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# Conic section from expanded equation: circle & parabola

## Video transcript

let's see if we could do a couple more of these conic section identification problems so I have this problem x squared plus y squared minus 2x plus 4y is equal to 4 and so the first thing I like to do is just try to figure out what type of conic section this will be and we have our would be these are this my x squared term my Y squared term they're on the same side of the equation and they both have positive coefficients so this tells me that we're going to be dealing with an ellipse and in this case in particular the coefficients are the same number they're both positive one so this is going to tell me that is that this is a circle so let's get it this into standard form and try to graph this circle so we're going to want to complete the square so let's take the x squared the X term since we get x squared minus 2x plus something to complete the square later on plus then just do the y squared terms y squared plus 4y plus something is equal to 4 and now what do we add here we take half of the minus 2 minus 1 square it that becomes plus 1 add a 1 we have nothing out here so we really just added 1 to the left hand side of this equation so we have to add 1 to the right hand side and here we take half of 4 half of 4 is 2 2 squared is 4 put a 4 here so you have to add a 4 to the right hand side as well and we actually did add just a 4 because there's nothing multiplying the 4 out here and so this becomes X minus 1 squared plus y plus 2 squared is equal to 4 plus 1 plus 4 is equal to 9 and there you have it we have in the standard form of a circle you remember that if a circle is centered at 0 the standard form would be x squared plus y squared is equal to R squared so this is R squared this is the radius squared so that tells us the radius of the circle three and was just shifted so that its origin instead of being at zero zero is at the point one minus two and the reason we've got one minus two we just have to think about what makes this whole expression equal zero in this case it was the origin in this case it's X equal one x equals one and what makes this whole expression equal zero in this case it was y is equal to zero in this case it's y is equal to minus two so that's our center that's our radius and we're ready to graph graph this circle so it's at let me see it is that I should graph the circle first that's fair enough if I actually want to just graph the circle so it's going to be at 1 minus 2 so 1 minus 2 so it's going to be like down here so this it's going to come this circle is going to start there yes Center let's try my best is going to be at 1 and then you go one two and so that's pretty close to its center maybe I should go one two that's its center right there at one minus two and then it's radius is three so this distance right here is three in any direction it's three and that is three fair enough that was a pretty straightforward problem circles in some ways or the straight are the simplest and remember I said it's going to be an ellipse and you say oh this is the standard formula for an ellipse just as a refresher if you divide both sides of this equation by nine what do you get you get X minus 1 squared over 9 plus y plus 2 squared over 9 is equal to 1 and then you see that the the horizontal axis I guess we're against the horizontal diameter is going to be 3 and the or the horizontal radius is going to be 3 and the vertical race it's also going to be 3 because the radius never changes in this ellipse which is really a circle let's do one more just so make sure you you know this stuff cold so I have 2x squared plus I plus 12 X plus 16 is equal to zero let's look at the x squared and the y squared terms there's an x squared term but I don't see a y squared term so this is a bit of a conundrum and this will lead us to the fourth of our conic sections which I talked about in the first video but we haven't really touched on yet with that's the parabola and how do I know it's a parabola well I mean you you're familiar and I'll go more in future videos on all the different ways that a parabola comes about and how is the you know is that all the points equidistant between one point in a line and all of that but you know just on very simple ways you recognize is kind of the most simple parabola is y is equal to x squared that parabola looks something like this right where this where it's minimum point or its vertex is that the origin or if you have a parabola like X is equal to Y squared that looks something like this where it's a sideways version of that one where once again its vertex is at the origin and just out of out of interest actually I won't go there yet but actually well no I'll just so we know that this is a parabola because we have a Y and we have an x squared right there different degrees there's no there's no there's no second degree term of the Y and just to put this in a form this familiar to you let's just subtract everything but the Y from the left hand side so you get Y is equal to minus 2x squared minus 12x minus 16 and this is kind of the the traditional form that you're familiar with you're probably even used to finding the zeros of this parabola we could do that right now we could say okay when does this equation intersect the x axis the x axis is when y is equal to zero set that equal to zero you get minus 2x squared minus 12x minus 16 or this is different than what we normally do normally I would immediately break into completing the square but I just want to figure out the zeroes of this parabola first so this is zero is equal to minus two times factoring out - do you get X plus six X plus eight so zero is equal to minus two times X plus two times X plus four just factor that and so for this whole thing to be zero either this is 0 or that is 0 and so it's either X plus 2 is equal to 0 or X plus 4 is equal to 0 so X is equal to minus 2 and X is equal to minus 4 that's the two zeros of this parabola so we immediately know one thing about this parabola and we've you've probably already done this in your algebra classes if we were to draw the x axis if we were to draw the x axis it intersects the x axis at 1 and -2 and 3 and minus 4 so that's all we know about this right now so let's see if we can use some of our completing the square skills with the conic sections we've done so far to come up with a little bit more information about this parabola so let's try to complete the square with it I'll rewrite it down here so it's y is equal to this is what I'm dealing with and let me just take the X terms by themselves and factor out the -2 minus 2 times x squared plus 6x I'm going to add something else and then I have a minus 16 over there to make this a cup of a perfect square I have to take half of this half of the 6 half of the 6 is 3 3 squared is 9 if I add 9 to the right-hand side of the equation remember I didn't just add 9 this 9 times minus 2 I'm adding so if I subtract that's minus 18 if I subtract 18 from the right-hand side I also have to do it from the left-hand side so subtract 18 there and not my equation becomes Y minus 18 is equal to minus 2 times what is this this is X plus 3 squared minus 16 and let's just get it in a form that we might start recognizing from our conic sections let's add 16 to both sides so you add 16 to both sides Y minus 18 plus 16 is going to be Y minus 2 put parentheses around that is equal to minus 2 times X plus 3 squared and now you might wonder why I put it in this Arman and and I did because this will help us this is kind of the same pattern that you see with all of the other conic sections like if I were to tell you to graph you know y is equal to x squared y is equal to x squared it would look something like let me draw the whoops let me draw some axes here y equals x squared looks something like this it looks like a poor I mean it's a parabola with the vertex at 0 or its minimum point and that's what the vertex is is the minimum point of or the maximum point of the parabola we'll talk more about that you learn a lot more about that when we when we go into calculus but I think you can kind of recognize at the bottom of the U or the top of the U if I were to draw if I wanted to draw Y is equal to minus x squared and you could plot some points but it looks something like that if I were to try to graph y is equal to 2x squared it would just be like the y is equal to x squared would go up twice as fast it would look something like that so the vertex at the origin and finally if I were to graph y is equal to minus 2x squared it would look something like this it would point it would open downward and go down twice as fast now this one this equation that we ended up with right here this is the same thing as Y is equal to minus 2 x squared and it's the same general shape but stead of its vertex or its center or it's kind of starting point or whatever you want to call it to be at the origin it shifted now you say what Y value makes this term 0 what's Y is equal to 2 just fly you know in this case what Y value makes this 0 well it's 0 here because we were at the origin and here what x value makes this 0 is X is equal to minus 3 so this gives us information about where the vertex is it's at - it's at X is equal to minus 3 y is equal to 2 so said X is equal to 1 2 3 y is equal to 1 - it's right there we already know those two points because we figured out it's 0 but even if we didn't know those two points we know that this has the same general shape as Y is equal to minus 2 x squared so it's going to open word like this and a little bit faster than Y is equal to minus x squared so it's going to look like this I mean oh it goes to that point and we know goes through that point there you go so we've touched on every every conic section and in the next few videos I'll go into a little bit more depth of kind of the theory behind conic sections and how they came about and all of that but I think now you're ready to tackle a lot of what you might actually see on your algebra tests see you soon