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# Intro to focus & directrix

CCSS.Math:

## Video transcript

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 an arbitrary axis right over here so it'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 could 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 comma B and then let's give ourselves a line for the directrix and actually let me do this in a different color and so just white because I did the coordinates in white so I will do it in this magenta color so that's a comma B is the focus and let's say y equals 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 is equal to 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 equals a you're going to get points anywhere along this curve which is a parabola and you we see wait I don't I don't get this I don't get why points along this curve are going to be equidistant well let's just I ball 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 and now this if we take this point right over here on 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 all as a set of all points that are equidistant to this focus and this directory 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 you could the distance can it'll be at a 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 would 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 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 we'll try to think about how do you relate these points the focus and the directrix to the actual to the actual equation or the actual equation for a parabola