07 - Equation of a Circle & Graphing Circles in Standard Form (Conic Sections) - Free Educational videos for Students in K-12

07 - Equation of a Circle & Graphing Circles in Standard Form (Conic Sections) - By Math and Science

Transcript


00:00	Hello . Welcome back to algebra . The title of
00:02	this lesson is called comic sections circles part one .
00:06	So a title at this this way because we're going
00:08	to be learning about circles for the next five or
00:11	six lessons , really diving into the details . But
00:14	I also want you to keep in mind that it's
00:15	part of the family of of of shapes that we
00:18	call comic sections . We've already talked about the fact
00:20	that we have circles . That's a comic section .
00:22	We have ellipses , we have parabolas and we have
00:25	hyperbole is we have four comic sections which can be
00:28	obtained from taking cones , just a regular cone and
00:31	slicing them in different ways . We get these different
00:33	shapes called the comic sections . So we're gonna talk
00:35	about probably the most important one . Circle . The
00:37	most beautiful shape . It's always been said throughout history
00:40	is a circle because it goes back in it joins
00:43	back to its starting point . All right , So
00:46	what we're gonna do in this lesson is I'm going
00:47	to give you the equation of a circle right here
00:49	in the beginning , we're gonna do a few quick
00:51	examples to show you how to graph or how to
00:54	sketch the shape of a circle . But really in
00:57	the beginning here , you won't understand why that's the
00:59	equation of a circle , right ? You're not going
01:01	to really understand that . But the next part of
01:03	this lesson , right after we get through with the
01:05	first examples , we're going to really dive in and
01:07	show why this equation I'm about to write on the
01:10	board actually does describe the shape of the circle because
01:13	I want you to understand what you're doing . I
01:15	don't want you to just blindly trust me , I
01:17	want you to understand . Hey , there's a ,
01:19	there's a logical way I can understand what this equation
01:22	really is . So let's first write this equation down
01:25	again , one of the most important equations . So
01:27	here we have what we call a circle and you
01:30	all know in general what a circle is . It
01:32	has a central point . In all of the points
01:35	on the circle are an equal distance away , which
01:37	we call the radius from the central point . So
01:40	what we say is if we have a center of
01:44	the circle at the point H comma K . Because
01:48	this circle we're going to draw , it can be
01:50	in the center of the xy plane , or we
01:53	can move the circle anywhere we want left , right
01:55	up down in the xy plane , so it has
01:57	to have some center . We call it H .
01:59	K . These are just placeholders . The center of
02:01	the circle can be one comma to the center of
02:04	the circle can be negative 19 comma seven . The
02:07	circle can be centered at 00 or any other value
02:10	can come up with . So it has a center
02:12	at H . Comma K . And a radius which
02:16	we denote by our which is obvious because it's a
02:18	radius . Right ? So what is the equation of
02:21	the circle ? And this is the famous equation of
02:24	a circle in general ? A circle has this form
02:26	all circles look like this X minus h quantity squared
02:30	plus why minus K . Quantity squared is equal to
02:35	r squared . Now I'm gonna let that sink in
02:38	for a little second even though you don't really understand
02:40	why it's the case . But I'm gonna I'm gonna
02:42	circle it on the board because we're gonna leave it
02:43	up there . Now in general were saying the center
02:47	is at H . Comma K . So we're saying
02:49	the center of the circle can be anywhere that we
02:51	want it to be . But for now let's just
02:53	pretend the circle is centered at the origin , right
02:56	? The circle is centered at the origin . If
02:58	the center then is at zero comma zero then this
03:01	is zero and this is zero and then the entire
03:03	equation boils down to x squared plus y squared is
03:07	equal to r squared . So the equation of a
03:10	circle is quite beautiful really . When you think about
03:12	it , you have the X variable squared plus the
03:14	Y variable squared is equal to whatever the radius is
03:17	squared . Now you don't understand yet why this actually
03:21	does describe the circle . In fact , this equation
03:23	looks really weird compared to any equation or function that
03:26	we have drawn in the past , right ? Because
03:28	usually when we grab things , we put why the
03:32	variable Y on one side of the equal sign and
03:34	everything else on the other side . So we have
03:36	for instance , Y equals mx plus B or three
03:39	X plus four for a line or for a parabola
03:42	will say why equals 16 X squared . Or for
03:45	any other polynomial why is equal to two X squared
03:48	plus three X plus four always why is on one
03:50	side of the equal sign and everything else is on
03:53	the other side of the equal sign . Which lets
03:54	us plot the thing . But this is different because
03:57	we have the X . Variable in the Y .
04:00	Variable variable which are both squared . Of course we
04:03	have the H . And the K . We'll talk
04:04	about that later but they're both squared and they're both
04:07	on one side of the equal sign . So it's
04:08	not set up like the other equations you've ever plotted
04:11	before . And that's because Connick sections are just a
04:13	little bit different . All of them are like that
04:15	. The equation of a circle is set up this
04:17	way it looks a little different . The equation of
04:19	their lips looks very , very different than other equations
04:22	you've plotted or looked at before . The equation of
04:25	hyperbole and parables also look different . So you're just
04:27	gonna have to get used to the idea that the
04:29	comic section equations look a little bit different than what
04:32	you've graft in the past , and that's because they're
04:34	defined in a different way . So in fact one
04:37	thing I want you to keep in the back of
04:38	your mind is one reason why circles look so different
04:41	. The equation of a circle looks different is it
04:43	turns out a circle is not a function . And
04:46	I know you might say well if it's not a
04:47	function , why do we care ? Well , lots
04:48	of things are important in our functions . And this
04:50	is one of the most important ones . The circle
04:52	is not a function because if you think about a
04:54	circle being circular figure , it fails that vertical line
04:57	test . Remember we said long time ago with functions
05:00	that in order for it to be a function there
05:02	needs to be a 1 to 1 correspondence between the
05:05	X . Value and the Y . Value , you
05:07	know ? So that means it passes that vertical line
05:09	test . If it's a function . Now with a
05:12	circle , of course you're gonna always cut through two
05:14	halves of the circle , no matter how you do
05:15	the vertical line test . So it's not a function
05:17	. Does it mean a circle is not important ?
05:19	Of course not . Circles are one of the most
05:21	important things . All right . So let's jump into
05:24	some examples to show you how to use this equation
05:26	really quickly and then we're gonna derive it . I'm
05:28	gonna show you where it comes from . All right
05:30	. So , let's take the most simple circle I
05:31	can probably think of Here is the simplest one in
05:34	existence . X squared plus Y squared is equal to
05:37	one . This is an equation of a circle .
05:39	You might say . How does he know ? It's
05:41	an equation of a circle ? Well , the way
05:42	you know is because it follows this form , you
05:45	have the X variable square plus the y variable squared
05:48	equals something squared . Yes , I know it's a
05:50	one but when you think about it , one is
05:52	really one squared . So the right hand side is
05:55	something squared , right ? And you might say ,
05:57	well wait a minute , it's different than this .
05:58	Well not really . If the center is H comma
06:01	K . Then we know from this that I can
06:04	write this . You know I I didn't give it
06:07	to you this way but I could write it as
06:08	hr minus zero squared . I'm sorry x minus zero
06:10	squared plus why minus zero square is equal to one
06:15	square . So I'm not gonna give it to you
06:17	like that on a test . I'm gonna give it
06:18	to you like this . And I'm gonna say hey
06:20	what is the center and what is the radius of
06:22	that circle ? And in your mind you need to
06:24	say well I can that's really a shift , a
06:26	shift of the centre by zero units and a shift
06:29	of the centre in the y direction by zero units
06:32	. And so what this really tells me is the
06:34	center of the circle is located at 0:00 and the
06:40	radius is equal to one . Why is it one
06:45	? Because the right hand side is r squared .
06:47	The radius is what we call our but the right
06:50	hand side is r squared . So really defined the
06:52	radius . You take the square root of one ,
06:54	and so the radius is equal to one . So
06:58	the bottom line for a circle , you read the
07:00	center directly out of how the X and Y variables
07:03	are shifted and the radius is read directly off the
07:06	right hand side . You take the square root of
07:08	whatever is on the right hand side . Because remember
07:10	the right hand side is the radius squared . So
07:13	to read the radius out , I take the square
07:14	root to get back the radius that I have .
07:16	So this particular circle , for instance , if I
07:20	wanted to draw or sketch it , I'm not gonna
07:22	sketch every one of these things , but we just
07:24	put it off in a little xy plane here .
07:27	The center is located at zero comma zero . That
07:29	means the center is right in the center of the
07:31	coordinate system and the radius is one . Now the
07:33	radius of a circle means all of the points on
07:36	the circle r the distance radius one radius unit ,
07:40	however far the radius is away from the center .
07:42	So the way the sketch it is , if this
07:45	is one and this is one and this is negative
07:48	one and this is negative one , then the circle
07:51	must go through all these points because the radius is
07:53	one . So if I'm gonna sketch and I'm gonna
07:55	go down , I'm gonna go down through here ,
07:57	I'm gonna go up through here and I'm gonna go
07:58	up through here . Is that a perfect circle ?
08:00	No , you can tell . It's lopsided the way
08:02	I've drawn it here because I didn't do my tick
08:04	marks perfectly . But you get the idea when you
08:06	sketch it , you're saying one unit away , one
08:08	unit away , one unit away , one unit away
08:11	. The center is read directly out . The radius
08:13	is read by taking the square root of the right
08:14	hand side . I do not expect you to understand
08:17	yet why this equation actually gives you this shape .
08:22	We haven't talked about that yet . For now I
08:25	want you to trust me . And let's do a
08:26	couple of quick little examples . And then when we
08:29	get through those , I will show you exactly why
08:31	this equation actually produces the shape . All right .
08:35	So for now let's let's just kind of plow ahead
08:38	and let's take a look at another circle . Let's
08:40	say I give you another circle X -1 quantity squared
08:44	plus why minus one quantity squared is equal to 25
08:50	. Now , first you ask yourself , does it
08:52	follow the form of what I have here ? And
08:54	the answer is yes , X minus some number squared
08:57	. Why minus some number squared radius squared . So
09:00	the way you do this is you read the center
09:03	directly out of the equation . So the center of
09:06	this circle is located at 1:01 and the radius Is
09:15	equal to whatever's on the right hand side . You
09:16	take the square root of 25 , so the radius
09:19	is equal to five . Notice I picked this number
09:21	nicely , so I could take the square root of
09:23	it uh there but you don't have to have a
09:26	nice number . If I put 10 on the right
09:27	hand side , then the radius we the square root
09:30	of 10 . The square root of 10 is an
09:32	ugly number but it's a number . If whatever numbers
09:35	on the right , you take the square root of
09:36	it . If you don't have a nice perfect square
09:38	, it's okay . You just have A radius that
09:40	some decimal units long . Of course you can have
09:42	a circle with a 1395 centimeter radius . Of course
09:47	that can happen . But in this case I'm choosing
09:49	nice numbers so notice what we're saying if the center
09:52	is that H comma K . It's x minus h
09:55	y minus K . So I just read the center
09:57	directly off this minus business comes directly from the idea
10:00	of shifting a function . Now we've talked about shifted
10:04	functions great length in the past and I don't want
10:07	to get into an entire lesson on why this shifted
10:10	version with these minus signs produces a shift to the
10:12	right . I've done that many many times in the
10:14	past but probably the easiest way to do it is
10:17	to compare this equation x squared plus y squared is
10:19	one with this equation x minus one squared y minus
10:23	one squared . And you can , in your mind
10:24	you can replace this with a one . Basically this
10:26	equation is shifted to the right one unit and up
10:29	one unit . Because if I put a one in
10:31	here and I put a one in here I'm gonna
10:33	get a zero square and zero squared . If I
10:37	put a zero here I'm gonna have a zero squared
10:39	in a zero square . So the bottom line is
10:42	the only way I can get these equations to kind
10:44	of like look similar to one another is if I
10:46	shift every X value to the right one unit because
10:50	one minus one is zero and every why value up
10:53	one unit because one minus one is zero . So
10:55	when you're comparing the basic equation of a circle ,
10:57	this is the basic equation of a circle . The
11:00	equivalent shifted version would look something like this , X
11:03	minus , let's say two quantity squared plus y minus
11:07	two quantity squared is equal to one . So I'm
11:10	claiming that this equation is exactly the same shape as
11:14	this , one has the same radius , but this
11:16	one is shifted to the right to units and up
11:19	two units . And the reason or I don't want
11:21	to get into a bunch of lecturing on it .
11:23	But the reason that it's that way with a minus
11:25	sign shifting to the right and a minus sign shifting
11:27	up is because when you compare these equations are exactly
11:30	the same . But if I want to for instance
11:32	, make this quantity zero here . The only way
11:34	I can do it is to feed X values in
11:36	here two units bigger than I'm feeding them in here
11:40	Because to -2 as zero . So if I want
11:42	these to be , I mean we know it's the
11:43	same shape , we know they have the same radius
11:45	, But I want to figure out how to make
11:47	them kind of equivalent . How do I go from
11:49	here to here . And the only way I can
11:51	do that is to feed x values and that our
11:53	two units bigger than that two minutes to giving you
11:55	00 being here . And also the same thing with
11:59	why if I envision feeding a Y value of zero
12:02	in here , the only way I can do it
12:03	is to feed a Y unit into this equation two
12:05	units bigger . So it's the same exact shape .
12:08	It's just shifted over to the right and up two
12:11	units over to the right to units and up two
12:13	units . Because the only way that I can feed
12:15	the same values as I'm feeding into the base equation
12:18	is to feed two units bigger into the shifted version
12:22	. That's why it becomes shifted over to the right
12:24	and shifted up there . All right . So anyway
12:27	, getting back to our problem is shifted over to
12:29	the right and up one . So the center is
12:31	11 The radius is five . Now , we're not
12:33	gonna do this for every single problem , but let's
12:35	go ahead and sketch this one as well . So
12:38	here is X . Y . Plane , right ?
12:42	And the center of this thing is that one comma
12:44	one . So here is one comma one . We're
12:46	gonna put a dot right here . Now the center
12:47	is no longer at the origin . The center is
12:49	shifted over to the right up one , but the
12:51	radius is five . So how do we do that
12:54	? Well , you have to be five units away
12:55	. So 12345 I'm gonna put a little tick mark
12:59	here , 12345 and put a little tick mark here
13:04	. I need to go to the left five units
13:05	one , 2345 And put a little tick mark here
13:09	and then I need to go down 12345 Need to
13:13	put a tick mark here . I put these little
13:15	tick marks here because that helps me guide my pencil
13:19	whenever I am drawing the circle . And of course
13:22	I'm not a good artist . So it's not going
13:23	to be a good drawing . But you see the
13:25	idea here , when you look here at the at
13:28	the uh , my pencil is a little bit too
13:30	long , but you can see the radius is five
13:32	units of constant five units away . So basically you
13:34	put your center down , you count five units down
13:37	five units up five units to the right , five
13:39	units to the left foot . A little tick mark
13:41	. And then you can of course see that .
13:43	This is 123456 This is six units away . This
13:46	is negative one negative two negative three . This is
13:47	negative four units away and so on . I can
13:50	count up and count down as well . All right
13:54	, we're gonna get a little more practice with the
13:56	circle business before we move on . What if we
13:58	have the circle X minus two plus , Why squared
14:03	is equal to 16 ? Now , a lot of
14:04	students look at a circle like this and say ,
14:06	well , it doesn't really look like a circle .
14:08	It's uh sorry , Mr squared there . It's um
14:11	it looks different . This one shifted in , this
14:13	one isn't Okay , well , that's fine . If
14:15	you want to in your mind transform this , That's
14:18	fine too . You don't have to do this .
14:20	But a lot of times what I'll tell people to
14:21	do , as I say , write it like this
14:23	instead of why squared ? Right ? It is why
14:25	zero squared . Now suddenly it mirrors exactly what the
14:29	form of the circle should be . If you were
14:32	going to read the center of the circle off of
14:34	this equation , what would the center be ? The
14:37	center would be ? There's a shift to the right
14:40	to units . That means that the center is two
14:43	units to the right , but the shift up or
14:45	down is really zero . So it's just two comma
14:47	zero , that's the center , Right ? The radius
14:50	is whatever is on the right hand side , you
14:52	take the square root of it . Because what's on
14:54	the right hand side of this equation is the radius
14:55	squared . So that means the radius is squared of
14:58	16 , which is four . Okay , again ,
15:02	we're not gonna do sketching of a million of these
15:04	things , but we will do a little bit here
15:08	. So the center here being at two comma zero
15:11	means the center is 12 units over comma zero means
15:14	it's right on the line right there and the radius
15:17	is four units . So I have 1234 units to
15:21	the right . 1234 units up . I'm gonna put
15:24	a little tick mark there because that's where my radius
15:26	is gonna be , 1234 tick marks to the left
15:30	and 1234 tick marks directly below . Now you see
15:35	I have a tick mark all through here that I'm
15:37	gonna try to go through . Is it perfect ?
15:41	No , but that's mostly what you're trying to do
15:43	. Two comma zero radius of four . You can
15:45	see it's a constant distance away from the center of
15:48	this guy . So the base equation of the circle
15:51	is just X squared plus y squared is r squared
15:53	. Whatever you're doing inside the parentheses , shifting the
15:56	thing left or right up or down is just determining
15:58	where the new center is . Again . Because when
16:02	I shift the thing , I have to feed units
16:04	have to feed X and Y values two units bigger
16:07	and why value two units bigger in order to give
16:10	me the same shape as my original base equation .
16:13	All right now I want to do a couple of
16:16	more and I think I want to try to squeeze
16:17	them in over here because I need to save the
16:20	last two boards for the probably the most important part
16:22	of this lesson here . Let's go and take a
16:24	look at X squared Plus y squared is equal to
16:29	100 . What would be I'm not gonna graph this
16:31	one , but what is the center and what is
16:33	the radius here ? So , you know , as
16:36	I said , you know , you can it's almost
16:39	exactly the same thing as what we've done up here
16:41	, X minus zero squared , you can write it
16:43	as why minus zero squared ? You can write it
16:45	as but you know you don't have to do that
16:47	, that's something that I tell people to do in
16:49	the beginning . But really when you don't see any
16:50	shift here then you know right away that the center
16:54	is just located at zero comma zero , right ?
16:57	And the radius is just whatever is on the right
16:59	hand side , but you have to take the square
17:02	root of it . So the radius is 10 units
17:04	. So if I wanted to sketch this I would
17:06	draw an X . Y plane , I would put
17:07	the center of the circle right in the in the
17:10	origin right at 00 and I would count 10 units
17:12	to the right . 10 units up 10 units to
17:14	the left . And that would basically draw my circle
17:16	which would be a pretty large circle . Alright ,
17:18	so that's just one more little example . I want
17:21	to do one more and have a little space too
17:22	graphic . Um Because it's a little more important ,
17:26	what if you have X plus two quantity squared plus
17:31	y minus one quantity squared is equal to nine .
17:35	Now this one looks a little different because the y
17:37	minus one is written just like it is here why
17:40	? Minus something ? But the ex here there's no
17:43	minus sign . So a lot of times students will
17:44	get confused . Okay when you're shifting functions and I
17:47	covered all of this when we did the original shifting
17:49	functions less than a long time ago . When you
17:51	shift things when you shift any function it could be
17:54	a circle , could be a line , could be
17:55	a parabola , could be cubic , could be a
17:58	square root , anything any graph , you can shift
18:00	anywhere you want on the xy plane . When the
18:04	X variable is shifted with a minus sign , it
18:07	means it shifted in the positive X . Direction .
18:09	When the Y variable as a minus sign shifted shifted
18:11	in the positive Y direction , but when you have
18:14	plus signs it's shifted in the opposite direction . Why
18:18	? Because you can write this um you can write
18:21	this X plus two , you can write it as
18:23	X minus and minus two if you want to you
18:26	don't have to I'm just trying to give you different
18:28	ways to think about it . Yeah this is perfectly
18:32	exactly the same as this . So this is a
18:34	shift to the right but negative two units . So
18:37	I'm shifting to the right but oh hold on a
18:38	second I'm actually shifting to the left because I'm shifting
18:41	negative two units to the right . So it's a
18:43	little bit of a convoluted way to think about it
18:45	. But the way to really think about it is
18:47	that when you see a plus sign you actually go
18:49	the opposite direction which is left . When you see
18:51	a minus sign you go to the right . So
18:53	the center of this guy , Yeah The center is
18:58	really located at -2 comma positive one like this .
19:04	And the radius is the square root of nine which
19:07	is three like this . Okay . And I will
19:11	try to do a little quick sketch here let's see
19:13	if we can we can fit it in right here
19:16	, we're going to do it we're gonna do it
19:18	right here we're gonna go and say okay we have
19:21	an xy plane where is the center negative two comma
19:25	one . So negative two for X . Here's negative
19:28	one negative two for X . One for why ?
19:30	So there's one for why ? So the the center
19:33	is no longer on the right hand side of the
19:35	plane . It's on the left hand side of the
19:36	plane , the radius is a . Three . So
19:38	you have to count up 123 units from there .
19:41	Put a tick mark 123 units left from there .
19:44	123 units down from there and 123 units from there
19:49	. So I'm putting a little tick marks there .
19:51	So there's 123123123123 Now I know these tick marks are
19:57	not perfect because they're not evenly spaced and I did
19:59	a little sloppy job with that . But this is
20:01	basically what you're looking for . The reason it looks
20:03	a little bit oblong is because these tick marks are
20:06	a little bit wider than those . But there's there's
20:09	a radius of 123 to the left , 123 to
20:13	the right . It's just that compressed things a little
20:14	bit but that's more or less how you how you
20:17	graph this thing . So now what we've done is
20:19	I've introduced the equation of a circle and you know
20:22	how to use it . But so far you don't
20:24	know why it works . You don't know why .
20:25	I mean when we learn how to graph lines we
20:27	make a table of values and we show that that's
20:30	the graph of the line . We do graph parabolas
20:32	, initially we do a table of values and I'll
20:35	show you that that's how you graph a problem .
20:37	But here for the circle business I just told you
20:39	trust me this is the equation of a circle but
20:41	you don't actually know why it's the equation of a
20:42	circle . So what we're gonna do now is we're
20:45	gonna go and explore that in more detail . It's
20:48	extremely important when you get into higher math to know
20:50	where things come from . Otherwise you're not doing anything
20:53	other than just repeating what you've been taught . So
20:57	what we wanna do is I want to do a
20:59	circle . I want to show a circle with center
21:05	at zero comma zero and a radius of four .
21:09	So what I want to do is draw this and
21:11	I want to figure out what the equation of that
21:13	circle must be . Now we already know what it
21:16	must be because I already showed you what the equation
21:18	of a circle is . But let's pretend we have
21:19	no idea what an equation of a circle is .
21:21	And I want to draw this and I want to
21:25	understand where everything comes from . So what we do
21:29	is we say , okay , I'm gonna go and
21:31	draw an X . Y plane , I'm gonna try
21:33	to draw at large so that we can see everything
21:35	. So there's X . Here's why And the center
21:39	of this circle is 00 right there . And the
21:42	radius is for that means that there must be 1234
21:46	, must be a little crossing there . 1234 crossing
21:50	there . 1234 crossing there , 1234 crossing there .
21:55	So the circle must look something like this , it
21:58	goes down and then it goes way down here and
22:02	it goes way over here and goes way up .
22:04	Is that perfect ? No , it's not perfect .
22:05	But that's basically what the circle centered at zero radius
22:08	of four looks like . Now , what we want
22:11	to do is we want to recognize that the blue
22:14	line here is the set of all points that make
22:17	up that thing that we call the circle . There
22:19	must be some kind of an equation that will predict
22:23	for lack of a better word . All the points
22:25	that lie on the blue curve , if we can't
22:27	figure out what the equation is of what's on the
22:29	blue curve , then we failed . So we need
22:31	to figure out what the equation is that describes what
22:34	the blue curve is . How do we do that
22:36	? Well , let's first go over here and say
22:38	, well , let me switch colors . Actually ,
22:39	uh up here , we're gonna say that the points
22:42	on this curve uh there's infinite points , of course
22:46	, on this blue curve , but I'm just gonna
22:48	pick one of them right here . And I'm gonna
22:50	say it's point P . Because that makes sense ,
22:51	point P . Right ? And this point P has
22:53	a value X . Comma y . Now we're saying
22:56	that the blue curve really is the set of all
22:58	points P . Right ? So there's a point P
23:00	. Here , another point P . Here , another
23:02	point P . Here there's an infinite number of points
23:03	all the way around . And every one of those
23:05	points have a different value . X comma Y .
23:08	Right . This one might be I don't know .
23:10	This is 123 comma 12.8 or something . So this
23:14	might be three comma 2.8 . Whatever you can see
23:16	that as you go around the circle , the different
23:19	values of P . R gonna all be different .
23:20	But there should be an equation to predict what those
23:23	points are . All right ? So what we're gonna
23:25	do to figure this out is the following . We
23:27	want to we want we know , let me put
23:30	it this way that between the center and all of
23:33	the points on here . The critical thing is the
23:35	distance here is what we call the radius . The
23:39	distance here is what we call the radius . And
23:41	we know that it's equal to four units . That's
23:44	the critical thing that lets us figure out the equation
23:46	of a circle , because we know that if the
23:47	point is here , the distance must be four units
23:50	. If the point is over here , the distance
23:52	must also be four units . If the distance is
23:54	here for units , distance here for units , no
23:56	matter where I point the thing , the distance from
23:59	the center must be point For units . Now remember
24:02	I'm saying the distance from the center must be four
24:04	units . The distance from the center must be four
24:07	units . Remember we learn something called the distance formula
24:10	. We know how to find distances anytime . We
24:13	have two points in the xy plane , we can
24:15	just stick it right into the distance formula and calculate
24:18	the distance . And we're saying that all of these
24:20	points have to be four units from the center .
24:22	So we can then write down the equation by using
24:26	that . Now . Remember the distance formula comes directly
24:28	from the pythagorean theorem , which we've discussed a long
24:31	time ago . All right . So if you think
24:34	about it , if this is the point X ,
24:38	I'm sorry , P which is X comma Y ,
24:40	then there must be some X value right here uh
24:46	corresponding for that value of P . And there must
24:48	be some wise guy . This is the X .
24:50	Comma Y . X . You read it from this
24:51	axis and why you read it from this axis right
24:55	? And the center , don't forget is located at
24:59	zero comma zero . That's what a center is .
25:02	It's located at zero comma zero . So how can
25:04	we write down what we need to figure out ?
25:07	We know the distance from zero comma zero to whatever
25:15	point we care about X comma Y is equal to
25:19	four . But for this particular circle four units ,
25:21	the distance from zero to whatever point . But I
25:24	drew the point here , but I could've drawn the
25:26	point here or here or here . It doesn't matter
25:28	. Whatever the point is , the distance between the
25:30	center and that is four units . So how do
25:32	we write that down ? The way we're gonna write
25:35	that down is called the distance formula . What is
25:38	the distance between 00 and X . Y . Now
25:41	I know we don't know what Xy is , that's
25:43	the whole point . We don't know what P X
25:44	. Y is but we know that there's some point
25:47	has an X . Value and it has some Y
25:49	value . So if if we were going to find
25:51	the distance between here and here , we would do
25:53	it exactly . As we always do remember , the
25:55	distance formula is what it's X two minus X .
25:58	One quantity squared plus Y tu minus y one quantity
26:03	squared . We take the square to that whole thing
26:05	, right ? That's what the distance formula is .
26:07	The only thing about it is we know what one
26:09	of the points is . But the other point we
26:11	don't really know what it is . We just know
26:13	it has some X and some Y value . So
26:15	then to find the distance between them going from here
26:18	, subtracting here , the distance would be whatever X
26:21	is minus zero quantity squared plus whatever . Why is
26:27	minus zero ? This is the distance between those two
26:31	points . Make sure you understand it ? Yes ,
26:33	I don't know what X and Y is . Of
26:34	course , I don't know what they are . Could
26:35	be , you know , 12 This would be two
26:38	point something , you know , whatever . Two point
26:40	something . Of course , I don't know what it
26:41	is , but I know it's xy so I take
26:43	X minus the X value . Why , minus the
26:46	Y value . I stick it into the distance formula
26:48	. And I know that this must be equal to
26:52	A distance of four units of four units . Why
26:57	? Because the radius of the circle is four units
27:00	. I know what the distance is . So I
27:01	calculate the distance between the points and I set it
27:04	equal to four because that's the radius of the circle
27:07	I drew . Now how do I do anything with
27:09	this equation ? I have this large square root but
27:12	I'm going to not deal with that right now .
27:13	Let's go ahead and just rewrite this . This is
27:15	just x minus zero squared . So we can write
27:17	it as X squared . This is why minus zero
27:20	squared , We can write it as this square .
27:21	We still have a square root is equal to four
27:23	. Now , in order to get rid of the
27:24	square and on the left , what do I do
27:26	? I take the square of both sides ? Right
27:29	? So I say x squared plus y squared is
27:32	equal to four . I have a square root .
27:35	But what I can do is I can raise the
27:38	left hand side to the power of to and since
27:40	it's an equation , I can do the same thing
27:42	to the right hand side , I can do anything
27:44	. I want to both sides of the equation .
27:46	Now on the left hand side , the square kind
27:48	of cancels with the square root . So all you
27:50	have is X squared plus y squared is equal to
27:53	four squared which is 16 . This is the equation
27:56	of the circle . This is exactly what we would
28:00	expect based on the problems that we did before .
28:03	Right , we said a center of circle always has
28:06	a center h comma K in a radius of four
28:09	. The center is located at whatever the shift is
28:11	here . But in this case the circle that we
28:14	drew on the board , we didn't have any shift
28:17	at all . So we expect it to be ,
28:18	X squared plus y squared is equal to something and
28:21	it has to be whatever the radius is squared ,
28:23	the radius is for you square it , you get
28:25	16 . So this is the equation of the circle
28:29	With a radius centered at 0:00 with a radius of
28:34	four . And you can generalize this . So let's
28:37	generalize this . Mhm . Because we drew the circle
28:42	is a radius of four , but we know if
28:43	we make the circle bigger or smaller , the only
28:46	thing that's really going to change is the distance formula
28:49	, we're going to change what's on the right hand
28:50	side . So even if you make it a generic
28:52	are all you're saying is that a circle has the
28:55	general form of X squared plus y squared is equal
28:58	to the radius squared . That's all we did on
29:00	the right hand side , we squared the radius .
29:02	So if the radius were 10 we would square it
29:04	and we would get 100 on the right . If
29:07	the radius were three we would square it and we
29:09	would get nine on the right . So this is
29:11	the equation of a circle Centered at 00 with a
29:15	radius of our whatever that radius is . So that's
29:19	how we go from . Here's a special shape called
29:22	a circle to what is the equation of that now
29:24	. It's not like a traditional equation that you've learned
29:27	before because the Y . Is not by itself on
29:30	the left hand side . It and notice it's not
29:32	a function either because it fails the vertical line test
29:36	, I'm cutting through two values of the relation here
29:40	every time I cut through like this . So back
29:43	to what is a function back we learned a long
29:45	time ago . It's not a function . But that's
29:47	okay . It doesn't mean it's not important . Circles
29:49	are one of the most important shapes in all of
29:51	nature . I mean , really ? And I mean
29:52	that's serious . I mean you get to calculus ,
29:54	you're using circles all the time , but it's not
29:56	a function . That's okay . So if you get
29:58	a trick question on the test is a circle of
30:00	function . No , it's not a function , but
30:02	it's an extremely important relation , which is what we
30:05	would call or equation . Now this I went from
30:08	a circle centered at the origin with a radius of
30:10	four and we figured out what the equation of the
30:12	circle must be . So we can generalize this is
30:14	why the equation of a circle looks like this .
30:16	But what if the circle is not centered at 00
30:19	? Let's quickly do that . If you can understand
30:22	that , which I know that all of you can
30:23	then the next one is not hard at all .
30:26	Let's shift let's shift the circle . Um two units
30:35	right , and three units down . So what we
30:43	want to do is go through the whole exercise again
30:45	. But with the circle in a slightly different location
30:47	. And we want to show that the thing that
30:49	you get out of it is actually this business .
30:51	We want to show that that describes a circle that's
30:54	shifted anywhere other than the origin . Mhm . All
30:58	right . So , we're gonna draw a xy plane
31:01	so we can always have a picture . I always
31:02	recommend that . So here is our xy plane like
31:07	this . Now we're saying basically that the center of
31:11	the circle is two units to the right and three
31:14	units down . So here's two units to the right
31:16	123 units down . That means the center of the
31:18	circle is here . So we say the center is
31:24	uh sorry , can still center , we say the
31:26	center is at two comma negative three . Right ?
31:30	And we say the radius is the same as the
31:31	radius before . We're not changing anything . We're saying
31:33	the radius is still four , Everything is the same
31:36	. Um But we're just shifting the thing here .
31:39	So , how do we draw ? Sorry , not
31:41	R equals R . Were saying R is equal to
31:42	four , same as before . We're just shifting the
31:45	thing . So how do we draw this ? Well
31:46	if the radius is four it's 1234 and then 1234
31:53	We're gonna put a little tick mark there then 1234
31:56	we put a little tick mark there , then 12
31:59	then 34 we'll put a little tick mark here .
32:02	So this should describe this circle something like this .
32:08	Now before we go on I want you to agree
32:10	with me that this circle is exactly the same as
32:14	this one . I mean I know I drew it
32:15	maybe it's the tick marks are a little different but
32:17	you see the idea this one's centered at the origin
32:19	radius of four . All I'm doing is shifting it
32:21	over two units three units down . It was originally
32:26	here over two units 3 units down . But the
32:28	circle is the same . I mean the shape of
32:30	the thing is the same . It's just located in
32:31	a different place . So how would I go about
32:34	doing that ? How would I go about figuring out
32:36	what happens here ? Well , we do the same
32:38	kind of thing . We pick a point on the
32:40	circle . Let's call it this one . Let's call
32:42	a point , pete . It's got X comma y
32:45	. The center here is now at two comma negative
32:48	three . That's what the center of the circle is
32:52	, right ? But we also know that the radius
32:56	From this point to this point is four units same
32:59	as before four units exactly the same thing . So
33:04	we're gonna do the exact same thing . We're gonna
33:05	figure out what is the distance , using the distance
33:07	formula from this point to this point . And no
33:10	matter where you are in the circle , the distance
33:13	must be four units . That's what the circle is
33:15	, that . That's what defines the shape of the
33:19	points that we call the circle . So if we're
33:21	gonna put that distance formula in there , we're gonna
33:23	do the same thing . It's X two minus X
33:26	one quantity squared plus y tu minus y . One
33:31	quantity squared . This is the distance between any two
33:34	points in the xy plane . But this point has
33:36	an X coordinate of , we're just calling it X
33:39	. This thing has an X coordinate now too .
33:41	So we have to subtract two . This thing has
33:45	a Y coordinate of why . But this thing has
33:47	a Y coordinate of negative three . So we do
33:49	have to subtract it , but you have a negative
33:51	three here and that's squared . And we take the
33:54	square were saying this distance is what we have written
33:56	down as the distance formula between those two points .
33:58	And we're saying that it's a radius of four units
34:01	away . So we have to put that the distance
34:03	between them is equal to four . Now let's crank
34:06	through this and see what we get . We're gonna
34:07	have X -2 quantity squared . Now we're going to
34:11	have this becomes y plus three quantity squared , We
34:17	have a square root here , and this is equal
34:19	to four . Now the same exact sort of thing
34:21	can happen in order to get rid of the square
34:23	root , we can just square both sides to make
34:26	it clear , I'm gonna say x minus two ,
34:28	quantity squared plus y plus three quantity squared . I'm
34:33	going to take the square root , I'm gonna have
34:35	four . Now , in order to make it clear
34:37	what I'm saying is it's an equation , I can
34:39	do what I want to both sides . I'm gonna
34:40	square the left hand side . So then I also
34:42	have to square the right hand side , so immediately
34:44	the square cancels with square root right here . And
34:48	so what you're going to have left over is just
34:50	what's underneath , X minus two , quantity squared plus
34:54	Y plus three quantity squared is equal to 16 .
34:58	Were saying this is the equation of a circle that
35:01	has a radius of four , meaning you take the
35:03	square root of the right hand side but located somewhere
35:06	other than zero comma 02 units to the right ,
35:09	three units down , let's go back and look and
35:11	see if that makes sense . Center is located .
35:14	H comma K radius . And of course I can't
35:17	spell radius . Sorry about that radius of our um
35:22	the shift of the white in the Y . Direction
35:25	is what do we have , what do we pull
35:26	out of it ? Uh Two units to the right
35:30	, in the X . Direction . And because there's
35:31	a plus sign , it's actually the opposite direction ,
35:34	three units down , that's exactly what we would predict
35:36	from here . It's two years to the right ,
35:38	this is a plus sign . So it was three
35:39	units down , so we can then generalize exactly as
35:43	we did before that . Really , the equation of
35:46	a circle is X minus h quantity squared plus why
35:49	minus k quantity squared is equal to whatever the radius
35:53	is . Because we just picked a circle here .
35:55	So this is the general circle , right ? And
36:01	all circles will look like that . You put the
36:03	equation the equation of a circle down on your paper
36:06	, you put the center in , you put the
36:07	radius in , you square the thing and that and
36:09	that's what it is . That's what the equation of
36:11	a circle is . It comes directly from the distance
36:13	from formula , knowing that the distance between the center
36:17	to any point on the circle has to be the
36:18	same for all the points . And so we just
36:19	crank through it . So make sure you understand this
36:23	. You should understand the general idea of what a
36:24	circle is , you should understand where it comes from
36:26	. We're not done not by a long shot .
36:28	We need to sketch circles , we need to do
36:30	more complicated problems . We need to do all kinds
36:32	of things . We have a lot to do ,
36:33	but this is the most important part of it all
36:35	. Follow me on to the next lesson will continue
36:38	conquering comic sections and specifically circles .

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