Focus & directrix of a parabola
- What I want to do in this video is understand two words that you might hear associated with parabolas. And that is, that is a focus. That is a focus of a parabola, focus. And a directrix. Directrix. Directrix. Directrix, right over there. So what are these things? Well, a parabola can be defined as the set of all points, let me draw arbitrary axes right over here. So that's my y axis, this is my x axis. This is my x axis. And so a parabola can be defined as the set of all points that are equidistant to a point and a line. And that point is the focus of that parabola, and that line is the directrix of the parabola. So what am I talking about? So let's give ourselves a point. So let's say this point right over here, and we can even say that that is the point let's say that's the point, let's say the x coordinate is a, and the y coordinate is b right over here. So that is the point (a, b). And then let's give ourselves a line for the directrix. And actually I'm going to do this in a different color instead of just white, cos I did the coordinates in white. So I will do it in this magenta color. So that's (a,b) is the focus. And let's say y=c is the directrix. So this right over here is the line, this right over here is the line y is equal to c. So this on the y axis right over there, that is c, this is the line y=c. So a parabola, what does it mean to be the set of all points that are equidistant between a point and this line? Let's think about what those points might be. Well, this point right over here would be halfway between, between this point, between the focus and the directrix. And then as we move away from x=a, you're going to get points anywhere along this curve. Which is a parabola. And you might be saying, wait, I don't get this, I don't get why points along this curve are going to be equidistant. Well, let's just eyeball the distances. So this one, this distance, and obviously I'm drawing it by hand so it's not going to be completely precise, that distance needs to be equal to that distance. Well, that seems believable. Now this, if we take this point right over here of the parabola, this distance needs to be the same as that distance. Well, that seems believable. If you take this point on the parabola, this distance, this distance needs to be the same as this distance. So hopefully you get what I'm talking about when I say that the parabola is a set of all points that are equidistant to this focus and this directrix. So any point along this parabola, this point right over here, this point right over here, the distance to the focus, the distance to the focus, should be the same as the distance, as the distance to the directrix. Now, what you might realize is when you're taking the distance between a point and a point, the distance can, it'll be at, I guess you could say it could be at an angle. This one's straight up and down, this one is going from the top left to the bottom right. But when you take the distance from a point to a line, you essentially drop a perpendicular, you essentially go straight down. Or if the parabola was down here, you'd go straight up to find that distance. These are all, these are all right angles right over here. So that's all a focus and a directrix is. And every parabola is going to have a focus and a directrix, because every parabola is the set of all points that are equidistant to some focus and some directrix. So that's what they are. In future videos we'll try to think about, how do you relate these points, the focus and directrix, to the actual, to the actual equation, or the actual equation for a parabola.