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

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 . |

#### 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.

This page not only allows students and teachers view Writing Equations of Ellipses In Standard Form and Graphing Ellipses - Conic Sections videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.

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