01 - Conic Sections: Ellipses - Graphing, Equation of an Ellipse, Focus - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back to comic sections . Were covering | |
| 00:02 | the topic of ellipses or studying the comic section known | |
| 00:05 | as the ellipse today . So by the end of | |
| 00:08 | this lesson , what I really want to get across | |
| 00:10 | to you is number one , what does the ellipse | |
| 00:12 | look like ? And number two , probably the most | |
| 00:14 | important thing is what is the equation ? The general | |
| 00:16 | equation of an ellipse . But most importantly how ellipses | |
| 00:20 | look very similar to circles . In fact , the | |
| 00:23 | shape of an ellipse looks like a stretched version of | |
| 00:26 | a circle . So we're gonna talk about that and | |
| 00:28 | the equation of a circle and the equation of an | |
| 00:30 | ellipse , even though they look kind of different . | |
| 00:33 | At first I'm going to show you by the end | |
| 00:34 | of this lesson that they actually are very , very | |
| 00:36 | closely related . So if you take the equation of | |
| 00:39 | a circle , you can kind of modify it to | |
| 00:41 | look like an equation of an ellipse because a circle | |
| 00:43 | is very closely related to an ellipse . So we're | |
| 00:46 | going to go through that logic . I want you | |
| 00:47 | to understand that the ellipse and circle , or cousins | |
| 00:50 | of one another and they're very closely related to one | |
| 00:53 | another . And as the problems and the lessons proceed | |
| 00:56 | , we will get into graphing ellipses . More will | |
| 00:58 | start shifting ellipses around in the xy plane . And | |
| 01:00 | so by the end of all of it , you'll | |
| 01:01 | understand where your lips has come from , how to | |
| 01:03 | derive the equation of the lips and so forth . | |
| 01:05 | So the first thing I want to do , this | |
| 01:07 | is a long lesson . I have a lot to | |
| 01:08 | get out . Okay , so I want to kind | |
| 01:10 | of get started . Uh the first thing I wanna | |
| 01:12 | do is draw the general shape of an ellipse . | |
| 01:14 | Now , I'm really not great at drawing ellipses , | |
| 01:17 | I'm sorry about that . They're almost always pancake shaped | |
| 01:20 | . So this is a terrible lips , I can | |
| 01:22 | tell . This is too fat , too skinny . | |
| 01:23 | It's it's okay though because it doesn't the accuracy doesn't | |
| 01:27 | matter so much for what I'm trying to get across | |
| 01:28 | . But you need to use your imagination and pretend | |
| 01:30 | that this is an ellipse , which is a stretched | |
| 01:33 | version of a circle . All right now , the | |
| 01:35 | thing you have to get used to with any lips | |
| 01:37 | is there are two special points inside of an ellipse | |
| 01:40 | . They're called Focus number one . And Focus number | |
| 01:43 | two . If you think about a circle , there's | |
| 01:45 | a special point to a circle . Also , it's | |
| 01:47 | just called the center of the circle . It's right | |
| 01:49 | in the middle and there's only one of those special | |
| 01:51 | points . We call it the center . But for | |
| 01:53 | any lips , there's not just one special point because | |
| 01:55 | it stretched out . There's two special points and we | |
| 01:58 | call them , we don't call them the center because | |
| 01:59 | they're not in the center . We call them Focus | |
| 02:02 | number one and Focus number two . So this would | |
| 02:04 | be the center of the ellipse . But the focus | |
| 02:06 | number one is somewhere over here , so I'm gonna | |
| 02:07 | go and put F number one right here And focus | |
| 02:11 | number two is somewhere over here . F sub two | |
| 02:13 | . There's two different focuses . We actually call them | |
| 02:16 | foe . See when you have two of them , | |
| 02:17 | plural of focus . Focus is is fuzzy . Now | |
| 02:21 | as the one thing we're gonna talk about mathematically is | |
| 02:24 | if you take a ellipse like this with Focus number | |
| 02:27 | one and Focus number two . If that were to | |
| 02:29 | start to cram this ellipse together and make it less | |
| 02:31 | and less elliptical by cramming it together . Eventually it | |
| 02:34 | will become a circle and when it becomes a circle | |
| 02:36 | , Focus one and focus to become right on top | |
| 02:39 | of each other , right in the center . So | |
| 02:41 | that's why a circle is a special case of an | |
| 02:44 | ellipse , where both of the focus is kinda line | |
| 02:47 | up right on top of each other . We're gonna | |
| 02:49 | talk about that more . We're gonna show the equations | |
| 02:51 | of why that's the case . But what I want | |
| 02:53 | to do now is I'm going to talk about the | |
| 02:54 | definition of an ellipse . What does an ellipse mean | |
| 02:57 | ? Why does it have this special shape ? All | |
| 03:00 | right , so the black curve is what we call | |
| 03:02 | any lips . There are points all along this curve | |
| 03:05 | . Uh And those points , I can label them | |
| 03:06 | anything I want , but I'm gonna call this point | |
| 03:08 | P . And this P has an X . And | |
| 03:11 | a Y coordinate . Because you have to imagine this | |
| 03:13 | ellipse in an xy plane with X coordinate and a | |
| 03:16 | Y coordinate . And every point on this black curve | |
| 03:19 | has an X and a Y coordinate . I could | |
| 03:21 | plot all those points and I would pop my ellipse | |
| 03:24 | . But what I want to get across to you | |
| 03:25 | now is what is the geometric definition of an ellipse | |
| 03:29 | ? Right ? If you think back to a circle | |
| 03:31 | , it's easy enough . I don't even have to | |
| 03:32 | draw a circle is a shape where the center is | |
| 03:35 | in the middle and all of the points on the | |
| 03:37 | circle are in equal distance , all the way around | |
| 03:40 | from the central point . So the definition of a | |
| 03:42 | circle might be something like the set of all points | |
| 03:46 | , an equal distance from the center . That is | |
| 03:48 | what a circle is . Well , an ellipse has | |
| 03:50 | a very similar definition , but because there's two special | |
| 03:53 | points instead of one center , the definitions a little | |
| 03:56 | bit more complicated . So I'm going to draw a | |
| 03:58 | distance from this focus to this point and then I'm | |
| 04:02 | gonna draw a distance from this focus to the same | |
| 04:04 | point . And what the definition of an ellipse says | |
| 04:07 | is that the black curve is the set of all | |
| 04:10 | points . Where if I take this distance to the | |
| 04:13 | black curve and this distance from the black curve to | |
| 04:16 | the second focus and I add them together , then | |
| 04:19 | the ellipse has a constant basically that that total distance | |
| 04:23 | when you add them all together is a constant for | |
| 04:26 | any point on the black curve . So a better | |
| 04:28 | way to probably talk about it would be to write | |
| 04:30 | it down . What I'm saying is much like a | |
| 04:32 | circle is defined in terms of the distance from the | |
| 04:35 | center , that's where all the points on the circle | |
| 04:37 | are . All of the points on the black curve | |
| 04:39 | are defined as like this . The distance from focus | |
| 04:42 | number one to the point P . The single point | |
| 04:44 | P . Right here , plus The distance from focus | |
| 04:47 | , number two to the same point P has to | |
| 04:51 | be equal a constant . Now this constant is going | |
| 04:54 | to determine the shape and the size of the ellipse | |
| 04:56 | . So if I say that this distance plus this | |
| 04:58 | distance has to be five , let's say let's make | |
| 05:01 | it five . Uh Then what it means is that | |
| 05:04 | this distance plus this distance must equal five . Then | |
| 05:07 | it also means if I go to a point over | |
| 05:10 | nearby than the distance between here and this point on | |
| 05:14 | the black curve . Plus this point down here must | |
| 05:17 | also be five . The distance from this focus to | |
| 05:19 | this point on the curve here , plus this distance | |
| 05:22 | here must be five . You see the pattern , | |
| 05:24 | the distance down here to this point plus this point | |
| 05:26 | . This distance must be five . This plus this | |
| 05:29 | must be five . This plus this must be five | |
| 05:30 | . This plus this must be five . So much | |
| 05:33 | like a circle is a simpler version of that . | |
| 05:35 | The definition of a circle is the distance from a | |
| 05:38 | central point . This is similar . It's just because | |
| 05:41 | we have two special points , it's that some of | |
| 05:43 | those distances must be a number constant . I'm picking | |
| 05:47 | the number five here , but just like the radius | |
| 05:49 | of the circle changes the shape of the circle . | |
| 05:52 | The some of these uh what we call focal radi | |
| 05:55 | I This is a focal radius number . One , | |
| 05:57 | focal radius number two has to be some number and | |
| 06:00 | the different number , we choose , just like the | |
| 06:01 | radius of a circle is going to determine how big | |
| 06:04 | or small , how the thing looks in general . | |
| 06:07 | All right , we're gonna do a lot more with | |
| 06:09 | that uh in a little bit we're gonna actually derive | |
| 06:12 | in the next lesson . The equation of the lips | |
| 06:14 | just from this definition right here . But I want | |
| 06:17 | you to kind of put that in the back of | |
| 06:18 | your mind right now , and just remember that's what | |
| 06:21 | the geometric definition of an ellipses and what I want | |
| 06:25 | to do now is turn our attention back to uh | |
| 06:29 | the equation of a circle because I want to focus | |
| 06:32 | this lesson mostly on how a circle . If you | |
| 06:34 | stretch it a little bit becomes any lips . And | |
| 06:37 | the equation of a circle can be shown to look | |
| 06:39 | like the equation of an ellipse . So we've already | |
| 06:41 | done circles . Let's go and transform them to make | |
| 06:43 | them into ellipses . Now , for those of you | |
| 06:45 | who don't like surprises , I want to show you | |
| 06:47 | kind of the punch line before we kind of go | |
| 06:49 | too far here . This is the general shape of | |
| 06:51 | any lips . So the punch line , we're going | |
| 06:53 | to talk about this in great detail during this lesson | |
| 06:56 | and the next lesson . But I have a summary | |
| 06:58 | on the board here . I don't usually like throwing | |
| 07:01 | summaries at you , but I want you to have | |
| 07:03 | the big picture in your mind , basically . This | |
| 07:05 | is an ellipse that's oriented horizontally , but you can | |
| 07:09 | also have the ellipse oriented vertically . So this is | |
| 07:12 | oriented where the long side is along the X . | |
| 07:14 | Axis and this one's oriented up and down where the | |
| 07:17 | ellipses oriented on the Y . Axis . For now | |
| 07:20 | just forget about this . And let's talk about this | |
| 07:22 | one up here . All this is saying is that | |
| 07:25 | for an ellipse that centered at 00 , that means | |
| 07:27 | that centre on the origin and the focus is let's | |
| 07:31 | just pretend it's at some number negative C and positive | |
| 07:33 | . See we're gonna talk a whole lot more about | |
| 07:34 | focuses later . Don't worry about some of the local | |
| 07:37 | radio . I will talk about that in the next | |
| 07:39 | lesson before an equation horizontally oriented like this . This | |
| 07:42 | is what the equation of the ellipse looks like . | |
| 07:45 | All right , So you have X squared , you | |
| 07:46 | have a Y square . Now stop right there and | |
| 07:48 | just remember back . The equation of a circle also | |
| 07:51 | has an X squared plus a Y squared , right | |
| 07:55 | ? We're gonna show that in just a second . | |
| 07:56 | But this one looks different because you're dividing by a | |
| 07:59 | square . You're dividing by B squared . So basically | |
| 08:01 | , whatever is underneath the X . Square term is | |
| 08:04 | the place where it crosses the X axis . So | |
| 08:07 | if you write it as a squared and then it | |
| 08:09 | crosses at the number A In other words , if | |
| 08:11 | this were X squared over three squared , the ellipse | |
| 08:14 | would cross over here and then of y squared over | |
| 08:17 | b squared whatever number is down here determines where it | |
| 08:20 | crosses on the y axis . So you can look | |
| 08:22 | at the equation and figure out exactly where the ellipse | |
| 08:25 | crosses on the X and the Y axis . Just | |
| 08:27 | by looking at it , you just take the square | |
| 08:29 | root down here , in the square root down here | |
| 08:31 | , and then that's where they cross . The focus | |
| 08:33 | will talk about how to calculate that comes from this | |
| 08:35 | equation , we'll talk about that part of it a | |
| 08:37 | little bit later . Now , if you rotate the | |
| 08:40 | ellipse so that it's vertically oriented , you have the | |
| 08:43 | same thing , you have a focus at negative C | |
| 08:45 | . And positive . See the focus always goes along | |
| 08:48 | the long direction of the ellipse , and you have | |
| 08:51 | the same kind of equation X squared over something squared | |
| 08:54 | , Y squared over something squared is one . All | |
| 08:57 | I'm saying here is that the number underneath the X | |
| 09:00 | variable is where it crosses on the X axis , | |
| 09:04 | the number under the y variable . You just take | |
| 09:06 | the square root and that's where it crosses on the | |
| 09:08 | y axis . And we have a long axis , | |
| 09:11 | we call a major axis and we have a short | |
| 09:14 | axis called a minor axis . So basically we haven't | |
| 09:18 | gotten into problems yet . But the bottom line is | |
| 09:20 | when you write the equation , I'm gonna lips down | |
| 09:22 | , it's gonna be really easy to graph them because | |
| 09:25 | all you have to do is look at these numbers | |
| 09:26 | that are under here and that's going to tell you | |
| 09:28 | exactly where the ellipses going across . And then we'll | |
| 09:31 | want to sketch the curve of it . So don't | |
| 09:34 | forget your geometric definition , uh how the shape comes | |
| 09:39 | about is that this kind of focal radius plus this | |
| 09:43 | focal radius has to equal a constant . When we | |
| 09:46 | go through the math , that we will go through | |
| 09:47 | the next section . We're gonna pop out with the | |
| 09:49 | equations that we just talked about there . Now , | |
| 09:51 | what I want to do for the rest of the | |
| 09:53 | lesson is I want to show you that the equation | |
| 09:55 | of a circle as a starting point can be shown | |
| 09:58 | to look like the equation of an ellipse and how | |
| 10:00 | they're very similarly related . We're also going to get | |
| 10:02 | some practice with graphic . So what do I want | |
| 10:04 | to do next ? I want to kind of put | |
| 10:07 | a little divider line right under here because we're going | |
| 10:10 | to start kind of a new a new little topic | |
| 10:13 | . We're going to talk about the equation of a | |
| 10:15 | circle . So let's say we have a circle , | |
| 10:19 | right ? We've done circles before , so let's take | |
| 10:21 | the equation of a circle , X squared plus y | |
| 10:23 | square is equal to three squared . Now this is | |
| 10:27 | an equation of a circle . Notice what it looks | |
| 10:29 | like , X squared plus y squared is some number | |
| 10:31 | squared . It doesn't really look like the equation of | |
| 10:34 | an ellipse because there's a one over here , so | |
| 10:37 | that's different . Plus there's these things you're dividing by | |
| 10:39 | , these are all different , but you do have | |
| 10:41 | an X squared and y squared in both cases . | |
| 10:43 | Alright , So and you do have a plus sign | |
| 10:45 | notice there's a plus sign here . Uh there's a | |
| 10:48 | plus sign here . So what I want to show | |
| 10:50 | you is that this can be shown to look like | |
| 10:53 | the equation of the lips , which I just wrote | |
| 10:55 | down over there by the following thing . Let's divide | |
| 10:58 | both sides by what is on the right hand side | |
| 11:00 | . So if I do that , I'm gonna have | |
| 11:03 | X squared plus y squared , I'll divide the entire | |
| 11:06 | left side by what I have on the right three | |
| 11:08 | square . And I know you know that that's nine | |
| 11:10 | , but let's um let's go with it . Um | |
| 11:14 | And on the right hand side you have three squared | |
| 11:16 | over three squared . And by the way , what | |
| 11:19 | does this equation look like ? What does the circle | |
| 11:22 | look like ? We've done this so many times , | |
| 11:23 | I almost forgot to graphic . But basically this is | |
| 11:26 | an xy graph . The right hand side is the | |
| 11:28 | radius squared . So basically what you do is you | |
| 11:32 | say what you have is this is x squared plus | |
| 11:35 | y squared on the right is going to be equal | |
| 11:38 | to nine which is three squared . So basically the | |
| 11:40 | radius is squared . On the right hand side , | |
| 11:43 | we have +123123 12 here is negative three and then | |
| 11:49 | 12 here is negative three here . And you should | |
| 11:51 | all know because we've done this so many times that | |
| 11:53 | the equation of this circle looks something like this , | |
| 11:56 | it's not a perfect circle . Um But you get | |
| 11:59 | the idea basically , it's a circle centered at the | |
| 12:01 | origin because there's no shifted , no shifting inside of | |
| 12:05 | these X and Y variables there . And the radius | |
| 12:07 | is just whatever is on the right hand side , | |
| 12:09 | square root of it , which is squared of nine | |
| 12:11 | , which is three . So it crosses notice that | |
| 12:13 | three distance units away all the way around . That's | |
| 12:16 | what the thing looks like . So it's very symmetrical | |
| 12:19 | . All right , So keep that in the back | |
| 12:20 | of your mind . Now , if we take this | |
| 12:22 | circle , divide by three squared divided by three squared | |
| 12:25 | . On the left hand side . When you have | |
| 12:27 | this guy , you can write this as X squared | |
| 12:31 | over three squared plus Y squared over three squared is | |
| 12:38 | equal to one . Why equal to one ? Because | |
| 12:40 | three squared over three squared is just the number one | |
| 12:43 | . And so when I have this here , I | |
| 12:44 | can break it apart . It's like if you think | |
| 12:46 | about this is a common denominator , I could add | |
| 12:48 | these back together . The common denominator would just be | |
| 12:50 | the three squared . The top would be this , | |
| 12:52 | this numerator plus this one . So getting a little | |
| 12:56 | bit closer here , notice what this looks like . | |
| 12:59 | Okay , I can leave it like this , but | |
| 13:00 | I can also write it if I want to is | |
| 13:02 | X squared over nine plus Y squared over nine is | |
| 13:06 | equal to one . So notice that this equation of | |
| 13:09 | a circle which looks very different than the lips actually | |
| 13:13 | can be shown to have the exact same form of | |
| 13:16 | an ellipse . It's x squared over a number plus | |
| 13:19 | Y squared over a number equals one . And that's | |
| 13:22 | exactly what I told you . It would look like | |
| 13:24 | X squared over a number plus Y squared over a | |
| 13:26 | number equals one . So all ellipses all the circles | |
| 13:31 | I should say can be shown to have the same | |
| 13:33 | form of an equation as the ellipse . The only | |
| 13:36 | difference is noticed , this thing is oblong , it's | |
| 13:38 | stretched in one direction , whatever the number A is | |
| 13:42 | . If it's like a really big number is going | |
| 13:43 | to stretch that ellipse out really far because A is | |
| 13:45 | where it crosses here . If B is really small | |
| 13:48 | , you're gonna have a really thin ellipse . But | |
| 13:51 | if this number under why happens to be bigger , | |
| 13:53 | then it's gonna stretch it in the Y . Direction | |
| 13:55 | . If the number under the X . Is small | |
| 13:57 | , it's gonna it's gonna shrink it in the X | |
| 13:59 | . Direction . So the A . And the B | |
| 14:01 | . Numbers in the equation of the lips determine how | |
| 14:04 | the thing is stretched right ? But if those A | |
| 14:07 | . And B . Numbers in this equation are is | |
| 14:09 | the same , then that means they're not stretched any | |
| 14:12 | different . They're stretched the same in both the X | |
| 14:14 | . And the Y . Direction . So if these | |
| 14:15 | numbers end up being the same what you end up | |
| 14:18 | with is a circle . And we show that because | |
| 14:21 | the equation of a circle when you divide through it | |
| 14:22 | has the exact form as the equation of an ellipse | |
| 14:25 | with the same numbers on the bottom . So this | |
| 14:28 | is where it crosses in the X . Direction at | |
| 14:31 | three distance units away . This is where it crosses | |
| 14:33 | in the Y . Direction at three distance units away | |
| 14:36 | . Yeah . So get that in your mind that | |
| 14:38 | the equation of an ellipse and the equation of a | |
| 14:40 | circle are basically the same . It's just that in | |
| 14:42 | the lips , those denominators and those fractions are different | |
| 14:45 | numbers . So what I would like to do now | |
| 14:47 | is now that you know that the equation of a | |
| 14:49 | circle can be made to look like this , let's | |
| 14:52 | start playing around with it . Let's stretch this circle | |
| 14:55 | in the X . Direction , let's leave the Y | |
| 14:58 | . Direction alone . Let's leave it at three distance | |
| 15:00 | units away . But let's stretch it in the X | |
| 15:02 | . Direction . So what I mean by that is | |
| 15:04 | let's go and draw another circle . I'm sorry , | |
| 15:07 | another ellipse over here and basically it's going to be | |
| 15:10 | a stretched version of the one that we have here | |
| 15:13 | . The one that we have here crossed three distance | |
| 15:15 | units in the Y direction . But I'm saying let's | |
| 15:19 | cross and stretch it out five units in the X | |
| 15:22 | . direction and five distance units in that direction as | |
| 15:26 | well . So you all know that this is three | |
| 15:29 | , this is negative three , this is five negative | |
| 15:31 | five . So what I want to do is draw | |
| 15:32 | any lips that looks it stretched exactly the same as | |
| 15:36 | that circle is in the Y direction . However , | |
| 15:39 | in the X direction it's stretched out . Whoops , | |
| 15:42 | that's that one up like that , something like that | |
| 15:46 | , it looks like a football doesn't really quite look | |
| 15:48 | right . But you see what I'm saying , I'm | |
| 15:49 | basically stretching it out like this . What would the | |
| 15:52 | equation of that ellipse look like ? What I'm saying | |
| 15:54 | is the equation of the ellipse is basically the numbers | |
| 16:01 | in the denominator of the X . Squared term is | |
| 16:04 | how much it stretches in the X direction . The | |
| 16:06 | denominator term for the Y determines how where it crosses | |
| 16:09 | in the Y . Direction for a circle . These | |
| 16:11 | numbers are the same , so it's a circular shape | |
| 16:13 | . But if we stretch it five distance units out | |
| 16:16 | and keep this at three distance units , what would | |
| 16:19 | that equation look like ? Because this one was X | |
| 16:23 | squared over nine . Why squared over nine is equal | |
| 16:25 | to one ? Well , for the wide square term | |
| 16:30 | it would be the same . Why squared over three | |
| 16:33 | squared ? And you're gonna have the equal one . | |
| 16:35 | Why ? Because we know it crosses it three and | |
| 16:37 | negative three . But for the X square term it | |
| 16:40 | doesn't cross it three , it crosses it plus minus | |
| 16:42 | five . So this has to be five squared . | |
| 16:45 | So this without any proof is the I haven't proved | |
| 16:48 | it to . You haven't haven't made a table of | |
| 16:50 | values . I'm just saying we know how to graph | |
| 16:51 | circles . We know the equation of a circle can | |
| 16:54 | be shown to be the same as the equation of | |
| 16:55 | any lives . And by changing these denominators , it | |
| 16:58 | just changes where things cross . That's it . So | |
| 17:00 | this crosses at plus -5 . This crosses at plus | |
| 17:03 | -3 . We took the same circle , we stretched | |
| 17:05 | it out . Okay , which means that this equation | |
| 17:09 | really is x squared over 25 plus Y squared over | |
| 17:15 | nine is equal to one , compare that to this | |
| 17:17 | one . This is X squared over nine , Y | |
| 17:20 | squared over nine . If one this is x squared | |
| 17:22 | over 25 Y squared over nine is equal to one | |
| 17:25 | . Everything from here on was the same in the | |
| 17:27 | equation , we just changed this and that's why it | |
| 17:29 | stretched it out . Now notice that this circle has | |
| 17:33 | a special point in the center , which we call | |
| 17:35 | the center , but ellipses have focuses , right ? | |
| 17:38 | Which are because we stretch it , we kind of | |
| 17:40 | take the center points and move them out . So | |
| 17:42 | I'm not going to calculate where the focus , number | |
| 17:45 | one and number two is in this guy here . | |
| 17:47 | But the focus is gonna be somewhere around here and | |
| 17:50 | somewhere around this year . Somewhere around here . I | |
| 17:55 | mean , I don't know , it could be a | |
| 17:56 | little bit closer , a little bit farther away as | |
| 17:57 | we do more problems , we're going to calculate exactly | |
| 18:00 | where the focus is . I'm not trying to calculate | |
| 18:03 | that . Now , I'm just trying to show you | |
| 18:05 | that when you stretch a circle out the special point | |
| 18:08 | , which is called the center , it splits into | |
| 18:10 | two focuses one On the left and one on the | |
| 18:13 | right . Yeah . Now we talked about this a | |
| 18:17 | minute ago , the long side of the of the | |
| 18:21 | ellipse is called the major axis . The short side | |
| 18:24 | of the ellipse is called the minor axis . So | |
| 18:25 | if you're ever asked on a test which accesses the | |
| 18:29 | major axis or whatever the X axis in this case | |
| 18:32 | is the major axis and the y axis is the | |
| 18:37 | minor axis , right ? Because the uh the major | |
| 18:44 | axis is basically we're stretched in the X . Direction | |
| 18:48 | longer in the Y direction . We're not stretched as | |
| 18:50 | much . So it's the smaller one , it's called | |
| 18:52 | the minor axis . Okay . One more thing I | |
| 18:55 | want to kind of talk to you about before I | |
| 18:57 | go on is that you can kind of think of | |
| 19:00 | ellipses , you know we have a circle , you | |
| 19:01 | only have one radius from the center point . But | |
| 19:04 | for any lips you can kind of think of each | |
| 19:07 | of these distances being different . Radius is different radi | |
| 19:09 | I so you almost have like two radius . The | |
| 19:12 | radius from this focus on the radius from this focus | |
| 19:15 | . Okay , so you can kind of think of | |
| 19:17 | these numbers underneath as being the radius in the X | |
| 19:20 | . Direction and the radius in the Y direction for | |
| 19:22 | the circle it's the same radius but for an ellipse | |
| 19:25 | , the radius in the X direction is five And | |
| 19:28 | the radius in the Y Direction is three . And | |
| 19:30 | that's just another way of thinking about circles happen to | |
| 19:32 | be a perfectly round , symmetrical version of an ellipse | |
| 19:36 | . So that's why we don't talk about radius one | |
| 19:38 | radius to for a circle . But for ellipses we | |
| 19:40 | do because of because of that , let me make | |
| 19:42 | sure I've caught up here . The faux side , | |
| 19:44 | the focus is always on the long direction of the | |
| 19:47 | ellipse . They're never on the short side , they're | |
| 19:49 | always on the long direction . We have a major | |
| 19:51 | access . We have a minor axis . And the | |
| 19:55 | takeaway here , is that an ellipse can be thought | |
| 19:57 | of as a stretched circle ? That has kind of | |
| 19:59 | two radius is one radi I in the X . | |
| 20:01 | Direction , A different radio icon of in the Y | |
| 20:03 | . Direction . Now what we did already is we | |
| 20:06 | took this circle , we stretched it in the X | |
| 20:08 | . Direction And we arrived at this . Now what | |
| 20:11 | I want to do is take this circle and now | |
| 20:13 | stretch it in the other direction . I'm gonna leave | |
| 20:15 | this alone . I wanted to cross it plus or | |
| 20:17 | -3 along the X . Direction . But I want | |
| 20:19 | to stretch it in the Y . Direction and then | |
| 20:22 | we're gonna take a look at what the equation looks | |
| 20:26 | like in that case . So let's go ahead and | |
| 20:29 | draw and access to . So we're all on the | |
| 20:32 | same page and we understand what we're doing . So | |
| 20:37 | what we're saying is we want to keep the original | |
| 20:39 | radius in the X . direction being at plus or | |
| 20:41 | -3 . But in the Y direction let's go up | |
| 20:44 | to let's make it big 1234567 So this is positive | |
| 20:49 | 71234567 This is negative seven , something like this . | |
| 20:54 | And so then if this is any lips that crosses | |
| 20:57 | at seven here and then goes down and crosses at | |
| 21:00 | three and then it goes down here , crosses at | |
| 21:02 | seven , crosses up three and then crosses up at | |
| 21:04 | seven . You see what I mean ? It's almost | |
| 21:06 | like there's two different radius is here , one in | |
| 21:08 | the X . Direction , one in the Y direction | |
| 21:10 | , the radius in the X direction is three . | |
| 21:12 | The radius in the Y direction is seven , right | |
| 21:15 | ? So then it's very easy to write down the | |
| 21:17 | equation of the ellipse because all you do is you | |
| 21:19 | say it's got to be X squared over something . | |
| 21:21 | So you look at how far is it stretched in | |
| 21:23 | the X direction , where does it cross ? It | |
| 21:25 | has to be over three squared . The crossing point | |
| 21:29 | on the y axis is basically what is its radius | |
| 21:31 | and kind of in the Y direction , which is | |
| 21:33 | seven squared And you just set it equal to one | |
| 21:37 | . All right ? So when you do the squaring | |
| 21:38 | , what you get is x squared over nine Plus | |
| 21:41 | y squared over 49 is the little one . And | |
| 21:44 | this would be the equation of this ellipse . It's | |
| 21:46 | centered at the origin because there's no shifting in the | |
| 21:49 | X and Y direction . Only the numbers in the | |
| 21:51 | bottom determine how the thing is stretched out . And | |
| 21:55 | you can compare it to the equation of a circle | |
| 21:57 | when we did all this math and we got down | |
| 21:59 | to this point . This is how much we're stretching | |
| 22:02 | , sort of the radius in the X Direction . | |
| 22:03 | This is sort of the radius in the Y direction | |
| 22:05 | . So we kept the X direction exactly the same | |
| 22:08 | . In this case we only stretched the Y direction | |
| 22:10 | and that's why it stretches up like this . And | |
| 22:13 | then you can say that the the why direction is | |
| 22:18 | called in this case the major access and this is | |
| 22:24 | the minor axis . And by the way I mentioned | |
| 22:28 | this in the previous lesson , but one of the | |
| 22:30 | coolest things about ellipses is all of the orbits of | |
| 22:33 | the planets in our solar system . You know , | |
| 22:35 | not just planets , but satellites , moons , anything | |
| 22:38 | orbiting our sun that's on a trajectory . That's not | |
| 22:41 | an escape trajectory . Just going around around . It's | |
| 22:44 | always there always elliptical . And what is special about | |
| 22:47 | these focuses these folks i is that for instance , | |
| 22:49 | if this were the orbit of Pluto , let's say | |
| 22:52 | the sun would be at one of the focus is | |
| 22:54 | one of the folks I of that ellipse . So | |
| 22:57 | all of the planets going around the sun are going | |
| 22:59 | in elliptical orbits with the sun being at the at | |
| 23:02 | the point we call focus number one . Focus number | |
| 23:05 | two way out in space . Doesn't have any any | |
| 23:08 | any meaning really , it's just empty . But the | |
| 23:10 | focus , the thing you're orbiting is actually at one | |
| 23:13 | of the , one of the fallacy of the ellipse | |
| 23:15 | that you have . Okay , so let me make | |
| 23:17 | sure I've got everything straight . We started with a | |
| 23:20 | circle . We showed that it really can be shown | |
| 23:22 | to look like an equation of the lips . We | |
| 23:24 | stretch in the X . Direction . And that means | |
| 23:26 | that we uh we have the number getting bigger down | |
| 23:31 | below the X . Squared term . So that stretches | |
| 23:33 | in the X direction when we have the number getting | |
| 23:35 | bigger in the Y direction . That we showed that | |
| 23:38 | the that the uh thing is stretched in the Y | |
| 23:42 | . Direction . And now what I want to do | |
| 23:44 | is talk a little bit more about these focuses because | |
| 23:46 | I didn't draw it here . But somewhere around here | |
| 23:48 | will be a focus and somewhere around here will be | |
| 23:50 | a focus . It always lies along the long end | |
| 23:53 | of the ellipse . Somewhere there's a focus one and | |
| 23:55 | a focus to . Now what I would like to | |
| 23:58 | do is talk a little bit more concretely about why | |
| 24:02 | an ellipse if you start to squish it down into | |
| 24:04 | a circular shape , really does become the equation of | |
| 24:06 | a circle . We've kind of shown that here , | |
| 24:08 | but I want to do one more little kind of | |
| 24:10 | thought experiment with you to make sure we're all on | |
| 24:12 | the same page , is it's kind of cool . | |
| 24:14 | All right , so let's think of a really long | |
| 24:17 | skinny lips . Something really , really long and skinny | |
| 24:20 | . If it's really , really long and skinny , | |
| 24:22 | then the number that appears under the X . Direction | |
| 24:24 | has to be really , really big because that means | |
| 24:26 | it's gonna cross really far away . The number under | |
| 24:29 | Y . When you take the square root of it | |
| 24:31 | , that's where it's gonna cross in the Y direction | |
| 24:33 | . So , if I want a really , really | |
| 24:35 | skinny lips , let's do that . Let's draw a | |
| 24:38 | long and skinny lips . Let's say I have any | |
| 24:41 | lips that looks something like this was way out here | |
| 24:44 | crosses way out here , something like this , that's | |
| 24:46 | not perfect at all . But let's say it crosses | |
| 24:50 | over here at negative 10 and positive 10 . And | |
| 24:53 | let's say it crosses at negative three and positive three | |
| 24:56 | . So it's really long and stretched out . What | |
| 24:57 | would the equation of that lips look like ? Well | |
| 25:00 | follows the same format centered at 00 So it's X | |
| 25:03 | squared divided by whatever the crossing point is squared , | |
| 25:09 | right ? And then plus Y squared divided by wherever | |
| 25:15 | the white crossing point is squared is equal to one | |
| 25:18 | . This is exactly what I have written on this | |
| 25:19 | board here , that whatever here is labeled as a | |
| 25:23 | square , but the crossing point is not a squared | |
| 25:25 | , it's basically the square root of that . Say | |
| 25:27 | whatever is on the bottom , you take the square | |
| 25:28 | root of it . And that is where it crosses | |
| 25:30 | , saying for the Y value . So here take | |
| 25:33 | the square root of this , I get 10 , | |
| 25:35 | take the square root of this , I get three | |
| 25:36 | . And that's how I know that that's the equation | |
| 25:38 | of this ellipse . So another way to write that | |
| 25:42 | would be X squared over 100 plus Y squared over | |
| 25:47 | nine is equal to one . This is the equation | |
| 25:50 | of the sea lips . Now , somewhere along the | |
| 25:53 | long direction is a focus number one . In a | |
| 25:56 | focus number two . So let's put it here , | |
| 25:58 | I don't know exactly where there are ways . We | |
| 25:59 | haven't calculated it , but here's the focus . We'll | |
| 26:01 | call it F one and here's the focus . We | |
| 26:04 | call it F two . Now , what I would | |
| 26:06 | like to explain or explore with you is as we | |
| 26:09 | squish this ellipse and get it more and more circular | |
| 26:12 | . Of course the boundary will get more circular , | |
| 26:14 | but these focuses will get closer and closer together as | |
| 26:17 | well and eventually we're gonna get to a circle . | |
| 26:20 | And when we get to that point then the equation | |
| 26:22 | of this ellipse should look like an equation of a | |
| 26:24 | circle and we're gonna show that right now . So | |
| 26:27 | let's do that . Let's draw another one of these | |
| 26:29 | guys . Let's make a slightly less elongated version of | |
| 26:34 | this . So here , I want to cross at | |
| 26:37 | let's say six and negative six . Probably not symmetrical | |
| 26:41 | . So let's do something like this , negative six | |
| 26:44 | , something like this . And let's go and leave | |
| 26:47 | it at plus or -3 . All I'm doing , | |
| 26:51 | I'm keeping this the same , but I'm showing you | |
| 26:53 | that we're basically making this thing slightly more circular like | |
| 26:58 | this . Now , somewhere in here is going to | |
| 27:01 | be a focus . Let's call it right here and | |
| 27:04 | right here , call this F one F two . | |
| 27:06 | But notice as we squished it . This focus has | |
| 27:08 | gotten closer now they're closer together . I'm not putting | |
| 27:11 | the actual numbers here , but you know that they | |
| 27:13 | have to move closer together . What would be the | |
| 27:15 | equation of this ellipse ? Well , it would be | |
| 27:19 | X squared over wherever it crosses squared , so six | |
| 27:23 | squared is 36 . Why squared take the crossing point | |
| 27:28 | square ? It ? Which is nine is equal to | |
| 27:31 | one . So this is the equation of this ellipse | |
| 27:35 | . So notice that this is 109 . So you | |
| 27:38 | can tell us really , really stretched . This is | |
| 27:40 | 36 and nine . It's not quite a stretch , | |
| 27:42 | but it's still pretty darn stretched . Now let's do | |
| 27:45 | another one and let's keep these guys , let's make | |
| 27:49 | them a little bit closer together . Let's make this | |
| 27:53 | over here now , at four And -4 . And | |
| 27:57 | then this will be at three and this will be | |
| 27:59 | at -3 . So this is going to be even | |
| 28:01 | more circular . This is not a good lips . | |
| 28:04 | I'm sorry about that actually , really terrible lumpy ellipse | |
| 28:06 | . But you can see it's getting more and more | |
| 28:07 | circular . And then of course , as we squish | |
| 28:10 | it closer together , the focus must be getting closer | |
| 28:12 | and closer together . And so what's the equation of | |
| 28:15 | the Philips X squared over the crossing point ? In | |
| 28:18 | the X direction squared , four square to 16 plus | |
| 28:22 | y squared divided by the three squared which is nine | |
| 28:26 | is equal to one . So you see what's happening | |
| 28:28 | is I get squish it closer and closer . These | |
| 28:30 | numbers are getting closer and closer to one another . | |
| 28:32 | So now what I'd like to do is I would | |
| 28:35 | like to get it really , really , really close | |
| 28:37 | to being a circle . Really , really , really | |
| 28:39 | close to being circular . So let's take it like | |
| 28:42 | this . Yeah , this is X . This is | |
| 28:45 | why . And let's say this is three , this | |
| 28:50 | is negative three . Now this would be three and | |
| 28:53 | they go to market three , right under it . | |
| 28:55 | Here , here's a three right here . So this | |
| 28:57 | is supposed to be a circle . But what I'm | |
| 28:58 | gonna do is I'm gonna make it a little bit | |
| 29:01 | just a tiny bit oblong . So it's going to | |
| 29:04 | cross it three , but it's not gonna cross at | |
| 29:06 | three here , it's gonna go just beyond it and | |
| 29:08 | across it three here it's gonna cross just beyond it | |
| 29:10 | . So it's very slightly egg shaped and I probably | |
| 29:14 | should have drew it even closer in . It looks | |
| 29:15 | a little bit like the one above , but you | |
| 29:17 | see if it were a circle , it would cross | |
| 29:19 | everywhere at three . Here , I'm crossing very slightly | |
| 29:21 | out . And because of that , the focus is | |
| 29:23 | now are really , really close to being on top | |
| 29:26 | of one another . What's the equation of this ellipse | |
| 29:28 | X . Squared over ? Let's just say for giggles | |
| 29:32 | that this crosses at 3.1 . So I'll have to | |
| 29:35 | square the 3.1 . The y squared will be three | |
| 29:39 | and positive negative three . So to be three squared | |
| 29:43 | . So what do I get for an equation here | |
| 29:45 | , X squared . What is 3.1 squared ? I | |
| 29:48 | get 9.61 just plus Y squared over nine is equal | |
| 29:53 | to one . So you see when the ellipse gets | |
| 29:54 | really , really close to circular , the bottom numbers | |
| 29:57 | are becoming basically equal . Eventually I'm gonna get to | |
| 30:00 | the point where they become um when they become the | |
| 30:04 | same . So let's do that right below here . | |
| 30:08 | And so here is three . Here is three . | |
| 30:12 | Here is negative three . Here is negative three . | |
| 30:15 | It's not gonna be perfect but I'm gonna try to | |
| 30:18 | draw a circle here , it's a lumpy circle . | |
| 30:23 | I know it's not perfect . But what's this equation | |
| 30:25 | looks like ? Well it's gonna look like X squared | |
| 30:28 | over three squared . I don't want to write this | |
| 30:30 | three squared , which will be nine plus y squared | |
| 30:34 | over three squared equals nine equal to one . Now | |
| 30:37 | this is the equation of the quote unquote ellipse . | |
| 30:39 | But now notice what's happened . Is that focus number | |
| 30:42 | one ? And Focus number two are so close to | |
| 30:44 | one another . They're essentially right on top of each | |
| 30:46 | other , right at the center . So both of | |
| 30:47 | the focus is kind of get closer to one another | |
| 30:49 | . This is the equation of the ellipse here . | |
| 30:52 | Now , what do you think is gonna happen if | |
| 30:54 | I multiply this equation by nine ? If I take | |
| 30:58 | this equation , uh let me kind of space this | |
| 31:01 | out here and multiply by nine . Sorry , I'm | |
| 31:04 | running out of space here , multiplied by nine here | |
| 31:06 | . What am I gonna get ? I'm going to | |
| 31:08 | distribute the nine into both of these . The nine | |
| 31:10 | is going to cancel with this . The nine is | |
| 31:12 | it distributes and will cancel with this one as well | |
| 31:14 | . So basically the nines are gonna disappear from the | |
| 31:17 | left . So what you're going to have is just | |
| 31:19 | X squared plus Y squared is equal to nine , | |
| 31:24 | which is the equation of a circle . Because what | |
| 31:26 | this is is X squared plus y squared is equal | |
| 31:30 | to the radius square , it's a radius of three | |
| 31:33 | . And that is the main idea of what I'm | |
| 31:36 | trying to get across to you in this lesson is | |
| 31:39 | that I can start with a really stretch your lips | |
| 31:41 | with two really far away . FocA folks , I | |
| 31:44 | right I can write the equation of the lips . | |
| 31:46 | These numbers are really different because it stretched really far | |
| 31:48 | in one direction . But as I bring them closer | |
| 31:51 | and closer than numbers because the crossing points are getting | |
| 31:53 | closer than numbers are getting closer and closer until finally | |
| 31:56 | I'm just on the other side of three and they're | |
| 31:58 | really close together . But if I ever make them | |
| 32:01 | exactly equal to one another , then it becomes . | |
| 32:03 | Even though you can write it as the equation of | |
| 32:05 | an ellipse , you can write it and multiply through | |
| 32:08 | and get it to look like what we think of | |
| 32:10 | as the equation of a circle with both Fosse lining | |
| 32:14 | up on top of one another . That's the most | |
| 32:17 | important thing to come up with . So we started | |
| 32:19 | the lesson by talking about the definition of the ellipse | |
| 32:24 | and we talked about the fact that the distance from | |
| 32:27 | the focus to any point on the ellipse , plus | |
| 32:30 | that distance to the other focus has to be a | |
| 32:32 | constant for every one of these points along the boundary | |
| 32:35 | . And then we showed that you can start with | |
| 32:37 | the equation of a circle , you can start stretching | |
| 32:39 | it and then you end up with what we call | |
| 32:41 | the equation of the ellipse . Now , what we're | |
| 32:43 | gonna be doing as we solve future problems . We're | |
| 32:45 | gonna use use these diagrams a lot . But the | |
| 32:48 | bottom line is , the crossing points for the ellipse | |
| 32:50 | are very easy because you read them right off , | |
| 32:53 | you have to take the square root of course , | |
| 32:55 | down below here , but it's basically right there in | |
| 32:58 | the equation and it's always gonna be centered at 00 | |
| 33:02 | unless you have some shifting going on , which we're | |
| 33:04 | going to get too much later . We're not moving | |
| 33:05 | the ellipses around yet , we're going to get there | |
| 33:08 | . So make sure you understand this . I encourage | |
| 33:10 | you grab a sheet of paper , try to take | |
| 33:13 | a lot of these . Take some of these ellipses | |
| 33:15 | that I had here and graph them yourself . Make | |
| 33:18 | sure you understand what's going on and you understand the | |
| 33:20 | concept that the folks I are getting closer and closer | |
| 33:22 | and closer together . Eventually , when you get to | |
| 33:24 | a circle , those two folks , I line up | |
| 33:26 | right on top of each other and the equation of | |
| 33:28 | that ellipse in this case then becomes the equation of | |
| 33:30 | a circle . So now you see circles are just | |
| 33:33 | a special case of the general thing that we call | |
| 33:37 | an ellipse . |
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