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Current time:0:00Total duration:9:11

Conic section from expanded equation: ellipse

Video transcript

the standard question you often get in your algebra classes they'll give you this equation and it'll say identify the conic section and then graph it if you can and the equation they give you won't be in the standard form because if it was you could just kind of pattern match with what I showed in some of the previous videos and you'll be able to get it so let's do a question like that and let's see if we can figure it out so what I have here is 9 x squared plus 4 y squared plus 54 X minus 8 y plus 49 is equal to 0 and once you I mean this this knows what this is it's not in the standard form and actually one quick clue to tell you what this is is you look at the the x squared and the y squared terms if there are I mean if there's if only there's only an x squared term and then there's just a Y and not a y squared term then you're probably dealing with a parabola and we'll go into that more later or if it's the other way around if there's just an X term and a y squared term is probably a parabola but assuming that there we're dealing with a circle and ellipse or hyperbola there will be an x squared term and the y squared term if they both kind of have the same number in front of it that's a pretty good clue that we're going to read dealing with the circle if they both have different numbers but they're both positive in front of them that's a pretty good clue we're probably going to be dealing with an ellipse and if they both have if one of them has a negative number in front of them and the other one has a positive number that tells you that we're probably going to be dealing with a hyperbola but with that said I mean that might help you identify things very quickly at this level but it doesn't help you graph it or get into the standard form so let's get into the standard form and the key to getting it in the standard form is really just completing the square and I encourage you to rewatch the completing the square video because that's all we're going to do right here to get it into the standard form so the first thing I like to do to complete this corner you have to do it in for the X variables and for the Y terms is group the x and y term so let's see the X terms are 9 x squared plus 54 X all right that's 9 x squared plus 54 X and let's do the Y terms in magenta so then you have plus 4y squared minus 8y and then you have this let me do this in different color plus 49 plus 49 is equal to zero and so the easy thing to do when you complete the square the thing I like to do is you know we have these it's very clear we can factor out a 9 out of both of these numbers if we could factor out a 4 out of both of those so let's do that because that will help us complete the square so this is the same thing as 9 times x squared plus 9 times 6 is 54 6x I'm going to add something else here but I'll leave it blank for now plus 4 times y squared minus 2y I'm probably going to add something here too so I'll leave it blank for now plus 49 plus 49 is equal to 0 so what we're going to add here there we're going to complete the square we want to add some number here so that this whole 3 term expression becomes a perfect square and likewise we're going to add some number here so that this 3 term number expression becomes a perfect square and of course whatever we add on this side we're going to multiply it by 9 because we're really adding 9 times that and add it on to that side and whatever we add here we're going to multiply it times 4 and add it on that side because if I put a 1 here it's really like as if I had a 4 here because 1 times 4 is 4 and if I have 1 here it's 1 times 9 it's a 9 there so let's do that so when we complete the square we just take half of this coefficient this coefficient is 6 we take half of it is 3 we square it we get a 9 but when we remember when it's an equation so what you do to one side you have to do the other so if we added a 9 here we're actually adding 9 times 9 to the left side of the equation so we have to add 81 to the right hand side to make the equation still hold and you could kind of view it if we go back up here this is the same thing just to make that clear is if I add a plus 81 right here plus 81 and of course I would have had to add plus 81 up here now let's go to the Y terms you take half of this coefficient sum - - half of that is - one you square it you get plus one one times four so we're really adding four to the left hand side of the equation so let's add four here and just to make just so you understand what I did here this is the equivalent is if I just added a four here and then I later just factored out this four and found it a four here out of four up here and so what does this become this expression is nine times what this is the square of you could factor this but we did it on purpose it's X plus three squared and then we have plus four times what is this right here that's Y minus one squared y minus one squared you might want to review factoring a polynomial or completing the square if that if you found that step a little daunting and then we have plus 49 is equal to zero plus 81 plus 84 is equal to 85 all right so now we have 9 times X plus 3 squared plus 4 times y minus 1 squared and let's subtract 49 from both sides that is equal to let's see if I subtract 50 from 85 I get I get 35 so if I subtract 49 I get 36 36 and now where it looks we're getting close we're getting close to a standard form of something but remember all the standard forms we did except for the circle we had a 1 we know this isn't a circle because we have these weird coefficients we're not weird but different coefficients in front of these terms so to get the 1 on the right hand side let's divide everything by 36 you divide everything by 36 this term becomes X plus 3 squared over C 9 over 36 the same thing as 1 over 4 and then you have plus y minus 1 squared for over 36 the same thing as 1 over 9 and all that is equal to 1 and there you go we have it in this standard form and you can see our intuition at the beginning the problem was correct this is indeed an ellipse and now we can actually graph it so first of all actually good place to start where is the center of this ellipse going to be it's going to be at X is equal to negative 3 right what x value makes this whole term 0 so it's going to be X is equal to minus 3 and y is going to be equal to 1 what makes what Y value makes this term 0 Y is equal to 1 that's our Center so let's graph that and then we can draw the ellipse see it's going to be in the negative quadrant so it's our x axis and this is our y axis and then the center of our ellipse is that 1 2 3 minus 3 and then positive 1 so it's that's the center and then what is the radius in the x-direction we just take the square root of this so it's 2 so in the x-direction we go to to the right so this is 2 let me go to to the left that's 2 and in the Y direction what do we do well we go up 3 and down 3 right the square root of this so let me do that we're going to go through to c1 1 2 3 so we're going to go up here I'm going to go 1 2 3 we're going to go down here that's the Y alright remember you have to take the square root of both of those so the D and so the vertical axis is actually this the major radius or the semi-major axis is 3 because that's the longer one and then this the 2 is the minor radius because that's a shorter one and now we're ready to draw this ellipse I'll draw it in brown so let me see if I can do this properly you have a shaky hand all right it looks something like that and there you go we took this kind of crazy looking thing and all we did is algebraically manipulate it we just completed the squares with the X's and the Y terms to get it into and then we divide both sides by this number right here and we got it into the standard form or you said oh this is an ellipse we have both of these terms where they're both positive we're adding we're not subtracting they have different coefficients underneath here so we're ready to go over lips and we just we realize that the center was at -3 1 and then we just drew the major radius or the the major axis and the Minor axis see in the next video