06 - Equations & Definition of Conic Sections - Circle, Ellipse, Parabola & Hyperbola - Free Educational videos for Students in K-12

06 - Equations & Definition of Conic Sections - Circle, Ellipse, Parabola & Hyperbola - By Math and Science

Transcript


00:00	Hello , Welcome back to algebra . The title of
00:02	this lesson is well I could title in various ways
00:05	, but the one the one that I wrote down
00:06	is what is the difference between the four comic sections
00:09	? Probably a better title would be definitions of the
00:12	comic sections . It's a kind of a weird title
00:14	but what I want to do in this lesson is
00:16	we want to go again through each of the four
00:18	comic section circle , ellipse , parabola and hyperbole but
00:22	want to go a level deeper than we did in
00:24	the last lesson in the last lesson . Just to
00:26	recap , we talked about the shape of these comic
00:28	sections but we mostly focused on what they're used for
00:31	. We talked about orbits every orbit in space ,
00:34	whether it's a circular orbit , an elliptical orbit ,
00:37	a space probe on a escape trajectory . They all
00:41	represent each of the four comic sections depending on how
00:44	much speed you have in your orbit . We also
00:47	talked about the idea that a parabola has a special
00:49	shape where it can take , take incoming radio waves
00:52	or light rays and focus them onto a focus .
00:55	So we kind of talked about the idea that a
00:57	parabola has a focus which is kind of near the
01:00	bottom of the Parabola that collects all of the light
01:03	rays , right ? So if you haven't done watch
01:05	that , I would like you to go ahead and
01:06	look at it again , make sure that you've you've
01:09	watched it . So here we're gonna go one level
01:10	deeper . We're gonna talk about the comic sections ,
01:12	we're gonna write down the shapes , but we're going
01:15	to go more deeply into why they have those shapes
01:18	. And we're gonna generally look at the equations of
01:20	these different comic sections and we're doing them all in
01:23	this one lesson , but in the next lesson and
01:25	the subsequent lessons were taking each of the comic sections
01:27	apart in more detail and solving tons of problems .
01:30	So you can still consider this to be kind of
01:32	an overview lesson of everything . So I want to
01:34	recap something very important . I'm gonna put my paper
01:37	down here that we did talk about in the last
01:39	section each of these comic sections can come from slicing
01:42	a cone . So the first comic section is the
01:45	one you already know about . A circle . A
01:47	circle comes from taking a cone and slicing it parallel
01:50	to the base like this , right across . When
01:52	you slice it and look at the cross section ,
01:54	you have that beautiful circular shape equal distant from the
01:58	the central point of the circle . So we all
02:00	know about circles already when I put that aside .
02:03	Um But what fewer people know is that you can
02:06	get the other comic sections from slicing a comb if
02:09	you take a comb and you slice it at some
02:11	random angle . This angle is not special at some
02:14	angle , I just happened to use my knife to
02:16	slice it at and you cut it apart and look
02:18	at it . The shape you get as the cross
02:20	section is an ellipse . So the help if I
02:22	turn it on the side , it's an elliptical shape
02:24	right now , of course the exact shape and size
02:27	of the ellipse will depend on how you cut it
02:29	. But you get a circle by cutting a cone
02:32	directly across . You get into lips by slicing it
02:34	at some random angle , right ? And of course
02:37	whatever angle that you have there is going to determine
02:39	the actual shape of the ellipse . And then we
02:41	take the same cone and we decide to slice it
02:45	slightly differently . In this case , you notice how
02:47	this side of the cone has an angle to it
02:49	, right ? So what we do is we slice
02:51	through parallel to this side . This line here is
02:55	as good as I can get parallel to this side
02:57	. So when you slice through parallel to the side
03:00	of the cone there than what you get , might
03:02	help if I turn it upside down is a parable
03:04	A . This is a parabolic shape , it comes
03:06	down , has a nice rounded bottom , but kind
03:09	of gets to a point and then it comes back
03:10	up . This is the shape of a standard problem
03:12	, of course , exactly where you cut it and
03:14	how you cut it , the shape of your cone
03:16	and all that is going to dictate exactly what it
03:19	looks like . But that's how you get the comic
03:20	section , known as a problem . The final comic
03:23	section we talked about is called a hyperbole . This
03:27	one is if you take uh your cone and instead
03:31	of slicing at an angle or parallel to the other
03:33	side , you look at the cone like this and
03:35	you slice it straight down so you slice straight down
03:38	. And when you do that you actually get something
03:41	that looks very similar to a parabola . In fact
03:43	, I'll go ahead and show you uh that they
03:45	are different . So so this one here is a
03:47	parabola . This one's a hyperbole to notice this one
03:50	has a much sharper point to it at the bottom
03:52	and it also has much more steep sides to it
03:54	. That's a hyperba . And then finally , whenever
03:57	you cut the hyperbole to , you need to really
03:59	envision it being really two cones . You have one
04:02	cone on the bottom and then one cone kind of
04:04	flipped upside down . When you slice through both of
04:06	them , you slice through the top cone and go
04:09	straight down and slice through the bottom cone . You're
04:11	actually gonna get two halves of the hyperbole . So
04:14	for a circle you make one slice you get one
04:16	shape , we call the circle for the ellipse .
04:18	You make one slice , you get one shape ,
04:20	we called in the lips for a parabola . You
04:22	make one slice , you get one shape , we
04:24	call a parabola for hyperbole . When you think of
04:27	the two cones , you really make one slice .
04:29	But you're slicing through two different cones . You get
04:31	two shapes which they kind of come in pairs called
04:33	a hyperbole or a hyperbolic shape . So what I
04:36	want to do now is I want to pick up
04:38	the toys here and we're gonna go through each of
04:40	these comic sections on the board drawing their shape ,
04:43	showing you exactly how the shape comes about writing down
04:46	some of the basic equations of the different shapes .
04:48	So you'll understand how the different comic sections relate to
04:50	each other in algebra , and of course by extension
04:53	into geometry . All right now we're ready to roll
04:57	up our sleeves and and get uh get serious about
04:59	this . So on the first board we're going to
05:01	talk about the comic section that everyone here already understands
05:05	. At least something about , it's called a circle
05:06	. We haven't talked about circles much in algebra yet
05:08	, we're going to talk about them at great length
05:10	here , you all have some idea what a circle
05:13	is , right ? So let me go ahead and
05:15	try to do that here to draw that for you
05:18	here . So here's my best freehand at a circle
05:21	. Is it a perfect circle ? No , it's
05:22	definitely a lumpy circle , so sorry about that ,
05:25	but this is the best circle I can draw freehand
05:27	. Now , the thing about a circle is what
05:29	defines a circle . Well , the center is the
05:32	thing that defines a circle . So I'm gonna put
05:33	this is the center of the circle and you all
05:37	know that every circle has what we call a radius
05:40	. So to draw that radius , I'm going to
05:42	kind of draw a little arrow here and I'm gonna
05:45	put a point P on the edge of the circle
05:48	here , and I'm gonna call this point P .
05:50	X comma , y why am I doing that ?
05:52	Because if you can I'm not going to draw a
05:54	grid here , but if you can imagine an xy
05:56	plane kinda where the circle is sitting on top of
05:59	the xy plane , then what we want to do
06:01	is we want to write an equation down that talks
06:04	about the shape of the circle . We want to
06:07	we all know circle has a center and radius .
06:09	Of course we know that , but I want to
06:10	be able to put it in a coordinate plane and
06:12	say , what is the equation of that circle ?
06:14	How can I plot that circle ? Is a circle
06:16	of function is a circle not a function . We
06:18	need to talk about those things . So we need
06:19	to write an equation down that describes every one of
06:22	these purple points here . So it shouldn't describe points
06:26	here and here and here and here . Because those
06:28	aren't on the circle . The only points on the
06:30	circle are on the purple line , the circle ,
06:33	the points related to the circle are not outside the
06:37	circle and they're not inside the circle . The only
06:39	points that describe the circle lie on this purple line
06:43	exactly . So these points we call P and we
06:46	say these points P have xy values . That's what
06:48	the X . Y . Means . So here's a
06:49	different xy value . Here's a different xy value on
06:52	this part of the circle and so on . As
06:54	you go around , you'll have a different xy value
06:56	for the points on on that curve . Now we're
06:59	gonna do a similar thing when we talk about a
07:01	parabola and in the lips and high purple . So
07:04	it's important for you to understand P X . Y
07:06	. Just means the set of points that describe the
07:08	shape that we call the circle . And they all
07:10	have X . Y coordinates . Right ? So the
07:13	way that we write down the definition of a circle
07:17	is in words , if you were to see it
07:19	in the text book , you would see something like
07:20	this . A circle is the set of all points
07:28	. I'm gonna put P . Meaning all of these
07:30	points . P there's an infinite number of points on
07:32	this purple line . So the points P the same
07:38	distance uh from the center . Yeah . Okay .
07:51	And so a way that you can write this in
07:53	terms of the definition , I mean this is inwards
07:56	a definition , right ? I mean this makes sense
07:58	. I mean this is what you all know ,
07:58	A circle is . It's a set of all the
08:00	points . The same distance . That means the same
08:02	radius from the center of the circle . You want
08:04	you all know that the points P are all the
08:06	same distance away from the center . That makes sense
08:08	, right ? But in mathematical talk or mathematical uh
08:13	you know parlance , you would say if this is
08:15	the center C and this is the point P ,
08:18	you would say that cp the distant cp must be
08:22	equal to a constant . Now this does not mean
08:26	C times P this does not mean the center times
08:29	P . This means the distance . Remember in geometry
08:32	, when we talk about distances , we have the
08:34	endpoints and we we write them as , you know
08:37	, you can think of it as a line segment
08:38	with a bar on top . The distance between A
08:40	and B is a B . The distance between X
08:43	, Y . Is x . Y . The distance
08:45	between the center and the point P is called C
08:47	. P . And this distance must be a constant
08:50	no matter what angle I'm looking at . If I'm
08:53	looking over here , the distance between the center and
08:55	this point must be the same as the distance between
08:58	here and here , which must be the same as
09:00	the distance between here and here . So we're saying
09:02	that the distance between C . And all of the
09:04	points P must be a constant number . What do
09:06	you think this constant is ? Well , that's the
09:08	radius of the circle , that's what describes the size
09:11	of the circle . So that's what a definition of
09:14	uh of a circle would be . Now . In
09:17	terms of um in terms of an equation right ,
09:22	we're not gonna talk too much about the equation of
09:24	a circle . Now we're gonna have an entire set
09:26	of lessons on it . But just to give you
09:28	a flavor , the equation of a circle would look
09:31	something like this . A general equation of a circle
09:34	would look something like this , X squared plus y
09:39	squared is equal to r squared . And what does
09:43	this mean ? Now this is a different kind of
09:46	an equation than we've seen before . I don't expect
09:48	you to understand this yet , I don't expect you
09:50	to say , oh yeah , that makes sense .
09:51	I expect you to believe me this is the equation
09:54	of a circle , but once we get through the
09:55	overview of comic sections , we're going to go back
09:58	to circles first and we're going to really dissect why
10:01	this is the actual equation of a circle for now
10:03	, I just want to get all the equations on
10:05	the board so that you can see how they kind
10:06	of are similar to one another . All of these
10:08	comic sections come from cones from slicing cones , so
10:11	all of the equations have a similar flavor to them
10:14	. In particular , you're gonna start seeing squares all
10:16	over the place in general when you see something squared
10:18	and math , it means something is curving in general
10:21	and a circle is the perfect shape of a perfect
10:24	thing that curves so it has squares everywhere . Yeah
10:27	, this are here is what we call the radius
10:30	, right ? So this thing is the radius .
10:35	So if you make are bigger than the circle is
10:37	gonna get bigger . So you can say from this
10:39	equation that are determines the size of the circle .
10:48	All right . So what I wanna do is I
10:50	want to put the circle on the back burner now
10:52	because I think we all understand the circle is the
10:54	same distance points of the same distance away from the
10:56	center , which means the center to the to point
10:59	P . No matter where the point P is must
11:00	be a constant . This is the equation the basic
11:03	basic equation of a circle . Of course we can
11:05	shift the circle around and change it a little bit
11:07	. But the basic equation is what we have written
11:09	here , but we're not gonna get into too much
11:12	of this yet . We're gonna do this later where
11:13	the radius determines the size . Now what I want
11:15	to do is get the next comic section on the
11:17	board and that is well you can put them in
11:20	any order you want , but I want to talk
11:21	about any lips next because in the lips , when
11:24	you think about it really looks kind of like a
11:26	circle , but it stretched out a little bit .
11:29	So the equation of an ellipse an equation of a
11:31	circle are kind of similar to one another . So
11:33	let's go over onto this board and talk about the
11:36	concept we put underline here , this is a circle
11:39	and we want to go over here and we want
11:40	to talk about any lips right ellipses . Very important
11:44	shape in in Math , we learned that all of
11:47	the orbits of all of the planets end up following
11:49	ellipses . Alright , so here's how we have to
11:53	do this . You have to kind of use your
11:55	imagination . I need to get some things on the
11:57	board before you totally understand it . So let me
11:59	just kind of draw a few things . So let
12:00	me try to draw my best guess and and the
12:03	lips , is this gonna be a perfect your lips
12:04	? No , it's not going to be close to
12:05	the perfect ellipse , I can tell you right now
12:07	, it's already too flat at the bottom , you
12:10	know , I should around the corner is a little
12:11	bit , but I'm not an artist , I never
12:12	claimed to be an artist . So you're just gonna
12:14	have to put up with that . So the shape
12:16	of this thing is going to look more like something
12:18	like this . Anyway , that's pretty close to any
12:19	lips . Now , the thing about it is when
12:22	you look at a circle , there's one special point
12:24	in the middle , We call it the center ,
12:26	right ? But when we have any lips and also
12:28	the other comic sections , We have not just one
12:31	special point in the center , we have two special
12:33	points and we don't call them the center because they're
12:36	not in the center , we call them the focus
12:38	. And when you have to focus is you call
12:40	them fosse , right ? So you have to get
12:43	used to the idea that there's actually going to be
12:45	kind of two special points in the centre of the
12:47	shape instead of one for the case of the circle
12:50	. So one of these focuses or one of these
12:52	folks i is located around here and the other one
12:55	is located around here . Now , I don't want
12:57	to bog myself down too much with how I know
13:00	exactly where the focus is . What's gonna happen is
13:02	we're gonna write all this stuff down and we're gonna
13:04	go in detail and talk about ellipses , and we're
13:06	going to derive the equations and talk about where they
13:09	come from . But for now , just know that
13:12	if you kind of take this ellipse and squish it
13:14	together , so it becomes a circle , then both
13:16	of these focuses or folks , I kind of end
13:19	up becoming on top of each other . And then
13:21	in that case you get a circle . So basically
13:24	an ellipse is really just a stretched version of a
13:26	circle . So when you take a circle and you
13:28	stretch it out , that one center point , that's
13:30	so special , the center of the circle becomes too
13:32	special points , and that's called Focus number one .
13:35	And Focus number two . Right , So we're going
13:38	to actually call this focus number one , and we're
13:42	gonna call this uh Focus number two , and I'm
13:46	gonna label it because it's probably one of the most
13:48	important things I want you to really learn here .
13:50	This thing is called a focus and this thing here
13:55	is also called a focus And this is focused number
14:00	two . And this is focused number one . Right
14:05	now , why do we have these special points called
14:07	the Focus ? Well , we need to kind of
14:08	get into that . Right ? So what we have
14:10	here is just like we had in the case of
14:12	a circle . The shape was described by the purple
14:14	curve . We call , we said it had points
14:16	P . And they had locations X . And Y
14:19	. Right ? We wrote down some stuff to talk
14:20	about where they where they are now in the this
14:24	ellipse . We have points as well and we're gonna
14:26	call it the set of points P . X .
14:28	Y . Again . So basically the purple curve represents
14:32	all of the points . There's an infinite number of
14:34	points on this curve right there , infinitely close together
14:37	, right ? They all formed the special thing we
14:39	call on the lips . These points have locations X
14:41	. Y . If you put this thing on top
14:43	of the coordinate plain and you could put a location
14:45	for every one of those points . So how do
14:47	we determine the special case of the special curve that
14:50	we call on the lips ? Well , in the
14:51	case of a circle , it was simple because the
14:54	circle meant just meant that it was every point had
14:56	to be an equal distance . So if you lock
14:58	down the middle and you trace your finger out ,
15:00	then you're gonna trace out all of the special points
15:03	. And the lips is a little more complicated .
15:05	So let me draw a couple of things and I'm
15:07	gonna draw a line segment from here to here ,
15:10	and I'm gonna draw a line segment from here to
15:12	here . All right , so I want you to
15:14	to kind of keep in mind I'm sorry about that
15:17	. Probably should have used a different color . So
15:19	this this is just saying Focus number two . So
15:21	here's a line segment from here to here . In
15:22	a line segment from here to here . So what
15:24	you have is let's see how do you want to
15:28	do this ? The set of points P . That
15:31	defines this purple , purple curving ellipse is the following
15:35	thing . The distance between F one mp , which
15:38	means this line here between F one and P plus
15:42	the distance here , which means from F to up
15:46	to pee . So this one plus distance . This
15:48	distance is equal to a constant . Think about what
15:53	that means for a second . The secret to what
15:55	an ellipses is all written down in this simple little
15:58	relation right here . What it means is I take
16:00	any two points in space , Call it focus .
16:02	One focus too . And I want to find the
16:05	set of points that gonna go around there so that
16:07	the distance between one of these points to my shape
16:11	, I'm calling an ellipse plus the distance down to
16:13	the other guy is equal to a constant . So
16:16	what this means is you can see how I've drawn
16:18	it here , that that that that works for that
16:20	one particular point . But let's draw another point .
16:22	Let's say we pick a point down here on this
16:25	point is on the ellipse as well . So this
16:27	is of course , p X comma y as well
16:29	. It's another point of course , but it has
16:30	X y coordinates . What we're basically saying , we
16:33	draw a little dot in line between this point and
16:35	this point and at this point to this point right
16:38	here . So you can imagine for any point on
16:40	the ellipse , you can draw these lines from the
16:42	ellipse to the border and then from the border to
16:45	the other folks , I'm sorry , from the focus
16:46	to the border and then from the border to the
16:48	other focus . So what you're basically saying is if
16:52	you have two points , call it Focus one and
16:54	focus to the ellipse is the set of points .
16:57	P so that as I walk around the ellipse ,
17:00	the distance from one focus up to the border and
17:04	back down to the other . Focus is the same
17:06	number . It's a constant . Just like the distance
17:09	here was a constant . No matter what , it
17:11	was really simple cause there was only one center point
17:13	. Now I spread the points out and I'm saying
17:15	, OK , if I move , let's say over
17:17	here , instead of looking at this point , I
17:19	look at this point here , then the distance between
17:21	here and here , that's a little longer . But
17:24	then the distance between here and here will be a
17:25	little shorter . So you add them up , you
17:27	get a constant number . So let's go over here
17:29	, let's say you take a point from here .
17:31	Way over here , over here , this is even
17:33	longer distance than what I've drawn in blue , but
17:35	then this is a shorter distance . So of course
17:37	you add them up , you get the same thing
17:39	. If I pick a point here compared to the
17:42	blue line , then this distance from F one to
17:45	my finger is shorter , but then the distance from
17:47	here to here is longer . So when I add
17:49	them up I still get a constant number . So
17:51	you see no matter where I walk on the edge
17:53	of this curve , if I even picked this one
17:56	, this is really , really short compared to the
17:58	blue line , but this distance is much much longer
18:00	, so I add them up and I still get
18:02	a constant number . So just like there's this thing
18:04	we call a radius when it comes down to circles
18:08	, Ellipses have a similar constant thing . It's just
18:10	that there's two centers , for lack of a better
18:12	word , we don't call them centers because they're not
18:14	in the center , we call them focus one and
18:16	focus to distance between Focus , one up to the
18:18	border and down to the other . Focus has to
18:21	be the same for every single point on that egg
18:24	shaped thing that we call an ellipse . So that
18:27	is what this equation means . The distance focus one
18:29	to the point . Focus to to the point has
18:31	to be equal to a constant . And that's why
18:33	I'm drawing it next to the circle because it's it
18:36	has a similar meaning . Now , if you wanted
18:39	to write in words , what it means , what
18:41	you would see in a book , an ellipse is
18:43	the set of points . P set of points .
18:49	But it's always gonna be the set of points .
18:50	P . Okay . Where the some I'm using shorthand
18:55	here . This means the set of points P where
18:56	the some of the distance from P two two fixed
19:10	points . Yeah . And these are called the fosse
19:14	are the same . Now when you read this kind
19:21	of thing in a book , the set of points
19:23	. P . So the some of the distance between
19:24	P two to fix points , folks are the same
19:26	. I mean even I know what it means .
19:28	I read it and I'm like what does that mean
19:29	? Because it's just too wordy . All it's trying
19:31	to say is you pick any two points . I
19:33	mean literally I can define in the lips with any
19:35	points I want , I can say here's a point
19:36	, here's a point . Okay , great . Two
19:38	points . How do I find an ellipse ? There's
19:40	got to be some egg shaped thing we call on
19:42	the lips points that surround those two . So the
19:45	distance from here to here is a number and the
19:47	distance over here is the same number . When I
19:49	add the individual little radi I up . So you
19:51	can think of this as radius one and radius to
19:53	you add them up . You get a constant number
19:55	. You can think for this point , radius one
19:57	radius to you add them up . You have to
19:59	get a constant number and so on for every point
20:01	around . Even if you pick a point here ,
20:03	the distance from here to here . Short compared to
20:05	this . But the distance is much longer compared to
20:08	this one , you add them together and you get
20:10	the same the same constant thing . Now when we
20:13	get to actually talking about the math here , we're
20:15	gonna derive this . We're going to talk about this
20:17	in a lot more detail . But for now I
20:20	think what I wanna do is write the equation of
20:23	a basic ellipse on the board and that is the
20:26	following . I don't expect you to know this yet
20:28	or understand this yet . But this is what it
20:30	is . X squared over A squared plus Y squared
20:34	over B squared is equal to one . This is
20:38	a basic basic equation of an ellipse and a .
20:42	And be determined . Mhm . The shape , what
20:50	I mean by that is an ellipse can be really
20:52	long and slender or it can be kind of fat
20:55	and and study . It could also be rotated to
20:58	go up and down instead of side to side .
21:00	And so A . And B . Are going to
21:02	determine how the thing looks , how stretched out is
21:05	it ? That's what we call the eccentricity . Is
21:07	it oriented vertical or horizontal ? A . And B
21:10	. Are going to determine that . But when you
21:11	look at it it looks similar to a circle .
21:13	There's an X squared plus Y squared . Here's an
21:16	X squared plus Y squared . So that part is
21:18	actually the same between the two . Now , in
21:20	the case of a circle we just set it equal
21:21	to R squared . In this case we set it
21:23	equal to one . I'll explain a lot more why
21:26	it's written this way later . But anyway , the
21:29	idea is you have X squared plus Y squared is
21:31	equal to something . And now for the ellipse we
21:33	have these A and B terms which really determine the
21:36	shape , how stretched out the thing is . All
21:39	right , So that's what an ellipse is . Now
21:42	, what I want to do is talk about Parabolas
21:44	. Now we've been talking about parabolas and algebra for
21:46	a very long time . You all know that the
21:48	function F fx is equal to X squared , defines
21:53	that basic parabolic shape . We've sketched them , we've
21:56	grabbed them , we've done tons of things with problems
22:00	because it's one of the most important shapes and one
22:02	of the most important functions in algebra . But now
22:04	we're diving deeper . Don't skip over this stuff .
22:07	We're diving deeper into how that special shape of a
22:09	parabola comes about remember we actually talked about in the
22:12	last lesson that the parable has that special uh special
22:16	characteristic that it focuses all the incoming parallel rays onto
22:21	that special spot we call the focus . We now
22:24	know ellipses have to focus . I parables only have
22:26	one focus , so we have to figure out where
22:29	the focus is going to be and how the shape
22:31	of the proble actually relates to that . And so
22:34	for the third connick section , we're gonna talk about
22:36	Parabola and I know your parabola uh I know that
22:42	you're maybe rolling your eyes because we've talked about Parabolas
22:47	so much , but I'm sorry , it's just something
22:49	that it requires a lot of work because Parabolas are
22:51	very , very important . All right . So ,
22:54	first , let's figure out what is the shape of
22:55	a problem ? Now , this is a free hand
22:57	Parable . I'm sorry , it's not gonna be perfect
22:59	. It's probably too broad at the bottom or whatever
23:01	. But you see it's the general shape of a
23:03	problem that , you know , and love and um
23:06	but now we want to talk about it in terms
23:08	of its definition , in terms of the equation and
23:11	how this shape is actually really defined in general .
23:14	All right . So the thing to note about a
23:16	problem is that it has a focus also and we
23:19	talked about that the focus is going to be somewhere
23:21	around here . The exact spot where the focus is
23:24	will come into focus . Later on . When we
23:26	talk more about the equations , I'm gonna write an
23:28	equation for parabola down , but we're gonna discuss parable
23:31	is that much more great length , a little bit
23:33	more down the road . So we have this special
23:35	spot that we called the focus that we call the
23:38	focus of the parabola . So that means all incoming
23:40	light rays bounce off and they get concentrated at that
23:43	point . But here's the part that we have not
23:45	talked about problems in the past . There's also a
23:48	special line associated with the problem . It's down here
23:52	, it's actually below the problem . It helps define
23:54	what the problem is . C . In the case
23:56	of a circle , all we needed was one point
23:59	to define all the set of , you know what
24:02	we needed , the center and the radius to define
24:04	the set of points . We call a circle for
24:06	the lips . We just needed to focus . I
24:09	and then whatever this constant is , is going to
24:11	determine the shape of the actual ellipse . Now for
24:15	a parabola , you need two things to define what
24:19	the parabola is , but it's not too fussy .
24:22	It's or two focuses . Its one focus . In
24:24	one line , this thing is called a focus .
24:28	So I'm gonna write down focus here . Right ?
24:31	So call it focus . F . But this line
24:33	has a special word . It's called directory ticks .
24:37	Mhm . Directory . So any time you see the
24:40	word directorates , you know , it's some line that's
24:42	the line associated with the Parabola . So , this
24:44	is something we haven't kind of talked about it all
24:47	yet . So , I need to make sure you
24:48	understand . In the past we've graphed Parabolas . We've
24:50	shifted Parabolas around . We've dissected parabolas everywhere . You
24:55	probably thought there was nothing left to learn about Parabolas
24:57	. But it turns out to define the special purple
24:59	shape called a parabola . You actually need two things
25:02	. You need the focus , which is the spot
25:04	where the rays are concentrated on if they bounce in
25:07	. And you also need the special line called the
25:09	Directory . So , we're gonna talk a whole lot
25:11	more about how we know where the directorate says later
25:14	. But there exists online underneath every parable you've ever
25:16	graft called the directorate's . We just never told you
25:19	about it because we didn't need to know it until
25:20	now . All right . So , we also have
25:23	another special point and this one you already know .
25:26	This is called the vertex . Let's see here .
25:29	No , but I want to put it over there
25:30	. I wish I would have done that on the
25:32	other side because I have some other stuff I need
25:33	to draw here . So it's over here . We're
25:36	gonna call this the vertex , the vertex is the
25:41	lowest point of the problem . Or if it's upside
25:43	down , it's the highest point of the problem .
25:45	So , given that there is a focus of a
25:48	parabola , a dot and a line . In fact
25:51	, given any dot in any line , you can
25:53	always define define a parabola that goes down in between
25:56	these two things . How do we define what it
25:59	is ? Well , which similar to what we've done
26:01	before ? We have to draw a few things .
26:04	All right . What we say is that from the
26:07	focus , the distance from the focus to the parabola
26:13	. Here's a dot here . Uh And the distance
26:16	from this point down here to the directory is the
26:20	same distance they're equal to each other . So I'm
26:22	gonna put a line through it . This is from
26:24	geometry , this line means this thing is congruent ,
26:27	which means it's equal to and in length this one
26:29	. Now , the way I've drawn it , this
26:30	line is actually slightly bigger . That's just because I'm
26:33	drawing it freehand . But what I'm trying to tell
26:35	you is that for every parabola you can make this
26:38	little argument that the distance between the focus to the
26:40	point P and the point P to the directorate is
26:43	the same distance . So , this little cross mark
26:45	means this is the same distance as this one .
26:48	And this relationship that these are equal is true for
26:51	every point on the problem . In fact , if
26:53	you look over here , if I draw , for
26:55	instance , maybe over here this distance um is the
26:59	same as this distance right here . And of course
27:01	that's not exactly right , because I drew this thing
27:03	freehand . The actual shape of the problem is not
27:06	quite the way I've drawn it . And also maybe
27:08	if you go up here , the distance here and
27:11	then also the distance down here , this should be
27:12	the same as well . And the one I look
27:14	at this , the more , the more it's not
27:16	quite right . And the reason that's not right is
27:17	because my parable is too narrow , it should be
27:19	, it should be drawn a little bit um wider
27:22	like this . So , you know what ? Let's
27:24	try to fix it . Why not ? We've got
27:26	we've got time and it's really important . So I
27:27	want to try to fix it for you . It's
27:29	not going to be exact , no matter how how
27:31	hard I try but let me go here and let
27:34	me kind of take these out , take these out
27:35	a little bit here . If I go and do
27:37	something like this and say OK , something like this
27:42	. Maybe that's slightly better shape of a problem ,
27:45	maybe something like that . Okay , so then if
27:48	you look , let me take these away . Okay
27:52	, what I'm trying to say is that the distance
27:55	between the focus and this spot right here is the
27:58	same as the distance down here . So these are
28:00	equal . This looks a little better . The distance
28:02	between here and here and here and here is equal
28:05	. It's a little better . It's not perfect .
28:06	The distance from here to here and then here all
28:08	the way down here , these are equal . So
28:10	it's pretty close . It's not exactly right , but
28:12	it's pretty close . You can see that the set
28:14	of points that are the purple points and actually I'm
28:16	gonna call them the same thing . I've been calling
28:18	them P . X . Comma Y . So there's
28:21	this is P all of these points are called point
28:24	P everywhere here this is point P . This is
28:26	point P . Different point , different coordinate X .
28:28	Y . But we all call him the set of
28:29	points P . The distance between the focus and the
28:32	point on the problem is the same as the distance
28:35	from the Parabola down to the directorate's . Now ,
28:39	we've never had to draw the directorate's before because I
28:42	was just giving you the equations of the Parabola before
28:44	and just telling you to draw a plot the points
28:46	and draw it . And I told you that f
28:48	of X equals X squared is a parabola . I
28:51	told you that , but I never told you why
28:53	it was a problem . I never told you what's
28:55	special about a problem . What is special about a
28:57	parabola ? Is the distance from the focus to any
29:00	point on the Parabola is the same as the distance
29:03	from the point on that . Parable it down to
29:06	this invisible line called the directories . Right ? So
29:09	if you wanted to write a definition , a mathematical
29:11	definition , definition of what this is , then the
29:14	way you would say it is the distance FP the
29:17	distance from the focus to the point P is the
29:19	same as from point P to point D . Where
29:24	we call this distance the directorate's here . So you
29:27	could say , you know , this is the one
29:29	point , the one point B to point D .
29:31	Three , you know , like this . So the
29:33	set of points on the directorate's is called the points
29:36	D . The set of points on the problem called
29:39	the set of points P . And the focus is
29:41	just one point . So the focus to the point
29:45	on the problem is the same as the distance ,
29:47	that same distance from the problem down to the directory
29:50	from Point P . Whatever point you pick on the
29:52	purple curve is going to be true . Even if
29:53	I go this way , the distance between here and
29:55	here is the same as between here and here .
29:58	If I go down below the distance between here and
30:00	here is the same as between here and here .
30:02	If I go way over here , the distance between
30:03	here and here is the same as the distance between
30:05	here and here . And that is what a parabola
30:07	is . It's a special shape that kind of uh
30:11	cuts through between the director and the focus in such
30:14	a way that those distances are the same . So
30:16	what you would see in a book um would be
30:19	something like this . A parabola is the set of
30:22	all points . Mhm . Equidistant from a fixed line
30:38	and a point . Mhm . Which is called the
30:42	focus not on the line , so Right . So
30:50	basically if I pick any point , call it the
30:53	focus and I pick any any line , call it
30:58	the directory , I can always find a set of
31:00	points that will cut down below the focus and above
31:03	the directorates and cut through like this . So the
31:05	distance between here and the curve is the same as
31:08	I've outlined it here . Now , what is the
31:10	equation of a parabola ? Of course , you all
31:11	know what it is because we've used it so many
31:13	times . But the basic basic equation of a parabola
31:16	is Y is equal to a times X squared .
31:20	So , notice that this one is different than the
31:22	other ones because I have a square on the X
31:25	. Term , but I do not have a square
31:26	on the Y term . Right notice here I have
31:28	a square in the X tournament square in the white
31:30	term . Yeah , there's some other junk here ,
31:31	but this is a closed shape because I have uh
31:34	squares on both of them like this and then I
31:36	have squares on both of them here as well for
31:38	the circle . So that's a closed shape . Um
31:41	Parabola is not quite the same . I don't have
31:44	a square on the Y term , I only have
31:46	a square in the X . Term . And of
31:47	course this variable A was very important . The A
31:51	determined the shape . So this variable A determined the
31:58	shape of the problem . Specifically if A . Is
32:02	really large . Like if it's 100 X squared ,
32:05	you have a really narrow parabola , really , really
32:08	tight , right ? If you have a very ,
32:09	very small value for a like 0.1 X square ,
32:13	then you're probably is enormously huge . It's like really
32:15	broad , really , really shadow shallow . So in
32:18	each of these equations of the comic sections , you
32:20	have certain little numbers that determine the shape of the
32:22	thing . And of course we know that we can
32:25	go in here and shift in this problem in the
32:28	X direction by putting the term inside of this X
32:31	variable . We can shift in the Y direction .
32:33	So we can of course shift the parable is the
32:35	circles and the parabolas ellipses and circles around by playing
32:40	around with what's inside of the variables here . But
32:42	these equations I'm writing down here are just representative basic
32:45	equations . Now , the circle , the ellipse and
32:48	the parabola are um I don't wanna say they're easy
32:52	, but they're easier to understand the hyperbole . A
32:55	actually is a little weird to wrap their brains around
32:58	, but it's not so hard to understand when you
33:00	understand where you're coming from here . You know ,
33:02	you have the set of points an equal distance from
33:06	the center . Okay then this is the set of
33:08	points where you're adding together these two kind of line
33:12	segments and whatever the some of them are you getting
33:15	a constant number . This defines the special shape .
33:17	We can pick any point we want on this purple
33:20	curve and I can take from here to that point
33:22	and from that point to here and I can add
33:24	it up and I'll get a constant number . That's
33:25	what the ellipse is . Keep that in mind as
33:28	we go and talk about what hyperbole is . So
33:33	we need to write a high purple and alan ,
33:34	I know we haven't talked about high purple as yet
33:36	much , but this is the introduction to what a
33:39	high purple is . Very important actually . All right
33:42	, so now we need to do is draw the
33:43	shape that we call a hyperbole to and we have
33:46	to kind of mirror images here . These are not
33:48	going to be great drawings . I'm sorry about that
33:50	. It's hard to do this free hand but I'll
33:52	try to do my best . But they basically form
33:55	these shapes . If you turn your head sideways ,
33:57	remember how we cut that cone ? It would produce
33:59	one shape angle down and one kind of shaped angled
34:03	up and that kind of face each other because the
34:05	cones were pointed toward each other . But anyway ,
34:07	that doesn't matter so much . We have this shape
34:09	called hyperbole . This purple shape is what we call
34:11	a hyperbole . Both halves refer to the single thing
34:14	called hyperbole . This thing is not a hyperbole and
34:17	this is a separate hyperbole . They both form this
34:19	thing called a hyperbole together . Now , just like
34:22	in the case of any lips we had um the
34:26	focus right ? Uh and also for a problem as
34:28	well . So here we have a focus for this
34:30	hyperbole . We have a focus here as well .
34:32	So we call this one Focus F one and we
34:35	call this one focus F two , you might imagine
34:38	because we have to focus is now this is why
34:40	it becomes a little bit weird to think about ,
34:44	I need you to think about the ellipse . So
34:45	we had the two focuses and what we did is
34:47	we said the distance from one focus to the point
34:49	on the ellipse . Plus that distance from the same
34:53	point down here was a constant . We had to
34:55	add those to kind of radi i together . If
34:57	you want to think of a kind of kind of
34:59	sort of like a radius , we add them together
35:00	and we get a constant for hyperbole as we need
35:03	to do subtraction . That's why it's a little hard
35:05	to visualize . So let's say there's some point here
35:08	on this um hyperbole and let's say it's we don't
35:12	want to draw this , Let's call it P one
35:15	X comma . Why ? Okay , so what I'm
35:20	saying here is that the distance between here and here
35:24	and then we have a distance here and here .
35:26	So in order to well it's probably gonna be easier
35:29	to to show with another point as well . Um
35:31	we have let's call it p sub two X .
35:37	Comma . Y . Right , so let me draw
35:40	another one as well , make it a little easier
35:41	to understand . So then we have the distance between
35:44	F1 up to here and then the distance between here
35:48	down to the focus as well . So what we're
35:51	basically saying here is that the distance F one up
35:58	to the point P one minus the point P .
36:03	Um P one down to F sub two . This
36:09	is the exact same distance as whenever you say Point
36:13	F one up to P two minus P 2 2
36:19	f . two . And another way to really say
36:23	this is the distance in general from Point F one
36:28	up two point any P on the parabola minus that
36:31	same distance point P . But to the other focus
36:34	is going to be equal to a constant . So
36:38	when you really think of it this way it's very
36:40	similar to the equation . I'm gonna lips , we
36:42	added together the distance between the focus and the point
36:44	on the parabola , we had to add them up
36:46	and we got a constant here , We're subtracting them
36:48	. Now . What I need you to understand is
36:50	more than just the math here . I want you
36:51	to intuit intuitively understand what it means . If I
36:54	look at this point in blue here , the distance
36:56	here and then if I subtract this distance here ,
36:58	that's going to give me some number , let's pretend
37:00	this is like six centimeters and this is like two
37:02	centimeters . I'm gonna subtract them , I'll get four
37:04	centimeters . But if I take the distance between this
37:07	and this other line , that's going to be a
37:08	longer distance than this line , it's longer . But
37:11	then I'm subtracting off a longer distance as well .
37:15	So when I do both sets of subtractions , I'm
37:17	going to actually get the same distance because no matter
37:20	where I walk on this parable , if I walk
37:22	all the way down here , but then I'm gonna
37:24	take this , but then I have to subtract off
37:26	this distance which is also longer , then I'm going
37:29	to get the same thing . That's why we say
37:30	it's a constant . So in general , the hyperbole
37:34	A is similar to the ellipse in the sense that
37:36	you're kind of taking two distances and you have to
37:38	get a constant out of them . But in the
37:40	case of the ellipse , we add those distances together
37:42	and we say that we get a constant in the
37:44	case of a hyper Bella we subtract those distances and
37:47	we say that we get a constant . So this
37:49	is what basically you get here now , in terms
37:51	of , you know what you might see in a
37:52	book , as far as like the definitions of hyperbole
37:55	, it's the set of points . P so that
38:04	the difference between the distance between , let's see such
38:14	that the difference between the distance from this is why
38:22	it gets hard to understand it from P two two
38:27	fixed folks . I these are points , right ,
38:32	is a constant . Yeah . Now if I give
38:36	this to anybody , just say , hey , here's
38:37	what a high purple is . You're not going to
38:38	understand it because it doesn't make , it doesn't make
38:40	intuitive sense the set of points piece . So the
38:42	difference between the distance between P two to fix fosse
38:46	is constant . It's so hard to understand . All
38:48	it's basically saying is pick two points . We call
38:51	it Focus one Focus two . There must exist to
38:54	purple curves in there so that the difference in the
38:58	distance between this to one of the points on the
39:00	curve and the point on the curve to the other
39:02	focus is the same and that very special shape so
39:05	that you always get the same number when you do
39:06	the subtraction , meaning it's a constant is called an
39:10	ellipse . I'm sorry , it's called a hyperbole .
39:13	I'm sorry about that . Now the equation for hyperbole
39:16	to looks like this . The basic equation X squared
39:19	over a squared minus Y squared over B squared is
39:26	equal to one . Now of course we can change
39:28	it . We can shift the hyperbole around by doing
39:30	shifting and all this . But notice this has a
39:32	very similar form to the ellipse and that's because the
39:35	constraint is real similar . So we had a plus
39:37	sign here when we're adding up these little segments to
39:40	get a constant . So we have a plus sign
39:41	here . Here we're subtracting the two segments so we
39:43	have a minus sign . So they're very similar to
39:46	one another , right ? But the idea is that
39:48	you have a subtraction going on . The A .
39:50	And the B term are gonna also determine A .
39:53	And B . Are going to determine the actual shape
39:54	of the hyperbole . Is it going to be really
39:56	broad or it's gonna be really really narrow coming in
39:59	and out and that . And also A . And
40:00	B will determine if it's tilted up and down along
40:02	the Y axis or if it's extended along the X
40:05	axis as well . Now the purpose of the section
40:08	was not to make you an expert in comic sections
40:11	that actually takes more time . But what I wanted
40:13	to do is give an overview of what the different
40:16	comic sections are in more detail than we did in
40:18	the last lesson to talk about mostly the definitions of
40:21	what the set of points for circle parabola , ellipse
40:24	hyperbole are and to also write the equations now .
40:27	Not so that you understand them yet , because I
40:29	don't think you really should understand them yet , but
40:32	just so you can kind of see the similarities .
40:34	So for a circle it's a set of all points
40:36	, an equal constant distance from the center . It
40:41	has the equation of this form . You have a
40:43	square in the X term , a square in the
40:44	right term , and you have a single number ,
40:46	we call the radius which determines the size of the
40:48	circle . For an ellipse , it's similar . You
40:51	have kind of the distance from the central point to
40:55	the point on the edge . We call it maybe
40:57	radius one if you want to and then you have
40:59	to add to it . The other kind of quote
41:00	unquote radius from the the other special point to the
41:04	edge . We add them together and we get a
41:05	constant thing . But in turn produces an equation like
41:08	this . You still have the X squared plus the
41:10	y squared . You have a number on the right
41:11	. But then you have two numbers . The two
41:13	numbers dictate the shape of the thing . Is it
41:15	going to be really long and stretched is going to
41:17	be fat and study . Is it going to extend
41:19	in the extra direction ? Is it going to extend
41:22	in the Y direction ? This will determine . And
41:25	also other things that we'll talk about later . The
41:27	shape of this ellipse . Then we had the famous
41:30	parabola . We never learned before that we had a
41:33	directorates that exist below every one of these parabolas .
41:36	You pick a special line called directorates . You picked
41:39	a focus and you can always dry curve that goes
41:42	between the two such that the distance from the focus
41:45	to the curve and the curve to the directorate is
41:48	the same . Right ? So the set of all
41:50	points equal distant blah , blah blah . And so
41:52	when you do the math , which we will do
41:54	in a few lessons , you'll find out the basic
41:56	equation of a problem looks like this . A single
41:58	number determines the size or the shape of the problem
42:02	. How squished it is , how broad it is
42:04	. Notice we only have one square in the X
42:06	term , We don't have a square on the white
42:07	term . And so the problem is opened up like
42:09	this . Next we finally have the hyperbole actually very
42:14	shares a lot of characteristics with the ellipse . You
42:16	can kind of think of a hyperbole as any lips
42:19	that's been stretched so much that it kind of flips
42:21	back on itself and kind of becomes opened up uh
42:25	into this hyperbolic shape like this . And it is
42:28	defined to be the distance from one focus to the
42:30	point on the curve minus the distance between that point
42:33	to the other . Focus is a constant and the
42:36	equation you get looks almost exactly the same as any
42:38	lips but you have a minus sign instead of a
42:40	plus sign . There are a lot more details to
42:42	this . We're going to get into all of them
42:44	. Now what we need to do is we need
42:45	to talk about circles in great detail , how to
42:48	graph them , how to shift them , how ,
42:49	what the size and shape looks like . We'll do
42:51	the same thing for Parabolas will do the same thing
42:53	for ellipses , will do the same thing for hyperbole
42:55	is um by the time you get to the end
42:56	of it you'll understand all the comic sections where they
42:59	come from , how to graph them , why they're
43:01	useful and how to in general solve almost any problem
43:04	that someone can throw at you with regard to comic
43:06	sections .

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