Writing Equations of Ellipses In Standard Form and Graphing Ellipses - Conic Sections - Free Educational videos for Students in K-12

Writing Equations of Ellipses In Standard Form and Graphing Ellipses - Conic Sections - By The Organic Chemistry Tutor

Transcript


00:01	in this video , we're going to focus on ellipses
00:04	. We're going to talk about how to graph the
00:06	lips and also how to find the coordinates of the
00:09	vortices and the full side of the ellipse . So
00:13	let's begin on the left . We have an ellipse
00:18	with a horizontal major axis on the right , the
00:22	major axis is vertical . The left of the major
00:26	axis is always Equal to two . A . And
00:32	that's the case for both types of ellipsis . Yeah
00:41	, this is the left of the minor axis which
00:44	is vertical on the left , but it's horizontal on
00:47	the right , the left of the minor axis ,
00:53	it's equal to to be for this one , the
01:01	minor axis is horizontal but it's still equal to to
01:04	be mhm . Now , whenever you have an ellipse
01:09	where it's centered at the origin , here's the equations
01:14	that you're gonna be dealing with for horizontal ellipse ,
01:18	it's X squared over a square plus Y squared over
01:24	B squared . For the ellipse on the right ,
01:30	it's X squared over b squared plus Y squared Over
01:36	a squared equals one . Now to determine which type
01:42	of ellipse that you have , whether it's elongated in
01:46	the X . Direction or elongated in the Y .
01:48	Direction . Look at your A . Value A .
01:51	Is always bigger than be when dealing with ellipses .
01:55	If the larger number is under X . Squared you're
01:58	dealing with an ellipse with a horizontal major axis .
02:03	If the larger number A . Is under Y squared
02:06	then he lifts his vertical . It has a major
02:08	vertical axis . So let's say this is the center
02:15	, this is going to be positive A . Or
02:18	just A . And this is negative A . This
02:20	is B . And this is negative B . So
02:25	here are the major vertex is and these two points
02:30	represent the mine advertises . So the graph on the
02:33	right , who are the major purchases . And here
02:38	are the minor purchases for the ellipse on the left
02:43	. The coordinates of the major versus is are plus
02:46	or minus a comma zero For the minor vortices it's
02:50	zero Plus or -7 . For the lips on the
02:54	right it's the reverse . The coordinates of the major
02:57	purchases are zero comma plus or minus B . And
03:03	for the minor purchases it's plus or minus a comma
03:09	zero . Now let's talk about the coordinates of the
03:14	folks I for the lips on the left . The
03:20	full side will be along the major horizontal axis and
03:28	it's C . Units away from the origin or the
03:33	center . So once you have the center if you
03:36	at sea you'll get the focus on the right .
03:38	If you subtracted by sea you'll get the focus on
03:41	the left . So the coordinates for the folks i
03:47	it's gonna be plus or minus C . Comma zero
03:50	for the lips on the left side . for the
03:52	lips on the right side it's zero plus or minus
03:56	C . So starting from the center need to go
04:00	up si units and down si units . And that
04:04	will give you the coordinates of the folks . I
04:09	Now what happens if we have in the lips that
04:11	is not centered at the origin ? If it's not
04:15	centered at the origin then the coordinates of the center
04:19	become H comma K . This equation changes to x
04:28	minus H squared over a squared plus . Why minus
04:35	K squared over B squared is equal to one .
04:42	And the same is true for this equation everything will
04:44	be the same . But we're going to replace X
04:46	with x minus H squared and this will be over
04:52	B squared . And then we're gonna replace why with
04:56	y minus K squared . And this will be over
04:59	a squared . Now let's focus our attention on the
05:03	horizontal ellipse . When the center is the origin ,
05:09	I mentioned that the coordinates of the folks i is
05:12	pleasure minus C . Comma zero . But now when
05:16	the center is shifted to some point H comma K
05:21	. The coordinates of the folk I or rather the
05:23	false I will be shifted . So all we need
05:26	to do is add the coordinates of the center to
05:30	the coordinates of the full side . So we're gonna
05:35	add plus or minus C two H . So it's
05:38	gonna be a church plus or minus C . And
05:40	then we're going to advocate a zero which will be
05:42	K . So this will be the new coordinates of
05:46	the fosse for horizontal ellipse . Now the coordinates of
05:53	the vertex is are plus or minus a common zero
05:58	and cyril Plus or -7 . So when it shifted
06:02	, all you gotta do is add the coordinates of
06:04	the center to the vertex is to get a new
06:06	one . So to get the major vergis is is
06:09	going to be a church plus or minus a comma
06:12	K . And for the mine advertises its going to
06:15	be a church comma . Kay Plus or -7 .
06:21	Now let's consider the situation if we have a vertical
06:24	lips as the center shifts from the origin to some
06:32	point . H comma K . Let's see what's going
06:35	to happen to the full sigh . So the coordinates
06:39	of the full sigh at the center is at the
06:41	origin is zero plus or minus T . When the
06:45	graph has shifted from the origin it's going to be
06:48	a church comma K . Plus or minus C .
06:55	The major vertex is will be zero plus or minus
06:59	A . The minor vortices is plus or minus B
07:03	, comma zero . So when it's been shifted this
07:08	becomes H comma K plus or minus A And this
07:13	becomes a judge plus or -7 comma . Okay ,
07:19	now let's work on some practice problems . Go ahead
07:22	and grab the lips and identify the coordinates of the
07:25	vortices and the false I . By the way ,
07:29	in order to find a full site , you need
07:31	to know how to calculate C . And here is
07:33	the equation to calculate C . It's a squared minus
07:37	b squared . We'll see scored is a square minus
07:41	B squared . Not a squared plus B squared .
07:43	As in the case of the pythagorean theorem for hyperbole
07:48	asses , C squared equals a squared plus b squared
07:51	. But for ellipses , c squared is equal to
07:53	a squared minus b squared . So that's how you
07:56	can calculate the using that former . So feel free
08:01	to try this problem . And this problem , there's
08:05	no H or k value . So the coordinates of
08:09	the center is 000 . A squared is going to
08:16	be the larger of the two numbers , so nine
08:19	is greater than four . That means a squared is
08:22	nine , Which means a . Is the square to
08:24	nine . So a is three . B squared is
08:28	the other number four , Which means B is the
08:31	square of four or 2 . So now that we
08:34	know the values of A . And B . And
08:37	the center , we can go ahead and graph the
08:39	ellipse . So first let's plot the center A is
08:53	under the X variable . So we're going to travel
08:56	three units to the right from the center along the
08:59	X axis And three units to the left . So
09:05	this gives us the coordinates of the major versaces which
09:10	is going to be plus or minus a comma zero
09:13	or plus or minus three comma zero . Now be
09:18	as to and B or b squared is under y
09:22	squared . So we're gonna go up two units from
09:26	the center along the y axis and down two units
09:31	. So this gives us the coordinates of the mine
09:34	advertises which is zero plus or minus B or zero
09:39	plus or minus two . So those are the coordinates
09:42	of the vortices of the lips . Now we can
09:45	go ahead and grab the lips by connecting these four
09:51	virgin seas together . So that's how you can graph
09:53	in the lips . Now , how can we find
09:58	the coordinates of the full side ? So the coordinates
10:04	of the full sigh , it's gonna be Plus or
10:08	-10 . It's going to be along the major horizontal
10:14	axis for this problem . So since A is along
10:19	the X axis . The folks , I will be
10:21	along the X axis as well . So we need
10:23	to determine to see , we know that C squared
10:28	is a squared minus b squared A squared is nine
10:33	, B squared is four . 9 -4 . Is
10:38	five . Taking the square root of both sides .
10:41	We get that C Is equal to plus or minus
10:45	the square root of five . So the coordinates of
10:48	the folks I will be plus or minus route five
10:52	comma zero . If you have your calculator and you
10:59	type in the square root of five . The square
11:01	root of five is 2.236 , approximately . So along
11:06	the X axis , that's going to be somewhere in
11:09	this region . So that's how we can plot the
11:13	coordinates of the fill site for this problem . Now
11:19	, let's try a similar problem to the last one
11:21	. So for the sake of practice , feel free
11:23	to pause the video and work on this example .
11:24	Problems go ahead and graph the lips and then find
11:31	the coordinates of the vertex is and the full side
11:36	. So the first thing we should do is identify
11:39	the coordinates of the center . There's no H or
11:41	K for this problem . So the center is going
11:43	to be the origin . It's 000 . Next we
11:47	need to determine are a squared value , Is it
11:51	nine or is it 16 ? A squared will be
11:54	the larger of the two . Numbers 16 is greater
11:57	than nine . So a squared is 16 . So
12:04	once we have a square we need to calculate a
12:06	a . Is going to be the square root of
12:07	16 , which is for B squared has to be
12:11	the other number nine . Thus B is going to
12:15	be the square to nine which is three . So
12:18	now that we have A . And maybe we can
12:21	go ahead and graft ellipse so be a story and
12:34	B squared is associated with the X . Variable .
12:36	It's under X squared . So what we're gonna do
12:39	is We're going to travel three units to the right
12:44	from the center And three units to the left from
12:47	the center . So this is going to give us
12:50	not the major verjus is but the mine advertises the
12:54	minor courtesies is always associated with B . So the
12:58	coordinates of the mine advertises for this problem is plus
13:02	or minus B . Coming zero . So that's plus
13:04	or minus three comma zero Now a . s .
13:08	four and it's associated with what a square is on
13:12	the white square . So it's gonna be along the
13:15	Y axis . So we're gonna travel four units from
13:19	the center And four units below the center . So
13:27	the coordinates of the major vertex is will be zero
13:30	comma four and zero negative four . Here's the center
13:37	so it's zero plus or minus A or zero plus
13:40	or minus zero comma plus minus four . So now
13:44	let's go ahead and grab the lips . So that's
13:52	how we can plot it . Now . The last
13:54	thing we need to do is find the coordinates of
13:57	the full sign . So because the major horizontal axis
14:03	is the Y axis , in this case the folks
14:07	, I will be along that major vertical axis .
14:13	But first we need to calculate C . Let's use
14:18	this form a C squared is equal to a squared
14:20	minus B squared , A square to 16 . B
14:24	squared is nine 16 . -97 . So C is
14:29	going to be plus or -17 . Now the square
14:34	root of seven , it's about two six . So
14:41	the coordinates of the folks I will be in this
14:45	region , it's in the same direction as plus or
14:54	minus A , which is along the y axis .
14:58	Now , the coordinates of the fill sinus problem is
15:01	going to be zero comma plus or minus C or
15:04	zero comma plus or minus the square root of seven
15:10	. So that's how you can find the coordinates of
15:12	the full side . As well as the coordinates of
15:16	the vortices . Number three , identify the coordinates of
15:22	the center vortices and fosse and then graft the ellipse
15:30	. So the center is going to be hK McCain
15:35	here we have X ministry , H is simply going
15:39	to be three . You just need to change the
15:44	sign here . We have Y plus two . So
15:47	H is going to be , I mean K is
15:49	gonna be negative too . So that's H and K
15:53	in this problem . So now that we have the
15:56	coordinates of the center , let's determine our A and
15:59	B values . So which one is going to be
16:03	a squared 16 or 25 ? Well , 25 is
16:07	larger than 16 . So a squared Will be 25
16:13	A . Is going to be the square root of
16:15	25 which is five . B squared is the other
16:18	number 16 . So B . Is going to be
16:21	for now let's go ahead and graph it . So
16:32	first let's plot the center Which is 3 -2 .
16:37	So we're gonna travel three units to the right And
16:40	we're going to travel down two units . So there
16:43	is the center now A . Is five and A
16:48	. Is associated with the Y . Variable . So
16:50	from the center we're gonna go up five units .
16:54	Right now we're at a Y . Value of negative
16:56	too . So we're gonna stop at three and then
17:01	we're going to go down five units . And so
17:13	the coordinates of the vertex is rather than being zero
17:19	comma plus or minus A . It's h comma K
17:25	. Plus or minus a . H . History K
17:31	is -2 . A . S five negative two plus
17:37	five is three And -2 -5 is -7 . So
17:46	here at this point is that 3:03 and this point
17:51	here Is 3 -7 . So those are the coordinates
17:56	of the major versaces . Now let's focus on B
18:07	bs four And its associated with the X . variable
18:10	. So we're gonna go four units To the left
18:13	and four units to the right , starting from the
18:15	center . So let's go four units to the right
18:21	. So we get this point here And then four
18:24	units to left will take us to this point .
18:31	So the coordinates of the minor viruses rather than being
18:36	plus or minus becomes zero . It's going to be
18:38	a church plus or minus B . Come . Okay
18:44	. Hs tree B is for K is -2 ,
18:49	3-plus 4 is seven And 3 -4 is -1 .
18:57	So this point here that's seven negative two . And
19:05	then this other one is negative one negative two .
19:10	So now let's go ahead and graph the ellipse .
19:27	So that's how we can graph this particular ellipse .
19:31	And we have the coordinates of the major and minor
19:34	overdoses . Now let's focus on finding the coordinates of
19:42	the folks . I so the major access left or
19:49	the major axis is vertical . That's the coordinates of
19:52	the full side will be along that line . So
19:57	rather than being zero comma plus or minus C ,
20:01	it's going to be h comma , K plus or
20:05	minus C . So let's begin by financing . C
20:10	squared is going to be a squared minus B squared
20:15	A squared is 25 . B squared is 16 .
20:18	25 -16 is nine . The square to nine .
20:22	History . So C is gonna be plus or minus
20:25	three By the way , technically a . is plus
20:31	or -5 And B is plus or -4 because to
20:37	get the coordinates of the vertex is we would have
20:39	to like Add five and subtract five from the center
20:44	. So it's technically more accurate to say plus or
20:46	minus four and plus minus five for A . And
20:48	B . So now that we have the value of
20:56	C , we said that C is plus or minus
20:58	street , let's use this to find the coordinates of
21:01	the full side . So H history K . Is
21:06	negative two and we're gonna add plus or minus street
21:08	to that negative two plus three is one And -2
21:16	-3 is -5 . So we have a focus at
21:21	31 which is here And we have another one at
21:26	3 -5 which is here . So technically what we
21:30	need to do is starting from the center , We
21:33	would go up see units or up three units to
21:37	get this point To become a one and then we'll
21:41	travel down three units To get the other focus which
21:44	is 3 -5 . Number four , identify the coordinates
21:51	of the center overdoses and fill see . And then
21:55	we'll graph the lips . So let's begin with this
21:58	center . So here we have a negative one in
22:01	front of X . We're going to change it to
22:03	a positive one and here we have a positive one
22:06	in front of why ? So this will become negative
22:08	one . So the center , is that 1 -1
22:15	? A square is going to be the larger of
22:17	the two numbers . So a squad is nine A
22:21	is going to be the square to nine which is
22:23	stream . If you want to write plus or minus
22:24	three , that's up to you . But if you
22:26	just put three , you can still graphic correctly and
22:29	you'll still get the correct coordinates B squared . That's
22:35	going to be for The square root of four is
22:38	too . Now let's go ahead and calculate C .
22:42	So C squared is a squared minus B squared ,
22:47	A squared is nine , B squared is four ,
22:49	So c squared is five . The sea is going
22:53	to be the square root of five and you can
22:55	write that as plus or minus route five if you
22:58	want to . Now let's go ahead and grab the
23:02	lips . Let's first put the appropriate markings on the
23:21	X and Y axis . So the center is that
23:32	1 -1 , which is here a history and A
23:39	. Is associated with the X . Variable . So
23:42	we're gonna travel three units to the right and three
23:46	units to laugh . B . As to and it's
23:52	B squared . It's under Y squared . So we're
23:55	going to go up to parallel to the Y axis
23:57	and then down to , so that's how we can
24:00	graph the lips in this problem . Now let's identify
24:08	the coordinates of the viruses . If you ever forget
24:11	the formula you can just look at your graph .
24:14	So here we can see that it's four negative one
24:20	. And here the X . Value is negative two
24:22	. With the Y . Value is negative one .
24:24	So it's negative two comma one . And you remember
24:27	the formula it's H . Plus or minus a calm
24:31	. Okay for a horizontal ellipse , H . Is
24:36	one A history K . is -1 . So you
24:40	get one plus three which is four And 1 -
24:44	Tree which is -2 . And this should be negative
24:48	to -1 . So those are the ways in which
24:52	you can find the verses of the ellipse . Now
25:05	let's find the minor courtesies of the lips . So
25:08	this point here has an X and Y . Value
25:11	of one at this point here has an X .
25:14	Value of one but a Y . Value of negative
25:16	three . Or you could use this formula is H
25:19	. Plus or minus B . Actually take that back
25:24	. It's a church comma K plus or minus B
25:29	. So H is one . K . Is negative
25:31	one , B is too . So it's negative one
25:35	plus two which is one And then it's -1 -2
25:39	which is -3 . Given us those two minute courtesies
25:46	, I want you to be familiar with this formula
25:48	because sometimes A . Or B could be a radical
25:52	value . And to get the right answer you need
25:56	to use that formula as well be the case of
25:59	the full sight in this example . So the coordinates
26:07	of the full side for horizontal lips , it's gonna
26:10	be a church plus or minus C . Come okay
26:17	, h is one . See is a square to
26:21	five . So it's gonna be one plus or minus
26:23	two Route five . Comedy . In this case we
26:27	can't simplify the coordinates of the folk . I I
26:29	mean full side . So we have to leave it
26:32	like this . So that's your answer for the coordinates
26:35	of the folks . I so starting from the center
26:42	, we would have to travel see units to the
26:44	right . The square to five is like to point
26:47	to something 2.236 . So going a little bit more
26:54	than two to the right would take us somewhere to
26:57	this point and a little bit more than two to
26:59	less will take us to that point . So that's
27:03	approximately where the two focal points will be . But
27:09	this is what you would report as your answer for
27:12	the coordinates of the folks . I so anytime you
27:14	have a radical , it's gonna be hard to identify
27:17	the points using the graph . You need to use
27:20	this expression . So that's why I want you to
27:22	learn how to use it because you're gonna need it
27:25	for some problems . Number five . Right . The
27:30	standard form of the equation for the ellipse shown below
27:35	. So first we need to realize that the center
27:38	is at the origin and this is a horizontal ellipsis
27:42	where we have a horizontal major axis . So a
27:47	square is going to be under X squared . So
27:50	this is the general formula of the equation that we
27:52	need to use X squared over A squared plus Y
27:56	squared over B squared is equal to one . So
27:59	if we could find the values of A and B
28:02	, we could write the equation in standard form A
28:08	is four . Because this point here the major vertex
28:12	is is four units away from the center . We
28:17	could see that be is to The minor purchases are
28:21	two units away from the center . So if a
28:25	is four , that means that a squared is four
28:28	squared which is 16 . And if B is too
28:32	B squared or two squared is four . So the
28:35	form of that we have is x squared over 16
28:38	square Plus y squared over four square Is equal to
28:42	one . So this is the answer for this problem
28:47	, number six . Right . The standard form of
28:50	the equation for the ellipse shown below . So we're
28:54	given the center of the ellipse which is negative three
28:57	comma two . And we could find A . And
29:02	B . To the distance between the center and one
29:06	of the vortices . We can see it's 12 345
29:12	minutes . So it's five units along the X .
29:16	Direction and along the Y . Direction It's 123 .
29:22	Let's use a different colour to count . So this
29:24	is 12 34 . So we went down four units
29:30	to get one of the my liver disease . So
29:35	a . Is going to be the bigger number .
29:36	That means AS55 B . Is for If a .
29:41	is five , a squared It's gonna be 25 .
29:45	B squared is 16 . So now that we have
29:49	the values of a squared B squared and we have
29:52	see our center we can write the standard form of
29:56	the equation . So A . Is horizontal . So
30:03	what we have is a horizontal ellipse when it lifts
30:05	with a horizontal major axis . So the formula we're
30:09	going to use is x minus H squared over A
30:12	squared . Since A is parallel to the X axis
30:18	plus y minus k squared over B squared Is equal
30:23	to one . H is -3 . K is too
30:29	, so this is going to be x minus H
30:33	. Let's use brackets for now , H is -3
30:38	. A squared is 25 and then why minus K
30:44	is positive too Be square to 16 . So this
30:50	becomes X plus tree . So we change negative 3
30:54	to positive three squared over 25 Plus Y -2 Squared
31:04	over 16 Is equal to one . So that's how
31:09	we can convert from the graph to the equation in
31:13	standard form .

Summarizer

DESCRIPTION:

This algebra video tutorial explains how to write the equation of an ellipse in standard form as well as how to graph the ellipse when in standard form. It explains how to find the coordinates of the foci, vertices, and co-vertices. This video contains plenty of examples and practice problems.

OVERVIEW:

Writing Equations of Ellipses In Standard Form and Graphing Ellipses - Conic Sections is a free educational video by The Organic Chemistry Tutor.

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