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# Intro to ellipses

## Video transcript

in the last video we learned a little bit about the circle and the circle is really just a special case of an ellipse and it's a special case because then in a circle you're always an equal distant away an equal distance away from the center of the circle while in an ellipse you're the the distance from the center of the circle is always changing and you know you know what an ellipse looks like it's I will actually do that in the first video it looks something like that what I mean that the radius or the distance from the center is always changing let me say this is centered at the origin so that's the origin right there you see here we're really if we're on this point on the raid on the ellipse we're really close to the origin this is actually the closest we'll ever get just as close as we'll get down here and when we're out here we're really far away from the origin and that's about as far as we're going to get right there so a circle is a special case of this because in a circles case the farthest we get from the origin is the same distance as the closest we get or in other words where we are always the exact same distance away from the origin well with that said let's actually go a little bit into the math so that the general or the standard form for an ellipse centered at the origin is x squared over a squared plus y squared over B squared is equal to one where a and B are just any two numbers they're going to I could have written this as C squared + d squared I mean they're just placeholders but just give you an idea of what this means if this was our ellipse in question right now a is the is the length of the radius in the X direction remember we have a squared down here so if you take the square root of whatever is in the denominator a is the X radius so this distance in our in our little chart right here on our little graph here that distance is a or that this point right here since we're centered at the origin will be the point X is equal to a y is equal to 0 and of course this point right here this will be a this distance is also a so this would be the point minus a comma 0 and then the radius in the Y Direction we be this radius right here and this is B so this point would be X is equal to 0 Y is equal to B likewise this point right here would be X is equal to 0 Y is equal to minus B and the way I drew this we have kind of a short and fat ellipse you can also have kind of a tall and skinny ellipse but in the short and fat ellipse the direction that you're short in that's called your minor axis and so B you could always forget the exact terminology but B you can kind of you call it your semi or the length of your semi minor axis why is it so let me write that semi minor axis and where do these that word come from well if this whole thing is your minor axis or maybe you can call it your minor diameter if this whole thing is your minor diameter it's called minor because it's the shortest of the all of the diameters of this ellipse and then the semi is half of that right semi just for 1/2 so this is the length B is the length of the semi minor axis semi minor axis that's B in this example just because as I drew this ellipse it just happens to be that B is smaller than a if B was larger than a I would have at all I would have a tall and skinny lifts let me actually draw one it could have been like this right I could have an ellipse that looks something like that in which case all of a sudden B would be the semi-major axis because the B would be greater than a that this would be taller than it is wide but let me not confuse the graph too much and in this case a is the length of the I think you've guessed it a is the length of the semi-major axis or you could even call it the length of the major radius I think that makes more sense major radius major radius and you can call this the minor radius I don't like that color minor radius minor radius so let's just do it an example and I think when I did an example with actual numbers it'll make it all a little bit clearer so let's say I were to show up at your door with the following if I were to say x squared over nine plus y squared over 25 is equal to one so what is your radius in the x-direction well your radius to the X this is your radius in the X Direction squared so your radius in the X direction if we just map it we would say that a is equal to three because this is a squared and if we were to just map it we say this is B squared then this tells us that B is equal to five so if we wanted to graph this and once again this is centered at the origin let me draw out the ellipse first so first of all we have our our radius in the Y Direction is larger than our radius in the X direction so it's going to lips is going to be taller and skinnier it's going to look something like that draw some axes so that could be your x axis it could be your Y axis so this distance this is this is your rate your radius in the Y direction so this distance right here is going to be five and so will this distance and this is your radius in the X direction so this will be three and this will be three that's it you have now plotted this ellipse nothing to fancy about it and actually just to kind of hit the point home that the circle is a special case of an ellipse we learned in the last video that the equation of a circle is x squared and a circle centered at the radius a centered at the origin x squared plus y squared is equal to R squared so if we were to divide both sides of this by R squared we would get and this is just a little algebraic manipulation we would get x squared over R squared plus y squared over R squared is equal to one now in this case your a is R and so is your B so your your semi minor axis is our and so is your semi major axis Azhar or in other words this distance is the same as that distance and so it will neither be short and fat nor tall and skinny it'll be perfectly round and so that's why the circle is a special case of an ellipse so let me give you a slightly it look a lot more complicated and this some something you might see on an exam but I just want to show you that this is just a shifting let's say we want to do let's say we wanted to shift this equation this this ellipse let's say we wanted to shift it it to the right by by 5 so we wanted to shift that to the right by 5 so the instead of the origin being at X is equal to 0 the origin will now be at X is equal to 5 right so way to think about that is what does this term have to be so that at 5 we this this term ends up being 0 well I'll actually draw it for you cuz I think that might be confusing so if we shift that over the right by 5 the new equation of this ellipse will be X minus 5 squared over 9 plus y squared over 25 is equal to 1 so if I were to just draw this ellipse right now it would look like this it would look I want to make it look fairly similar to the ellipse I had before look it would look just like that just shifted it over by 5 and the intuition we learned a little bit in the in the circle video where I said oh well you know if you have X minus something that means that the new origin is now at positive 5 and you could memorize that you can always say oh if I have a minus here that the origin is at a you know is is at the the negative of whatever this number is so it would be a positive 5 and you know if you get a positive would be the opposite of that but the way to really think about it is is now if we go to X is equal to 5 when X is equal to 5 this whole term this whole term X minus 5 will behave just like this X term will here when X is equal to 5 when X is equal to 5 this term is 0 just like when X was 0 here so when X is equal to 5 this term is 0 and then Y squared over 25 is equal to 1 so you have Y has to be equal to 5 just like over here when X is equal to 0 Y squared over 25 have to be equal 1 so Y is equal to either positive or minus 5 so in either case positive minus 5 but I really want to give you that intuition and then let's say we wanted to shift this equation down by 2 all right so if we wanted to shift it down by 2 so our new ellipse look something like this and this isn't anything this isn't you know when a lot of times you learn this in conic sections but this is true of any function when you shift things you shift it this way if you shift it if you shift this graph to the right by 5 you replace all of the X's with X minus 5 and if you were to shift it down by 2 you would replace all the Y's remember we're shifting down by 2 so you replace all the Y's with y plus 2 so let me draw our new ellipse first just to show you what I'm doing so it's all good it's going to be a little bit it's going to be shifted down by 2 so our new ellipse it's going to look something like it's going to look something like that so I'm shifting it down by 2 I'm shifting the yellow ellipse down by 2 so this equation if I shift it down well well the X is still where it was before X minus 5 squared over 9 plus y plus 2 squared over 25 is equal to 1 and once again the reason I know this is because now when Y is minus 2 this whole term is 0 right 0 when y equals -2 and when this term is 0 it behaves the same thing as when this term was 0 so when y is equal to minus 2 you get the same behavior you're the same point you're at the same point in the curve right here actually as you are when y equals 0 and this one so here so it's not the same point you can kind of view the same part of the ellipse right you're at kind of the maximum you're kind of the maximum with point on the ellipse here and here when y is equal to 2 and you were here at y equal to zero and that's because when sorry when y equals -2 right this is minus 2 and that's because when you put y equal minus 2 here this whole term is 0 just like when y was 0 here I don't to hit I don't want to make it too confusing but just to kind of wrap it all up sometimes you might see something like you know graph the following y minus 1 squared over 4 plus X plus 2 squared over 9 is equal to 1 and so the first thing you could say is okay this is just like the standard ellipse Y squared over 4 plus x squared over 9 is equal to 1 it's just like this but it's shifted over right this one's origin is 0 0 well this one's origin would be what it would be the point X is minus 2 and Y is 1 so if you were to graph this your radius in your Y Direction is 2 right 2 squared is equal to 4 your radius in your X Direction is 3 3 squared is equal to 9 so let's see so your X radius is actually larger than your Y radius so it's going to be a little bit of a fat ellipse it's going to be a little actually let me draw the axes first so that's my whoops let me draw it like this that's my vertical axis this is my x axis and so my center is now at -2 1 so 1 2 that's minus 2 and I go up 1 so it's up 1 that's the center of my lips and now in the X Direction this is the X term my X radius is 3 so the lips will go 3 this is kind of the it'll go 3 in that direction this is its widest point will be 3 in that direction and then in the Y direction it'll go 2 so it'll go up 1 2 so that's there and then one two and it's there so if I were to draw that ellipse it would look something like this to my best shot it would look something like that a little bit fatter than it is tall and that's because your x radius is larger than your Y radius right this distance right here is 3 this distance right here is 3 this distance right here is 2 this distance right here is 2 and if this is the point I'm you can figure out what these points are I won't do all of them right now just for the sake of time but this right here is the point minus 2 1 so if you go 3 more than that so if you add 3 2 in the X Direction this is the point 1 comma 1 if you would take 3 way this would if you take 3 away from the X Direction this would be minus 5 comma 1 and you can figure out the other points that's might be a good exercise for you anyway that's a little bit on ellipses and in future videos we'll do really hairy problems we have to simplify it into this form so that we know that it definitely is an ellipse