have x minus

00:08:13.240 --> 00:08:15.120
something that means that the
new origin is now

00:08:15.120 --> 00:08:16.680
at positive 5.

00:08:16.680 --> 00:08:18.010
And you could memorize that.

00:08:18.010 --> 00:08:19.580
You could always say, oh, if I
have a minus here, that the

00:08:19.580 --> 00:08:23.880
origin is at the negative of
whatever this number is, so

00:08:23.880 --> 00:08:25.570
it would be a positive five.

00:08:25.570 --> 00:08:27.420
You know, if you had a positive
it would be the opposite that.

00:08:27.420 --> 00:08:30.950
But the way to really think
about it is now if you go to x

00:08:30.950 --> 00:08:37.490
is equal to 5, when x is equal
to 5, this whole term, x minus

00:08:37.490 --> 00:08:41.550
5, will behave just like
this x term will here.

00:08:41.550 --> 00:08:47.100
When x is equal to 5 this
term is 0, just like

00:08:47.100 --> 00:08:48.910
when x was 0 here.

00:08:48.910 --> 00:08:51.750
So when x is equal to 5, this
term is 0, and then y squared

00:08:51.750 --> 00:08:56.290
over 25 is equal 1, so y
has to be equal five.

00:08:56.290 --> 00:08:59.470
Just like over here when x is
equal is 0, y squared over 25

00:08:59.470 --> 00:09:01.930
had to be equal to 1,
y is equal to either

00:09:01.930 --> 00:09:04.690
positive or minus 5.

00:09:04.690 --> 00:09:06.550
And I really want to give
you that intuition.

00:09:06.550 --> 00:09:08.970
And then, let's say we
wanted to shift this

00:09:08.970 --> 00:09:10.780
equation down by two.

00:09:14.250 --> 00:09:17.250
So our new ellipse looks
something like this.

00:09:20.530 --> 00:09:22.310
A lot of times you learned
this in conic sections.

00:09:22.310 --> 00:09:23.880
But this is true any function.

00:09:23.880 --> 00:09:26.960
When you shift things,
you shift it this way.

00:09:26.960 --> 00:09:31.340
If you shift this graph to the
right by five, you replace all

00:09:31.340 --> 00:09:34.010
of the x's with x minus 5.

00:09:34.010 --> 00:09:38.860
And if you were to shift it
down by two, you would replace

00:09:38.860 --> 00:09:44.780
all the y's with y plus 2.

00:09:44.780 --> 00:09:46.610
So let me draw our new
ellipse first, just to

00:09:46.610 --> 00:09:47.580
show you what I'm doing.

00:09:52.770 --> 00:09:59.170
So our new ellipse is going
to look something like that.

00:09:59.170 --> 00:10:03.050
I'm shifting the yellow
ellipse down by two.

00:10:03.050 --> 00:10:08.310
So this equation, if I shift it
down, well, the x is still

00:10:08.310 --> 00:10:16.160
where it was before. x minus 5
squared over 9 plus y plus 2

00:10:16.160 --> 00:10:20.050
squared over 25 is equal to 1.

00:10:20.050 --> 00:10:23.840
And once again, the reason I
know this is because now when

00:10:23.840 --> 00:10:29.180
y is minus 2, this
whole term is 0.

00:10:29.180 --> 00:10:31.040
0 when y equals minus 2.

00:10:31.040 --> 00:10:34.370
And when this term is 0, it
behaves the same way as

00:10:34.370 --> 00:10:36.200
when this term was 0.

00:10:36.200 --> 00:10:39.400
So when y is equal to minus 2,
you get the same behavior,

00:10:39.400 --> 00:10:45.560
you're at the same point in the
curve, right here actually, as

00:10:45.560 --> 00:10:48.770
you are when y equaled 0
in this one, so here.

00:10:48.770 --> 00:10:50.050
So it's not the same point.

00:10:50.050 --> 00:10:52.880
You can kind of view it as the
same part of the ellipse.

00:10:52.880 --> 00:10:57.960
You're at kind of the maximum
width point on the ellipse here

00:10:57.960 --> 00:11:01.650
and here when y is equal to 2,
and you were here at y equal to

00:11:01.650 --> 00:11:04.530
0-- sorry, when y
equals minus 2.

00:11:04.530 --> 00:11:05.540
This is minus 2.

00:11:05.540 --> 00:11:07.390
And that's because when you
put y equals minus 2 here

00:11:07.390 --> 00:11:08.980
this whole term is 0.

00:11:08.980 --> 00:11:10.490
Just like when y was 0 here.

00:11:10.490 --> 00:11:13.130
I don't want to make
it too confusing.

00:11:13.130 --> 00:11:16.350
But just to kind of wrap it all
up, sometimes you might see

00:11:16.350 --> 00:11:24.490
something like graph the
following: y minus 1 squared

00:11:24.490 --> 00:11:33.430
over 4 plus x plus 2 squared
over 9 is equal to 1.

00:11:33.430 --> 00:11:35.760
And so the first thing you
could say is OK this is just

00:11:35.760 --> 00:11:42.410
like the standard ellipse y
squared over 4 plus x squared

00:11:42.410 --> 00:11:44.040
over 9 is equal to one.

00:11:44.040 --> 00:11:47.040
It's just like this,
but it's shifted over.

00:11:47.040 --> 00:11:55.170
This ones origin is 0,0, while
this ones origin would be what?

00:11:55.170 --> 00:12:01.170
It would be the point x
is minus 2 and y is 1.

00:12:01.170 --> 00:12:06.110
So if you were to graph
this, your radius in

00:12:06.110 --> 00:12:08.680
your y-direction is 2.

00:12:08.680 --> 00:12:10.000
2 squared is equal to 4.

00:12:10.000 --> 00:12:12.590
Your radius in your
x-direction is 3.

00:12:12.590 --> 00:12:15.900
3 squared is equal to 9.

00:12:15.900 --> 00:12:18.870
So your x-radius is actually
larger than your y-radius.

00:12:18.870 --> 00:12:22.860
So, it's going to be a little
bit of a fat ellipse.

00:12:22.860 --> 00:12:24.265
Actually, let me draw
the axes first.

00:12:28.360 --> 00:12:30.290
Let me draw it like this.

00:12:30.290 --> 00:12:38.120
That's my vertical axis,
this is my x-axis.

00:12:38.120 --> 00:12:41.450
And so my center is
now at minus 2, 1.

00:12:44.920 --> 00:12:49.040
That's minus 2,
and I go up one.

00:12:49.040 --> 00:12:50.830
That's the center
of my ellipse.

00:12:50.830 --> 00:12:54.560
And now in the x-direction,
this is the x term,

00:12:54.560 --> 00:12:57.850
my x-radius is 3.

00:12:57.850 --> 00:13:03.780
So the ellipse will go
three-- in that direction.

00:13:03.780 --> 00:13:07.210
This is it's widest point
will be 3 in that direction.

00:13:07.210 --> 00:13:10.700
And then in the y-direction,
it'll go 2, so it'll go up

00:13:10.700 --> 00:13:15.030
1, 2 so that's there and
then 1, 2 and it's there.

00:13:15.030 --> 00:13:17.140
So if I were to draw that
ellipse it would look something

00:13:17.140 --> 00:13:18.770
like this through my best shot.

00:13:23.930 --> 00:13:27.170
A little bit fatter than it
is tall, and that's because

00:13:27.170 --> 00:13:30.590
your x-radius is larger
than your y-radius.

00:13:30.590 --> 00:13:35.440
This distance right here is 3,
this distance right here is 3,

00:13:35.440 --> 00:13:41.530
this distance right here is 2,
this distance right here is 2.

00:13:41.530 --> 00:13:42.950
You could figure out
what these points are.

00:13:42.950 --> 00:13:45.590
I won't do all of them right
now just for the sake of time.

00:13:45.590 --> 00:13:49.410
But this right here is
the point minus 2, 1.

00:13:49.410 --> 00:13:52.810
So if you go three more than
that-- so if you add 3 to

00:13:52.810 --> 00:13:57.730
the x-direction this is
the point 1 comma 1.

00:13:57.730 --> 00:13:59.445
If you would take three away
from the x-direction, this

00:13:59.445 --> 00:14:00.740
would be minus 5 comma 1.

00:14:00.740 --> 00:14:02.260
And you could figure
out the other points.

00:14:02.260 --> 00:14:03.900
That might be good
exercise for you.

00:14:03.900 --> 00:14:05.910
Anyway that's a little
bit on ellipses.

00:14:05.910 --> 00:14:07.970
In future videos we'll do
really hairy problems where you

00:14:07.970 --> 00:14:11.480
have to simplify it into this
form so that we know that it

00:14:11.480 --> 00:14:12.550
definitely is an ellipse.