Focus & directrix of a parabola
- This right here is an equation for a parabola and the role of this video is to find an alternate or to explore an alternate method for finding the focus and directrix of this parabola from the equation. So the first thing I like to do is solve explicitly for y. I don't know, my brain just processes things better that way. So, let's get this 23 over four to the right hand side. So let's add 23 over four to both sides and then we'll get y is equal to negative one-third times x minus one squared plus 23 over four. Now let's remind ourselves what we've learned about foci and directrixes, I think is how to say it. So, the focus. If the focus of a parabola is at the point a, b and the directrix, the directrix, directrix is the line y equals k. We've shown in other videos with a little bit of hairy algebra that the equation of the parabola in a form like this is going to be y is equal to one over two times b minus k. This b minus k is then the difference between this y coordinate and this y value, I guess you could say. Times x minus one squared plus b plus k. I'm sorry, not x minus one. I'm getting confused with this. x minus a squared. x minus a squred plus b plus k over two. The focus is a,b and the directrix is y equals k and this is gonna be the equation of the parabola. Well, we've already seen the technique where, look, we can see the different parts. We can see that, okay, this x minus one squared. Actually, let me do this in a different color. This x minus one squared corresponds to the x minus a squared and so one corresponds to a, so just like that, we know that a is going to be equal to one and actually let me just write that down. a is equal to one in this example right over here. And then you could see that the negative one-third over here corresponds to the one over two b minus k and you would see that the 23 over four corresponds to the b plus k over two. Now the first technique that we explored, we said, "Okay, let's set negative one-third "to this thing right over here. "Solve for b minus k." We're not solving for b or k, we're solving for the expression b minus k. So you got b minus k equals something. And then you could use 23 over four and this to solve for b plus k. So you get b plus k equals something and then you have two equations, two unknowns, you can solve for b and k. What I wanna do in this video is explore a different method that really uses our knowledge of the vertex of a parabola to be able to figure out where the focus and the directrix is going to be. So let's think about the vertex of this parabola right over here. Remember, the vertex, if the parabola is upward opening like this, the vertex is this minimum point. If it is downward opening, it's going to be this maximum point. And so when you look over here, you see that you have a negative one-third in front of the x minus one squared. So this quantity over here is either going to be zero or negative. It's not going to add to 23 over four, it's either gonna add nothing or take away from it. So this thing's going to hit a maximum point, when this thing is zero, when this thing is zero, and that's just gonna go down from there and when this thing is zero, y is going to be equal to 23 over four. So our vertex is going to be that maximum point. Well, when does this equal zero? Well, when x equals one. When x equals one, you get one minus one squared. So zero squared times negative one-third, this is zero. So when x is equal to one, we're at our maximum y value of 23 over four which five and three-fourths. Actually, let me write that as a . Actually, I'll leave just that's our vertex. 23 over four and it is a downward opening parabola. So actually, let me start to draw this. So we'd get some axis here. So we have to go all the way up to five and three-fourths. So. Let's make this our y, this is our y axis. This is the x axis. That's the x axis. We're gonna see, we're gonna go to one. Let's call that one. Let's call that two. And then I wanna get, let's see, if I go to five and three-fourths, let's go up to, let's see one, two, three, four five, six, seven. We can label 'em. One, two, three, four five, six and seven and so our vertex is right over here. One comma 23 over four, so that's five and three-fourths. So it's gonna be right around right around there and as we said, since we have a negative value in front of this x minus one squared term, I guess we could call it, this is going to be a downward opening parabola. This is going to be a maximum point. So our actual parabola is going to look is going to look something it's gonna look something like this. It's gonna look something like this and we could, obviously, I'm hand drawing it, so it's not going to be exactly perfect, but hopefully you get the general idea of what the parabola is going look like and actually, let me just do part of it, 'cause I actually don't know that much information about the parabola just yet. I'm just gonna draw it like that. So we don't know just yet where the directrix and focus is, but we do know a few things. The focus is going to sit on the same, I guess you could say, the same x value as the vertex. So if we draw, this is x equals one, if x equals one, we know from our experience with focuses, foci, (laughs) I guess, that they're going to sit on the same axis as the vertex. So the focus might be right over here and then the directrix is going to be equidistant on the other side, equidistant on the other side. So the directrix might be something like this. Might be right over here. And once again, I haven't figured it out yet, but what we know is that because this point, the vertex, sits on the parabola, by definition has to be equidistant from the focus and the directrix. So. This distance has to be the same as this distance right over here and what's another way of thinking about this entire distance? Remember, this coordinate right over here is a, b and this is the line y is equal to k. This is y equals k. So what's this distance in yellow? What's this difference in y going to be? Well, you could call that, in this case, the directrix is above the focus, so you could say that this would be k minus b or you could say it's the absolute value of b minus k. This would actually always work. It'll always give you kind of the positive distance. So if we knew what the absolute value of b minus k is, if we knew this distance, then just split it in half with the directrix is gonna be that distance, half the distance above and then the focus is gonna be half the distance below. So let's see if we can figure this out. And we can figure this out because we see in this, I guess you could say, this equation, you can see where b minus k is involved. One over two times b minus k needs to be equal to negative one-third. So let's solve for b minus k. So we get we get one over two times b minus k is going to be equal to negative one-third. Once again, this corresponds to that. It's going to be equal to negative one-third. We could take the reciprocal of both sides and we get two times b minus k is equal to, is equal to three, is equal to three. Now we can divide both sides we can divide both sides by two and so we're gonna get we're gonna get b b minus k is equal to is equal to, what is that, three-halves, three-halves. b minus k is equal to, oh, let me make sure that has to be a negative three, so this has to be negative three-halves. And so if you took the absolute value of b minus k you're gonna get positive three-halves, or if you took k minus b, you're going to get positive three-halves. So just like that, using this part, just actually matching the negative one-third to this part of this equation, we're able to solve for the absolute value of b minus k which is going to be the distance between the y axis in the y direction between the focus and the directrix. So this distance right over here is three-halves. So what is half that distance? And the reason why I care about half that distance is because then I can calculate where the focus is, because it's going to be half that distance below the vertex and I could say, whatever that distance is is going to be that distance also above the directrix. So half that distance, so one half times three-halves is equal to three-fourths. So just like that, we're able to figure out the directrix is going to be three-fourths above this. So I could say the directrix, so let me see, I'm running out of space, the directrix is gonna be y is equal to the y coordinate of the focus. Sorry, the y coordinate of the vertex. I might be careful with my language. It's gonna be equal to the y coordinate of the vertex plus three-fourths, plus three- fourths. So plus three-fourths, which is equal to 26 over four, which is equal to, what is that, that's equal to six and a half. So this right over here, actually I got pretty close when I drew it is actually going to be the directrix. Y is equal to six and a half and the focus, well, we know the x coordinate of the focus, a is going to be equal to one and b is going to be three-fourths less than the y coordinate of the directrix. So 23 over four minus three-fourths. Gonna be 23 over four 23 over four minus three-fourths which is 20 over four, which is just equal to which is just equal to five. And we are done. That's the focus, one comma five. Directrix is y is equal to six and a half.