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Ellipse

Added May 03, 2018 •

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Intro to Ellipses

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today we're going to be talking about not going to write down our equations for an ellipse so there's two types of ellipses that one is horizontal has a horizontal major axis the other passing protocol major axis first we're going to write down the equation for horizontal major axis and it looks like this it's parentheses x - 8 clothes per square over a squared parentheses y - K squared squared equals one it looks as big as always have to equal one and just a little sign out here that in a horizontal with a horizontal major axis a squared is always going to be greater than b squared just like in problems we have Focus as Focus point but we have a focus point and it looks as though as well we actually have to focus points and so the plural of focus is Foci and to find the foci major axis and in this case since its horizontal the foci a r letter seeds right here the folk are always on the longer access sorry quotations who that's okay is plus or minus, K and we'll talk in a minute about how we find that c are on the longer axis and the Cobra tissues are always on the shorter access which we call the minor axis so are vertices which in this case would be those two on them Traxxas the coordinates are h plus or minus a, k and the co vertices are h, k plus or minus B alright so copy that down noticed Center still at HK like it was first Circle two are vertical major axis ellipse equation for this parentheses x - 8 squared over B squared + y - K squared over a squared still always equals wants Ellipsis have to eat squared which is still greater than the B squared is under the Y so that means it now this is bigger so the why is bigger we've got a longer why here where is up here under the exit the accent speaker on the X always tell by looking at an equation if it's going to be horizontal or vertical by where the bigger number is so just a little pecan example here since list of the 25 is larger than the 9 we know that the bigger numbers under the X so it's going to be bigger on AR x axis of this equation over here is going to create a horizontal major axis talk about our Focus for this one is Prince c h, k + 4 - c it's still should be a little curious about that will get there and then are vertices now are vertices here there on the major axis of the ones further away from the center so in this case it's going to be h, k plus or minus oh I just read that wrong at 6, K plus or minus a that's easy to fix and the other one is for cover disease is 8 + 4 -, k now back to that stuff that we were wondering about it there's a formula equation that we can use to find c squared minus B squared Shekinah remind you of the Pythagorean theorem a squared plus b squared equals c squared but said plus it's a subtraction sign alright so those are the fun this were going to need in order to graph an l parabolas all the formulas are helpful you can use the formulas but sometimes it's easier to just look at the equation and think about what that looks like so I'm going to show you how to do that with this. all right now let's take a look at this example problem that we have here I can tell that it's going to be an ellipse because of you have an x squared and a y squared and you're adding those two times together it has to equal 1 so that's so we can go ahead and use our ellipse formula we learned to decide which letter is r squared this is a squared and this is B squared five spice what is 25 equals 3 my Center is that this ellipse the center is my H&K think of it in terms of the formulas over here you can just say whatever we did for the circle the one here positive one in this is a y + 2 so my y coordinate of the center will be negative 2 I have my center of my lips going to go ahead and place that I'm going to go over one and down to next I'd like to decide if this is going to be a horizontal or vertical ellipse so have a horizontal major axis or a vertical major skin because the larger know I was under the text we said earlier that we know it's going to be a horizontal lips if it had been the other way around and we had 25 years and it would have been a vertical Italy lip stand here in the set of formulas I need to find my vertices the vertices of ellipse are right here and the Cobra tissues are here so let's go ahead and try to find are vertices formula we can use and I'll talk about the formula I just want to show you another way about it it's really easier way so if you you can say what the x-axis I'm going to go over so when you graph it because we know what's the longer side is horizontal you go to your Center and we go over 5 each way from the center so I can go over 1/2 one two three four five right here but just think of it as a horizontal Direction. You're going down under the Y single why the y axis goes up and down so under the why we're going to be going up and down it whatever RB value was so be was three going to go up Co vertices vertices where at over six down to and over here which was negative for negative 2 and Mike over to seize the ones along the minor axis the shore one one and one what was that 1 + 1 + 1 2 3 4 5 1 -5 because I didn't use the formula to find the find the vertices are co-vertices I just that I knew what the center was under the X we went over 5 each why we go up and down three but I think that's the easy way but I'm still going to show you the formula H plus or minus my Center is H&K so I can say anxious 1 + 4 - AA we set up here was 5 -2 1 + 5 is 6 so 1 + 5 is 6, -2 + 1 - 5 is -4, 2 the formula works if you're an equation person or former person you can use the formulas if you don't want to just think about going over 5 and Up and Down by 3 what's chapstick over to see for me the cover to see formula right here was h, k + 4 - B Hy-Vee, K is -2 + 4 - Bebe let's see if that checks out with what we got when we just counted over so -2 + 3 is -1 the formula for work let's talk about the actual graph now we've got our major axis and plants are vertices and are mine Iraq and all you have to do is but they're still what time we haven't talked about yet and that is this just like our problem had a focus Point ellipse Ellipsis have to focus points so the plural of focus point is called Foci so often wear a status information find something else to us use our Focus Formula Prince e c h + 4 - Kama k for a Focus Formula we know what H&K are H&K are the center of our foot guy so I can the center of are you less so are Spencer we said was up here at 1 -2 sah s1k is -2 so all I really need to find put plus I minus all I really need to know guess what we have a formula for C right here equation for C star formula to find C is a squared minus B squared equals c squared and we know what we are we already said a squid from a bus was 25 minus B squared was 9 equals c squared all right here my 25 is my a squared and my nine is my Beast grits I'm just plugging in the 25 in the nine 25 - 16 equals c squared which means C equals for now I could put plus or minus 4 but since I'm plug it in right here to plus or minus I don't really need to read that twice so I'm just going to put for right there? So that's out those up 1 + 4 is , -2 + 1 - 4 is -3, negative 2 about to be my guy by go over 5 no sir Michael Crichton the focus should always be on the major axis that checks out on the Nexus and we found all the parts we need to find out we found the center the vertices The Cove if you are quite catch all that you can watch this video I was going to turn to our worksheet so let's look at Eclipse worksheet and we're going to go down the right-hand column and then I'm going to let you try the take a closer look here at number two so the first thing we want to decide when graphing is which one is are a squared and which one is Arby's secret now we said a square is always larger than a squared so I know that this is my a squared and this is my be square is the number 6 + 6 squared - 36 MB is 3 * 3 squared + 9 whatever you're adding or subtracting to your ex's and your lies so with this x I'm not adding anything so my Center is zero and simple I'm not adding anything so my Center so my center of this because the larger number was under the Y that means I'm going to go up and down on my y-axis whatever number I squared to get 36 so underneath I'm going to go up and down 6 1 2 3 4 5 1 2 3 4 5 6 what's the number it's hard to listen and write at the same time so let me just go over that once now again as well you since you should have all the points plotted already so under the X was 9 which means on the X we go over 3 where does night under the why was 36 which means a was 600 under the on the y-axis we go up and down six we can list are vertices we don't have to use that formula that we wrote down on the other side the vertices are always the longer ones in this case are ellipses a semi vertices are the the law 0 + 6 and / 0 and down 6 now my Cobra 2 C's are the points on the shorter access the minor axis and here we went over 3 and up Siri straighten up Sarah that's usually what I am going to use are my focus for I know that Foci you look too at our formula sheet that we have the soap guy for a vertical major axis which is this what this one is is this formula I'm here h, k + 4 - 3 Tannerite that out right below at h, k plus or minus sees my Foci so I need to find see I already know 18K H and K are my Center points coordinates that's fine see we have a formula for C the formula was a squared minus B squared equals c squared well we know that a squared was 36 minus B squared which was not equals c squared track Santa back to c squared equals a 27 that's not so pretty which means that c will be square root of 27 technically I guess we could say the plus or minus and then we'll get C equals this factors down breaks down into three groups of three again I don't really have to put the plus or minus here since it's here in our formula so we're going to have zero, K10 so it's the 0 plus or minus C & C S3 screen alright let's see how that so let's go ahead and connect our points first and make our nice lips so there's our lips need the foci to sketch the graph but if you ask for the foci you need to find so good three squares if that's hard for me to think about what that might wear that might going to be easier for me to think about where to 27 for the 27 I know the square root of 27 is between the square root of 25 and the square root of 36 this would be 5 and this is 6 so I know we're close service somewhere 5 and 6 closer to the five sides and 25 is close to 27 and 36 so you can be a little bit more than 5 so when you grass this week and then up and down a little bit more than 5 so I'm going to go up 1 2 3 4 5 2 3 4 5 and a little bit more who are vertices number for can you tell me a number to oh no it's just a lips because it doesn't I don't want to have a negative y squared terms so I'm going to add this over to the other side so I'm going to rewrite this as 7 x squared keep that sin x squared but then when I move this over it's going to be come up positive 5y squared equals 35 not supposed to always how to make it 35 but the mass Commandment that says Thou shalt due to one side without does to the other applies here as well if I divide this by 35 I have to divide everything on this side by 35 as well I decided to write it out as two separate ones because that's going to be easier when I simplify 7/35 clean up to become x squared over $0.05 7 goes into 35 v x + 53057 * so look at y squared 7 equals fun are your lips formula you got an a squared plus y squared so you can manipulate an equation into the ellipse formula so let's see our Center again we have nothing with the X and nothing with the Y that we're adding or subtracting so it's who's going to be there I have to decide which one of these is an which one is B will a square is always larger so this is going to be my a squared in the five is going to be my b squared which is a little bit of a pain because if a squared equals 7 then a equals the square root of 7 what's 4 minus and if B squared equals 5 then b equals this plus or minus the square root of 5 so I'm going to go ahead and graph what I can I can grab the center 00 under the X so that means on the x-axis I'm going to go over 4 minus the square root of 5 so I can I go for my be in the square root of 5 so I know I'm just going to go over to a little bit more left a little bit more let me see one more time so under the ex was 5 so I know that might be squared is 5 so be was plus or minus the square root of 5 so that means I'm going to go over 5 on my ex and over over the square root of 5 vs little bit more tales of the square Forest to and then I'm going to go left the square to 5 now on my taxes under the why is seven ended up being plus or minus the square root of 7 and the square root of 7 between the 9th and it will I'm going to go one two and a lot more but not a lot more but not quite all the way to -3 under the Y because the score of 7 is under your why that ellipse you can connect which looks like this almost looks like a circle because those numbers were so close together but really it's a little bit longer on the y-axis because the Y was a little a little bit larger underneath it which means it's a horizontal or vertical lips what do you think rent a car so are you going to be using vertical farming when we get to talk about let's let's go ahead and list out the vertices here for the vertices I went over to and how much more I went over to and I want the score of the five positive square, Sarah find the vertices of the longer ones are right here that's going to be my Uber to see the square root of 5, 0 so let's think about anime over to see vertices going to be over here which will be negative score the 5, 0 Unger access my major axis was right here which means I went the square root of 7 in / 0 and what happens when you make a live video all right moving on to the foci the foci is the formula it's a squared minus B squared equals c squared we know a squared was 7 B squared equals tattoo equals c squared or C is going to equal plus or minus the square root of 2 so what this is when I go to my Foci formula so I'm going to look at we said this was a vertical ellipse the longer on the Texas my vertical ellipse is going to be right here so my Focus Formula is h, k plus or minus States I'm going to write that off to the side age, k+ and then we can plug in our values agent care about 0 7 0, 0 + 4 - c which is plus or minus the square root of 2 now the square root of 2 is a little bit as well as Square to is between the square root of 4 so I'm going to go somewhere between one and two little closer to the one side I'm going to go on my major axis the longer up one a little bit more and a little bit more and then download a little bit more the closer I get to your Center still in a lips but it looks a little bit more circular go ahead number 6 yay it's already in our standard form here we've got an x squared plus y squared equals one brilliant why it's larger so I know that 25 is going to be my a squared + 9 is my b squared so that means that he has to be a face cord is 25 a s B squared x 9 PS3 I sent her as my H&K but I don't even think of it as agent a so much I just think of it as whatever is with you it's my next cornet we have a -2 going to go in there 1 2 3 4 lights just went out in the classroom live video here CDOT dot your tensions of your day Jamie now you know something weird happened oh right now we're back thinking here so are vertices now we need to find so we have to decide is a tapered horizontal or vertical ellipse so the larger numbers under the Y we're going to have to do one that's not margarine is a license anyway I access or the vertical axis go up and down 5 1 2 3 4 5 1 2 3 4 5 under the ex was 9 so on the x axis or the horizontal axis I'll say I'm going one two three one two three tell Mommy recap that again so under the why was 25 so with my the Y runs been down so I'm the I went up and down here at 5 and down 5 under the X which is horizontal I went along the horizontal art through my center and I went over 3 and down 3 write 3 and less 3 and that's all I really need the grass at the vertices are always the ones that are on the longer access the major axis which is right here so we go to the origin and counter 1 2 3 4 1 2 3 so my vertice is for three and my other one be over for down 1 2 3 4 5 6 7 setup for -7 vertices are done that shorter access here than minor axis so my account / go / 1 Down 2 -2 in / 1 2 3 4 5 6 7 10 2 unless we just need to find the foci using our formula a squared minus B squared equals c squared we know that a square 25 B squared is 9 equals c squared When You Subtract that c squared equals 16 what's a -4 set for my Focus Formula which again is another vertical we're going to use the same friendly we use in the last two h, k plus or minus each was formed for my Center, Seminary -2 + 4 and then four, -2 - 4 is -6 those two points follow along our major axis like they should if we go over for 1 2 3 4 and up to and then we go over four and down six which that's good because your phone I should always be under longer access and they should always be inside the ellipse if you get a Foci out here and you know you did something wrong next I would because that looks like it's another phone number five looks like it's switched up a little bit number five we can describe the Y squared equals number is 9 so that's my a-squared which means that's my b squared 3 squared my center with my ex's and my wife is here so I do opposite signs with the X I have to with the wife naked woman supposed so it's go ahead and over to now under the ex was a 9 so along the horizontal access along the x-axis are parallel to the x-axis agency we're going to go over 3 as a is 3 / 1 2 3 from the center and 1 2 3 from the center again under the X we go parallel with RX when it's under the guy we're going to go y-axis here so far under the why was for that means parallel to the y-axis we go up to and down to that's all we need to grab it answer my vertices remember those are always the ones on the longer access the major axis so this is my major axis right here this is my minor axis of I would actually throw attitude this is my major Jackson minor axis so my vertices are here and here which appear to be at the coordinates -1 one and then one two three four five one and Mike over to CSR along the minor access so it looks like I went over to and download Andover 2 and up one two three now we just need to find Foci so Foci we have the formula a squared minus B squared equals c squared and help us find that Ice Cream Sandwich 9 minus B squared equals c squared c squared equals 5 for a horizontally lips looking at your sheets a horizontal ellipse foci formula is right there H + 4 - c, k swina where's my Center agent case 2 + 4 - c which is the score to 5, KS 1 just leave it like that is one quart phone call here stop that so much going on when I'm filming at night okay so let's see here we're going to go back to my soap I okay so I can List It office 2 plus the square 10 - 2 squared + 5, 1 that's fine for your phone but it's nice to see that there actually are what's the score of the final score of this is going to be that's right here plus the square root of 5 will the square root of 5 where to find what's the score of the 9 so I already went over to and now I need to go over little bit more over to and then one you and a little bit more be right back in here and then up one this one time I added too so adding to takes me here but then I subtract the square to 5 so then I have to go left to in a little bit more when she takes me back here and then up one remember the foci should always be on your major axis all right pause that for a minute. That was kind of a little bit tricky with that Foci they think about that okay let's move on tonight number 7 cards by 60 if you do that you'll end up with you'll end up with x + 3 squared / 16 class and then 4 goes into 16 four times so you could cancel that's where I can make that at 4 so then that b y - 2 squared / 4 equals 1 equation you're going to be using where you're a squared is 16 your B squared is and from there you can go ahead and find the center you can find vertices went by figuring out what and I'll let you let you try that from there I want us to take a look at it hear that gives you the graph and you have to come up with the equation let's pick what's the number 9 together does it look like a horizontal or vertical ellipse well it looks like it's going longer along the horizontal axis is going to be a horizontal ellipse so when I look at my formula sheet going to be using major axis formula squared plus find minus K squared over B squared equals what KNBR and it appears in this case the center is right in the middle at 0-0 so H&K are there is there I'll just Pet Center 06 so I'm going to Freddy x - 0 squared I'm just going to put a squared plus y squared over something equals 1 so now I just have to figure out what my NPR will a is always the one on longer access our major axis I went over 1 2 3 so as33 when you square a squared is 3 squared 9 is your minor axis 30 squared B squared 1 squared is just one and there we have our equation the grass like that just to recap my longer access so hair went over what's my shoulder my minor access I went up when I'm down have you tried number 10 on your own but I want to take a look over here at number 11 11 another horizontal Center is not at zero zero like the last one so I'm going to write the same formula squared over a squared plus y minus K squared over B squared equals 1 now my Center this time appears to be at left -1 so is that are Center at negative one one which means this is my age and this is my K so when I plug those in I'll get x - + -1 or x + 1 cuz we know it's always the opposite sine squared over something plus then we're going to have y -1 squared over something always equals one so now we just have to figure out which one is my and my b get as always the larger one so it looks like I went over he looks like I went up and down who won so since it was longer on my X-type my larger number under the X which is going to be my a squared so we squared to square this for and there we have my question so what I would like you to do for homework is YouTube finish number 10 as well as numbers 1 3 7 and 8 so you've got four 5 problems to do all together we will work on this some in our next lesson get to work be great students

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Ellipse

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