The Best Of Writing an Ellipse in Conic Sections

in this video I went through my back catalog and found the best videos on writing the equation of an ellipse when given certain pieces of information like the center the fosi the vertices the major vertices the cover vertices the major axis the minor axis as well as eccentricity all of these examples are provided inside of this video and if you go through this video step by step following my process you are going to have a better understanding of how to write the equation of an ellipse I hope you enjoy so ladies and gentlemen um hey there's no gentleman here today look at that you know I told you guys about my first job I'll say quick my first interview yeah it's on video it's my first interview though was in a school in Jacksonville and before I even hired before I even accepted on the Mandarin they asked me the only position that hel open was for a math teacher but it was for like the early like pregnancy so it's like for all the girls in the city like that would you know they ever having a baby that I would be like their math teacher so you know just fresh out of college just fresh out of college and they say oh you know do you think that would be a good fit I'm just like sure classroom full of ladies the women that are all ready to have a baby that sounds like a good idea Unfortunately they didn't give me a call back so I wound up here but are you now yeah I be a fun intro I guess anyways um so we're giving the verd -6 or plus or- 6 comma 0 and the fze is plus or- 2 comma 0 and I'm asking you to write the equation so ladies andent remember the parabis when I said I was giv the information the first thing I said is just you you there's only two um equations it's either opens up or down or left or right right the only way to figure that out is to plot the information and see what's going to make sense exact same thing with this so let's just plot the information and see what makes sense so our vertices are Plus orus six so 1 2 3 4 5 6 1 2 3 4 5 6 folky plus 22 so we have FY FY verticy verticy now remember ladies and gentlemen there's only two ellipses that we're dealing with one that's vertical and one that's horizontal now just out of I don't know a random guess if you were to say this is a these are the points of one of these ellipses do you would you recognize that it to be a ellipse where your major axis is vertical or where your major axis is horizontal horizontal right it's pretty obvious you can notice that you know that the vertices the fosi all have to be on the major axis and since this major axis is going horizontally then you have a horizontal ellipse right so let's just eliminate this right here all right now remember when we're looking at this if I give you the very veres and the fosi remember the vertices and the fosi are plus or minus the same distance from your center right if I say your verticy is six units to the right then it was also six units to the left correct so therefore what is going to be our Center because we need to figure out what this Center is it's always in this case the center is 0 0 yes no oh the problem actually said oh that's nice you're right I just say the center's of the ordin so that's good good right you're right cuz all the way up to 35 then they change it to n okay so we have the center is 0 0 so that helps us because now when we write our equation we know our formal was x - H for a horizontal then it' be a 2 plus y - k 2 over b^2 = 1 however do we have an h and and K yes there zero right so now we can just say X2 over a 2 + y^2 over b^ 2 equals 1 okay so now let's just go and see if we can figure out what our A and B are remember a is going to be this distance right doesn't matter if it's going to be your positive or your negative what is this distance what is this distance squared well the distance was six plus or minus 6 you square a negative 6 you square a positive six it doesn't matter you're going to get 36 36 good so now we have y^2 over um now we have y^2 over b^2 now our problem is we don't know what b^2 is right but the other information they gave us was they gave us C and we remember we talked about a and C are related to each other where we can say A2 = b^2 + c^2 so if I know A2 is 36 equal b^ 2 + c^ 2 so C is 2 so c^ 2 is going to be 4 - 4 I get 32 = b^2 Square TK Square TK B = plus or minus take the um simplify this into 4 radical 2 yes but we don't need to do that right because all they're asking us is what is b^ s so we can just leave it like that b^2 = 32 okay because remember you're looking for b s so we don't need to solve for b we just need to know what b s is so answer like standard form of like that graph where's B where's B yes no no no no no in the gra like the relationship B is your distance these are like if you call this like a major verticy call this like a minor verticy it's just it's where the it's like the minor vertices on the minor axis all right any last questions 33 so by plotting the information I go over negative 8 0 1 2 3 4 5 6 7 8 8 4 0 1 2 3 4 those are my two vertices now yes the two verticies are on the outside end of your ellipse and yes the uh the vertex is in the exact middle right because remember the distance from the vertex to a verticy is a and a major axis is 2 a right so it's just double that so yeah we can just look at this and say um you know you could count how large it is which would be what uh 12 and then divide that by two to find the distance of a which is six um or you can just count and find the middle but anyways I get it to be at I get my vertex to be at negative or at -20 would everybody agree with me on that so that's at -20 all right cool so now we get into the most important piece of information that I told you from the beginning we now have the two veres and the vertex that lie on an axis so we know that that axis must be the major axis right and what else lies on the major axis Garrett you have the vertices the vertex and the vertex vertices and the fosi right fosi folai whatever you want to call them so yes that's exactly right and we can show that by obviously looking at the fosi which is at -3 and 1 Z all right so now we see that this major axis is horizontal do you guys see all that important points lies on the horizontal axis so then it makes sense Aaron if you have to decide should it be a horizontal axis ellipse or a vertical major axis ellipse and you can see all right well if I if Mr mogan's asked me to graph what the equation would be then it's going to be very easy for me to see I need to use this equation so I'll write out the equation x - H well let's not write out the equation because I already have it written down let's write down the information we know we know that the vertex is at -20 so that's going to be x + 2^ 2ar plus y^2 = 1 we don't know it's because it's Yus 0 right which I can just write as Y is everybody okay with that sometimes I confuses um so now Shan if we know can we find the dis can we find the value of a Sean because we need to now now we need to figure out what A and B are right yes we need to figure out what A and B are a remember is the distance from the vertex to the versy so do we know one of those distances yes and what it it is what what is a what is the from a Vertex to a verticy here's the vertex and we want to go vertex to the verticy six right now here's the question though is it negative six or is it positive six but think about it it doesn't really matter in this case P was always the case of we always go from P was the distance from the um the vertex to the focus but P would also go in the opposite direction but yeah in this case guys we're squaring it right so it doesn't matter the value of a the absolute distance from a from the vertex to the vertices absolute distance is six so when I square that I get 36 and I know that that distance has to go under the X because that's what Caroline was talking about as far as when the or zoi was talking about or Michelle was talking about that when you have the larger number under the a that tells you it's elongating it horizontally all right so now we need to figure out B and we have an issue with B all right JD do you know the value of c one one and JD could find that value C by going from the vertex to the fosi that little distance right distance from the vertex to the fide was C so right which is equal to one now C is not in the equation so that's an issue correct yes C is not in the equation but e do you remember there was an equation that I provided that could help us with this do you remember what that equ equation on it's written on the board yeah it's right there c^2 = Aus b s yeah it's that one yeah yeah so if I have C and I have a is it possible for me to text my friend and find the value of B yeah okay it is possible so I can say c^2 = a^2 minus b s where c^2 is 1^2 = a 2 which is 36 x - b^2 all right and then I'll just add the b^2 to the other side and so b^2 equal 35 I'm not going to do the work because I'm kind of running out of whiteboard space but yeah you just subtract one on the other side too right so b^2 equal 35 so there you go that's it that's the equation all right oh and then obviously if we wanted to find so b squar square of b um if you wanted to find B you could take the square root of 35 that would be how far you'd go up there if you're going to graph it all right but we're not going to take the square root of 35 and do that yet not right now not for what I've been going over welcome ladies and gentlemen so what I like to do is show you how to determine um how to write the equation uh given the fosi and the co vertex of an ellipse so the first thing we want to do is identify you know do we have a major axis it's horizontal or vertical so to do that we need to at least right in the information that we have okay so the fosi which is at 44 and 414 so 1 2 3 4 one two three four and then at 414 four up four five six seven eight nine 10 11 12 13 14 okay so remember these are the fosi and then the covert one co-vertex is at 09 1 2 3 4 5 6 7 8 n Okay now what's nice about the information that we have is notice that the major axis what lies on the major axis is the vertices the fosi and the center so therefore I know that this has to be my uh major axis therefore my major axis is vertical since I know my major axis is vertical I'm going to write the equation x - h^ 2 + y - k a 2 equal 1 since it's vertical I'm going to have b^ 2 Under The X and A S under the Y I'll get to the A and the b in just a second um the next thing here is one cover verticy now notice that again when we're looking into trying to determine the center right and the center of an equation is in the form of HK well what's nice about the center is the center is the half distance between your two fosi as well as the half distance between your two co-vertices and the half distance between your two vertices which we don't have anything for yet um so if I wanted to find just the half distance of my two fosi um I can reason that my you know Center I probably say the half distance is going to be somewhere right on here now if since the covertex is 1 2 3 4 four units away from the center or at least the major axis it has to be four units to the other side one two three four on the other side all right um the next thing to also understand is that this is what we call the minor axis and the minor axis and the major axis have to be perpendicular to each other now let's get into some fun stuff the distance from the center to our co-vertex is what we call which we give the value of B the distance from the center to the um fosi or one of the um focuses is going to be C so we don't have C in an equation but we're going to need to know C to help us figure out what a is um the center we did determine is going to be at 4 comma 9 so I'll say Center is at 4 comma 9 that's the X and the y coordinate for my Center the value of B the distance from the center to one of my co- vertexes is going to be Z is going to be four so I'm sorry let's write out R so I can say B is equal to 4 and a is going to be equal to the distance um from my Center to my verticy which I don't have however there is a relationship of a b and c so we have c^2 is equal to a2+ I'm sorry A2 minus b^2 so c^2 uh which we know is the distance from the center to the fosi which is 1 2 3 4 5 so I have 5^ 2 is equal to A2 which we're trying to figure out minus b^ 2 which would be 4^ 2 all right remember C is the distance from the center to fosi so that becomes 25 is equal to a^ 2 - 16 now I'll add 16 add 16 and I get uh 41 is equal to A2 so therefore ladies and gentlemen um a would be the square < TK of 41 so we know b^2 is equal to 16 and A2 is equal to 41 so now we know what a squ is b^ squ is and our H and our K which is our Center we can now write the equation which would be x - 4 um divided by 16 plus y - 9 oh I'm sorry these are all squared / 41 equal 1 so there you go ladies and gentlemen that is how you write the equation of ellipse given your fosi and a covertex thanks welcome ladies and gentlemen what I'd like to do is show you how to write the equation uh of an ellipse um given the major axis is horizontal with the length of eight as well as the minor axis is vertical with the length of four so when we're writing an equation of ellipse there's two different equations one where the major axis is horizontal and one where the major axis is vertical and thankfully in this problem they say that it is a horizontal so therefore we know that a um a squ which a represents the distance from the center to the verticy um vertices is going to be um under the X since it's horizontal so therefore the standard equation that we're going to write our equation in is x - h^ 2 all over a^ 2 + y - k^ SAR all over B ^2 = 1 um now remember the Cent is represented by H and K so likely we already have that information right so that's good so remember it's X opposite of H though so here H is -2 so opposite of -2 would be positive2 so I write x + 2 squar plus y opposite of 3 would be -3 squar = 1 now the only problem is we don't know what a squ and we do not know what b^ squ is um but let's go ahead and graph a ellipse here and let's pretend here's the center here's verticy and here's verticy remember I said the distance from the center to a Vertex or one of the vertices is a well that distance has to be the same so therefore the distance from one verticy to the other verticy or one vertex to the other vertex is going to be or one ver uh I don't ah whatever um is going to be 2 a so the length of the major axis is equal to 2 a a so therefore I can say 2 a is = to 8 IDE 2 IDE 2 a = 4 well 4 squared is going to be 16 and then B that's going to be the length of the minor axis which again b b so I can say 2 B is equal to 4 IDE 2ide by 2 b = 2 and again 2^ squared will equal four so there you go ladies and gentlemen that is how you write the equation of the lips when given the length of the major and minor axis as well as the center thanks um what we have here ladies and gentlemen is we're given the fosi of 0 comma 0 and 0a 8 and we're also given a major axis length of 16 and what they're asking us to do is to find the standard form of the ellipse all right so whenever you're given just points and they're asking you to find the standard form yes uh I believe this is number 41 so gave in the book so what we're going to do for this problem is the first thing is ladies and gentlemen it's just plot the points right a nice little makeshift graph that you can and plot the points because the main important thing we want to do is do we have a ellipse that's going to have a major axis symmetry that's going to be horizontal or we going to have a major axis symmetry that's going to be vertical because if we can determine between those two that will tell us which formula we're going to want to use so what I have is I plot one F side 0 0 and the other one at 08 1 2 3 four five six seven eight that's up not eight over eight that's 1 2 3 four five six seven eight so we have two fos siiz right now knowing that your understanding of ellipses um remember our fos siid have to lie on our major axis our major axis right so therefore here is our major axis so therefore is our major axis going to be vertical or horizontal Al vertical right now it says the major axis length of is 16 um do you guys remember so now we have a an ellipse that looks something like this we know that the distance to our folky is C and the distance to our verticy is a right but the whole distance of the major axis is how far how long well here to here is a 2 it's 2 a right so if I want to figure out what 16 what a is I can just say 16 = 2 a divide by 2 a = 8 all right but here's where it gets a little point where is our center now the important thing is remember that the distance the absolute distance of C is where your folky or your folky on fi is um a distance away from your Center so therefore the center is directly in between your two FID so I'm given two F my two FID I know that my Center is exactly in the middle right it's not near one FID than the other one it's exactly in the middle so I look at this distance and I say 1 2 3 4 5 6 S8 so I can say that 8 equals 2 C therefore C = 4 so therefore if I know that Cal 4 all I need to do is travel four units from my fosi to find the center 1 2 3 four so we can now say that the center is at 0a 4 and if the center is at 0 comma 4 and my a is at eight that means I need to travel eight units from my Center to find each verticy so therefore my vertices are at 1 2 3 4 5 6 7 78 and 1 2 3 4 5 6 7 8 do that kind of makes sense what I have done so far okay now they're just asking us to find the standard form so we don't really need to find all those vertices but it was just kind of good idea for you guys to understand where each verticy is in comparison to finding the fux side and finding the center but the main important thing we need to figure out was what the center was because to write the standard form you have to know the center and we have to know the distance of a and C because in the standard form so now we have a vertical ellipse so we know that the formula we're going to use is x - H 2ar + y - K SAR / um b^2 / a 2 equals 1 right so what we're going to do is we're now going to plug in that information the one last piece of information we need to figure out though is we don't know what B is is doing but we now know what A and C are and we talked about the relationship between a and C it's going to be a squ = b^2 + c^2 right so therefore we could say 8^ 2 = b^2 + 42 so therefore we have 64 = b^2 + 16 - 16 and therefore we're going to have um 48 = b^2 take the root take the root um and that will be 4 three I'm sorry what am I doing we don't need to I keep on I'm in such a habit of rationalizing or simplifying my radicals we don't care what B is we just want to know what B squared is we're not trying to find um how wide our ellipse is so we just have have so now we know what b^2 is so to plug everything in h is going to be zero so we're just left with X2 plus y - 4 SAR all over b^2 is 48 all over a 2 which is 64 = 1 so there you go that's your s okay that make sense yes um I was conf it say we have a major AIS of 16 so that means that like the length obviously is 16 right so why would like the vertices is more than 16 is what I'm saying like you know what I'm saying like from verticy to verticy it's more than 16 16 it should be 16 is it 1 2 3 four five 6 7even 8 9 10 11 12 13 14 15 16 so yeah all I did was that's exactly what I did I found the center I took 16 I divided it by two and then from the center I added and subtracted um a and then you just added how far the center you just added C to the FY you know I mean yeah well what I did is I here I'm given the two fosi so what I did is I divided it by two to find the center and then once I found the center then I could move back up because the same thing the center evenly divides the distance of the fos siiz and the vertices so right the center is right in between all those points which all lie on the major axis all right um so when doing a problem like this ladies and gentlemen where they do not give you a formula the best thing I want you guys to do just like when we had the parabas and they gave us information the best thing I would ask you guys to do is just take a little separate sheet and just plot what information we have so it's just given us points right we need to obviously understand what those points are well let's just go and plot them so we say vertices are at 02 and and at 42 then we have the end points of the minor axis are at 23 and at 21 okay so remember ladies and gentlemen there's two types of vert or two types of ellipses we have we have a horizontal ellipse and our vertical ellipse remember that the larger length is what we call the major axis so we're given and what that's at the end of our major axi is what we call our vertices so here are our two vertices right that's our end point the vertices are at 02 and 4 comma 2 so if I had a parabola that's horizontal your vertices would look like this and if it was vertical the vertices would look like this so just by giving you that information implied in those points what type of ellipse do you think we have a hor one with a major axis symmetry that's horizontal or a major axis with horizontal or vertical horizontal horizontal horizontal right so automatically now that we know that it's horizontal let's go and write out the formula for it because they're asking us to find the standard form of the equation so we have x - H squar since it's horizontal a squ going below it plus x- K or sorry Yus K so it's horizontal you use that yes if you have if you at major axis is horizontal okay so we have that wait what about the major the major axis which is your a this right here okay if it's horizontal then that's one to use if it was vertical then your A and B would be swapped all right so do we know what a is or at least what a s is well remember a ladies and gentlemen is this distance since we're not dealing with the vertical we can erase this remember this distance from the center to the vertices is what we call a and notice this is also a so what is this distance which is this total distance here because that total distance is going to be what 2 a so let's count we can just simply just count what is the distance between our two vertices one 2 3 4 so we could say 2 a is equal to 4 so therefore a equals two right and therefore we could say A2 equal 4 and then what's important about that is if I know that a is two that means if I go to a verticy if I travel two units where do I go to the center which is going to be right around there all right not yet um but let's actually gra the vertices was the the ones Beyond let's graph this a little bit larger okay so now we're given also the distance of B so remember from this point from your minor axis from here to here is B and from here to here is B so therefore the total distance of the minor axis is what 2B so let's count how far how long is this distance 1 2 so we could say 2 B is equal to 2 therefore for b = 1 and we could say b^2 = 1 Yes No Maybe So So I guess could say b^2 is equal to 1 all right you do again so we figured out a was distance from here from here to here right yes B is the distance from here to here the minor axis so B is on your minor axis if a is from here to here B is from here to here so B is the minor axis the smaller axis okay so since from here to here here to here is one and from here to here is one so therefore we could say 2B which is double the distance is equal to two so therefore b equal 1 um so now we have a point for our our Center so now we can say we know b^2 is 1 a is 4 and we can say now our Center is 1 2 2 at 2 comma 2 so we know Center is 2A 2 b^2 = 1 A2 = 4 and do we need to figure out what C what C is where the f I are they're just asking us to find the standard equation do we care do we need to know what c is right now no no if they ask us what's the F side then yeah we had to figure out what C is but we don't need to figure C we already have all the information we need to create a formula now I'll just ask this question one more time because I always like to see here the answer make sure everybody understand it why don't I need to figure out what X and Y are I figured out what h k A and B are why do I not need to figure out what X and Y are not zero a Coe varable yes they are variables X and what do X and Y represent and Y oh intercepts I don't know yeah X and Y ladies and gentlemen represent all the points on the ellips right how many points are there on the lips infinite many right how many centers are there one how many vertices are there two right so those are those are finite points but X and Y is infinite right so that we're not that's why we're not solving for an X and Y Point those are just given into your formula so therefore let's just plug it in so we have our Center so it's going to be x - 2^ 2 over a 2 which is 4 + x - 2^ 2 over 1 which we don't really need to write but I guess itals one okay that's it welcome so um what I like to do is show you how to uh determine the end points uh or I'm sorry write the equation of an ellipse when given the end points of the major axis and the end points of the minor axis now remember the end points of the major axis are going to be our vertices and the end points of our minor axis are going to be our co-vertices now the reason why that's important is because remember the major axis if it's horizontal is going to produce one type of equation and if it's vertical is going to type a different type of equation um and that's really the different types is really where the values of A and B are going to be where a represents the distance from the center to A verticy and B represents the distance from a center to the co one of the co-vertices so what we want to do is first plot this information to be able to help us kind of determine first where is the center and two what are the values of A and B so I'm going to me my major end points are at two two and eight two 1 2 3 4 five six seven eight two so I'm going to label those as vertices vertices and then my major extra end points are at 5 3 1 2 3 4 five 1 two 3 and at five one cover verticy cover verticy so you can see in this case that my major axis the axis um the line between my vertices is horizontal and my Axis between my cover vertices is vertical so therefore the formula that I'm going to use is x - h^ 2 + y - k^ 2 equals 1 where since it's horizontal my a squ is under my X and my b^2 is going to be under my y so now what I need to do to be able to determine um uh what my value of a and b are I first need to identify where the center is and the center is going to be where my vertices and my co-vertices um the axes the minor and the major axis intersect it's also half the distance between the two vertices and half the distance between two co-vertices which is at 1 2 3 4 5 1 2 so you can say that the center is at 5 comma 2 um so here is my center now remember the distance from the center to the verticy is a and the distance from the center to the co- verticy is B so all I simply need to do is say all right so from Center to a verticy is 1 2 3 and from Center to the cover verticy is one so now I'm going to enter the information so I say a = 3 and b = 1 now to solve for my equation I have x - H uh which is my X opposite of H which is my H is 5 squar + y opposite of K which is 2 s = 1 / a 2 which is 3^2 which is 9 / b^ 2 which which is one um one so there you go ladies and gentlemen that is how you write the equation of the ellipse when given the end points of your major and minor axis thanks as long as you guys know what the formula for eccentricity is which eal C A which is not the hardest thing for you guys to memorize right you guys have memorize a lot of like crazy stuff Ecentric C over a pretty sure you can say it to yourself five times and you'll have it memorized so that's nice because I now know 2 is C and and three is a well guys to write an equation all we need is the center which they already gave us and we need to know a b and the center and then also we need to know where our major axis is which they tell us it's a horizontal major axis right so this is like literally they're just giving us all the information so we know that our formula is going to be x - h^2 all over a^2 because this one has a horizontal major axis so that's our equation the only thing we need to make sure we know is what is our a or I'm sorry our B so again we go to our formula c^2 = a^2 minus b^2 c^2 is 4 = a 2 which is 9 - b^ 2 so 5 equals b^2 well guys I have my H and my K I have my b s my a is three so if a is equal to three that means a squ is equal to 9 so look it I got everything I need b s which is done