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Conic section from expanded equation: ellipse

Sal manipulates the equation 9x^2+4y^2+54x-8y+49=0 in order to find that it represents an ellipse. Created by Sal Khan.

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  • spunky sam blue style avatar for user kaninikbaradi
    Where can i find the videos about equations of elipses, circles and parabolas? I'm afraid i don't really understand those completely.
    (11 votes)
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  • mr pants teal style avatar for user owen
    Is it just me, or do you ever stop after finishing an equation as long and complicated as this and just feel really intelligent? I know in the long run this equation isn't super hard, but the feeling of completing something that crazy-looking is super empowering.
    (4 votes)
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  • aqualine sapling style avatar for user YMarshall
    I understand that an equation for an ellipse is always set equal to one, not zero. Why is this?
    (4 votes)
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    • piceratops ultimate style avatar for user Ed Miller
      You could subtract 1 from both sides of the equation, and then it would be set equal to zero.
      It is just one of several conventions for the equations of circles, ellipses, and hyperbolae to be presented in this form, whereas the equations of parabolae tend to be presented in the form ax² + bx + c = 0.
      However, the general form for the equation of any conic section is:
      Ax² + By² + Cxy + Dx + Ey + F = 0
      Therefore, depending on context, you may see different conventions followed.
      For an ellipse, there does need to be a non-zero constant term in order to obtain a curve of any kind (as opposed to a single point). Whether you place the constant on the left or right side of the equation is a matter of taste, but the x² and y² terms are both necessarily positive, so a constant on the oposite side of the equation must be positive, while if placed on the same side it would have to be negative, otherwise there would be no real solutions to the equation, and therefore no points on the coordinate plane could be plotted and there would be no ellipse.
      (1 vote)
  • leaf green style avatar for user Sungmin
    How do I know if an equation is representing an ellipse but not a circle?
    (2 votes)
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    • leafers ultimate style avatar for user Cody Capella
      An easy way to tell the difference between an ellipse and a circle is if their radii are the same when the equation is in standard form (the way it was after Sal completed the square). For example, if the number under the (x-h)^2 and the number under the (y-k)^2 are equal, then you have a circle. If they are not equal, the radii are different lengths, so the equation is an ellipse.
      (5 votes)
  • marcimus pink style avatar for user Imani
    How exactly, though, does Sal know that it is an ellipse? By just looking at it? I think I need a bit of clarification in terms of knowing what figures are which by just looking at the equation. Thanks.
    (2 votes)
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    • male robot hal style avatar for user aditya.chincholi
      In the conic sections -
      Hyperbola - the 2nd degree variables(x^2 and y^2) need to have opposite signs
      Parabola - needs to have only 1 of the variables(x or y) as square while the other is degree 1(just x or y).
      Ellipse - needs to have both variables in degree 2
      Circle - special ellipse
      Looking at the above terms we can easily rule out Parabola and Hyperbola. So Sal says that it is probably an ellipse.
      (4 votes)
  • winston baby style avatar for user Neal Rame
    what about in the last equation - it equals 1 but if you have x and y -3 and 1 doesn't it have to be 0?
    (1 vote)
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  • blobby green style avatar for user ciaobella1078
    at , where did you get the 81 to add to the right hand side?
    (1 vote)
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  • aqualine ultimate style avatar for user Uri
    I'm not sure if something is wrong with just my computer or something but it seems like this video is exactly the same as the "Conic Section from Expanded Equation: Circle and Parabola" I think there might've been some kind of error in the upload?
    (1 vote)
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  • mr pants teal style avatar for user Hollerdog
    What is standard form?
    (2 votes)
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  • leafers tree style avatar for user Heather
    I don't understand how to put this in standard form.
    What type of conic is this?
    (1 vote)
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    • leaf green style avatar for user kubleeka
      y(-4x+25)= -x²-3x+6
      y=(x²+3x-6)/(4x-25), for x≠25/4

      This is not a conic section, just a rational function. It has a vertical asymptote at x=25/4, goes to +∞ to the right and -∞ to the left.
      (2 votes)

Video transcript

The standard question you often get in your algebra class is they will give you this equation and it'll say identify the conic section and 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'd be able to get it. So let's do a question like and let's see if we can figure it out. So what I have here is 9x squared plus 4y squared plus 54x minus 8y plus 49 is equal to 0. And once again, I mean who knows what this is it's just not in the standard form. And actually one quick clue to tell you what this is you look at the x squared and the y squared terms if there are. If 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 it's just an x term and a y squared term, it's probably a parabola. But assuming that we're dealing with a circle, an ellipse, or a hyperbola, there will be an x squared term and a y squared term. If they both kind of have the same number in front of them, that's a pretty good clue that we're going to be dealing with a 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. 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 it in the standard form. And the key to getting it in the standard form is really just completing the square. And I encourage you to re-watch 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 the square, and you're going to have to do it for the x variables and for the y terms, is group the x and y terms. Let's see. The x terms are 9x squared plus 54x. And let's do the y terms in magenta. So then you have plus 4y squared minus 8y and then you have-- let me do this in a different color-- plus 49 is equal to 0. And so the easy thing to do when you complete the square, the thing I like to do is, it's very clear we can factor out a 9 out of both of these numbers, and we can factor out a 4 out of both of those. Let's do that, because that will help us complete the square. So this is the same thing is 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 is equal to 0. So what are we going to add here? We're going to complete the square. We want to add some number here so that this whole three term expression becomes a perfect square. Likewise, we're going to add some number here, so this three term number expression becomes a perfect square. And of course whatever we add on the side, we're going to have to multiply it by 9, because we're really adding nine times that. And add it on to that side. Whatever we add here, we're going to have to multiply it times 4 and add it on that side. 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 had a 1 here it's 1 times 9. So 9 there. Let's do that. 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. Remember it's an equation, so what you do to one side, you have to do to the other. So if we added a 9 here, we're actually adding 9 times 9 to the left-hand 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 as if I added plus 81 right here. 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 is minus 2, half of that is minus 1. You square it, you get plus 1. 1 times 4, so we're really adding 4 to the left-hand side of the equation. And just so you understand what I did here. This is equivalent as if I just added a 4 here, and then I later just factored out this 4. And so what does this become? This expression is 9 times what? This is the square of-- you could factor this, but we did it on purpose-- it's x plus 3 squared and then we have plus 4 times-- What is this right here? That's y minus 1 squared. You might want to review factoring of polynomial or completing the square if you found that step a little daunting. And then we have plus 49 is equal to 0 plus 81 plus 84 is equal to 85. All right, so now we have 9 times 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 35, so if I subtract 49, I get 36. And now we are getting close to the standard form of something, but remember all the standard forms we did except for the circle-- we had a y-- and we know this isn't a circle, because we have these weird coefficients, well 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. If you divide everything by 36, this term becomes x plus 3 squared over see 9 over 36 is the same thing as 1 over 4, and then you have plus y minus 1 squared 4 over 36 is the same thing as 1 over 9 and all of that is equal to 1. And there you go. We have it in the 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's the center of the this ellipse going to be? It's going to be x is equal to negative 3. What x value makes this whole terms 0? So it's going to be x is equal to minus 3, and y is going to be equal to 1. 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. It's going to be in the negative quadrant. This is our x-axis and this is our y-axis. And then the center of our ellipse is at minus 3 and positive 1, so 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 two to the right. We go two to the left. And in the y direction, what do we do? Well we go up three and down three. The square root of this. Let me do that. Remember you have to take the square root of both of those. The vertical axis is actually the major radius or the semi-major axis is 3, because that's the longer one. And then the 2 is the minor radius, because that's the shorter one. And now we're ready to draw this ellipse. I'll draw it in brown. Let me see if I can do this properly. I 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. And then we divided both sides by this number right here and we got it into the standard form. We said oh this is an ellipse. We have both of these terms, they're both positive, we're adding we're not subtracting, they have different coefficients underneath here. So we're ready to go over the ellipse, and we realized that the center was at minus 3,1, and then we just drew the major radius, or the major axis and the minor axis. See you in the next video.