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# Equation of a parabola from focus & directrix

CCSS.Math:

## Video transcript

what I have attempted to draw here in yellow is a parabola and as we've already seen in previous videos a parabola can be defined as the set of all points that are equidistant to a point and a line and the point is called the focus of the parabola and the line is called the directrix of the parabola what I want to do in this video it's going to get a little bit of hairy algebra but given that definition I want to see and given that definition and given a focus at the point x equals a y equals b and a line of directrix at y equals K to figure out what is the equation of that parabola actually going to be and it's going to be based on A's B's and K's so let's do that so let's take a arbitrary point on the parabola let's say we take this point right over here and its x-coordinate is X that it's y-coordinate is y and by definition in order for this to be a parabola it has to be equidistant to its focus and its directrix so what does that mean that means that the distance to the directrix which I'm drawing here in blue has to be the same as the distance to the focus which I am drawing in magenta and when we take the distance to the directrix we literally just drop a perpendicular I guess you could say that is the that's going to be the shortest distance to that line and then when we but the distance to the focus well--that's we see that set a that's a bit of an angle and we might have to use the distance formula which is really just the Pythagorean theorem so let's do that this distance has to be the same as that distance so what's this blue distance well that's just going to be our change in Y it's going to be this Y it's going to be this Y minus K it's just this distance so it's going to be Y minus K now we have to be careful the way I've just drawn it yes Y is greater Y is greater than K so this is going to give us a positive value and you need a non-negative value if you're talking about distances but there are you can definitely have a parabola where the y coordinate of the focus is lower than the y coordinate of the directrix in which case this would be negative so what we really want is the absolute value of this or you could say we could square it and then we could take the square root the principal root which would be equivalent to taking the absolute value of Y minus K so that's this distance right over here and by the definition of a parabola in order for X comma Y to be sitting on the parabola that distance needs to be the same as the distance from X comma Y to a comma B to the focus so what's that going to be well we just apply the distance formula or really just the Pythagorean theorem it's going to be our change in X so X minus a squared plus the change in Y Y minus B squared and the square root of that whole thing the square root of all of that business now this right over here is an equation of our parabola it doesn't look like it it looks really hairy but it is the equation of a parabola and to show you that we just have to simplify this and if you get inspired I encourage you to try to simplify this on your own is just going to be a little bit of hairy algebra but it really is not too bad you're going to get an equation for a parabola that you might recognize it's going to be in terms of kind of a general focus a comma B and a general directrix y equals K so let's do that so the simplest thing is to start here so let's just square both sides so we get rid of the radicals so if you square both sides on the left hand side you're going to get Y minus K squared you're going to get Y minus K squared is equal to is equal to X minus a squared plus y minus B plus y minus B squared fair enough now what I want to do is I just want to end up with just a Y on the left hand side and just X is a B's and KS on the right-hand side so the first thing I might want to do is let's expand each of these expressions that involve with Y so this blue one on the left hand side that is going to be that's going to be Y squared minus 2y k plus K squared and that is going to be equal to I'm going to keep this first one the same so it's going to be X minus a squared and now let me expand let me expand I'm going to find a color expand this in green so plus y squared minus 2y B plus B squared all it is I multiplied y minus B times y minus B now let's see if we can simplify things so I have a Y squared on the Left I have a Y squared on the right well if I subtract Y squared from both sides so I can do that well that simplifies things a little bit and now I can I can let's see what I can do well let's get the K squared on this side so let's subtract K squared from both sides so subtract K squared subtract K squared from both sides so that's going to get rid of it on the left hand side and now let's add 2y B to both sides so we have all the Y's on the left-hand side so plus 2y B that's going to move this that's going to put us give us a 2y be on the left-hand side plus 2y B so what is this going to be equal to and I'm trying to run into my graph so let me give myself a little bit more real estate over here so on the left-hand side what am i going to have this is the same thing as 2 y be -2 YK which is the same thing actually let me just write that down that's going to be 2y do it in green G well yeah why not green that's going to be actually we start a new color that's going to be 2y B minus 2 y K you can focus you can factor out a 2 y and it's going to be 2y times B minus K so let's do that so we could write this as 2 or we could write it 2 times B minus K Y if you factor out a 2 and Y so that's the left-hand side so that's that piece right over there these these things cancel out now on our right-hand side I promised you a little bit of hairy algebra so hopefully you see that I'm delivering on that promise on the right-hand side you have X minus a squared and then let's see these characters cancel out and you're left with B squared minus K squared so these two are going to be B squared minus K squared plus B squared minus K squared now I said all I want is Y on the left hand side so let's divide everything by 2 two times B minus K so let's divide everything times two times B minus K so two times B minus K and I'm actually going to divide this whole thing by two times B minus K now what now obviously on the left hand side this all cancels out you're left with just a Y and then it's going to be y equals y is equal to one over two times B minus K and less B minus K is the difference between the y coordinate of the focus and the y coordinate I guess you could say of the line y equals K so it's one over two times that times X minus a squared X minus a squared so if you knew what B minus K was R this would just simplify to some number some number that's being multiplied times X minus a squared so hopefully this is starting to look like the parabolas that you remember from your childhood if you do remember parabolas from your childhood all right so then let's see if we could simplify this thing on the right and you might recognize B squared minus K squared that's a difference of squares that's the same thing as that's the same thing as B plus K times B minus K times B whoops times B minus K so the B minus K is canceled out and we are just left with and we deserve a little bit of a drumroll we're just left with 1/2 times B plus K so there you go given a focus at the point a comma B and a directrix at y equals K we now know what the formula of the parabola is actually going to be so for example so for example if I had a focus if I had a focus at the point I don't know let's say the point 1 comma 2 and I had a directrix I had a directrix at why Y is equal to and let's make it Y is equal to negative 1 what would the formula for what would the equation of this parabola be well it would be Y equal to one over two times B minus K so 2 minus negative 1 that's the same thing as 2 plus 1 so that's 2 times that's just 3 2 minus negative 1 is 3 times X minus times X minus 1 whoops X minus 1 squared plus 1/2 times B plus K 2 plus negative 1 is 1 so 1 and so what's this going to be you're going to get Y is equal to 1 over 1/6 X minus 1 squared plus 1/2 there you go that is the parabola with a focus at 1 comma 2 and a directrix at y equals negative 1 fascinating