– [Voiceover] 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 gonna 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, a 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,

and its 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 say, that is, that’s going to be the shortest distance to that line, but the distance to the focus, well we see that’s at 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 gonna

be our change in 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 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 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, we could square it, and then we could take the square root, the principle 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,y) to be

sitting on the parabola, that distance needs to be

the same as the distance from (x,y) to (a,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 gonna 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 a 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, it’s just gonna be a little

bit of hairy algebra, but it really is not too bad. You’re gonna get an

equation for a parabola that you might recognize, and it’s gonna be in terms

of a general focus, (a,b), and a gerneral directrix, y equals k, so let’s do that. So the simplest thing to start here, is 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 gonna get y minus k, squared is equal to x minus a, squared, 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’s, ab’s, and

k’s 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 y squared minus 2yk, plus k squared, and that is going to be equal to, I’m gonna keep this first one the same, so it’s gonna be x minus a, squared, and now let me expand, I’m gonna find a color, expand this in green, so plus y squared, minus

2yb, plus b squared. All I did, 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 simplified things a little bit, and now 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 from both sides, so that’s gonna get rid of

it on the left-hand side, and now let’s add 2yb to both sides, so we have all the y’s

on the left-hand side, so, plus 2yb, that’s gonna give us a

2yb on the left-hand side, plus 2yb. So what is this going to be equal to? And I’m starting 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 2yb minus 2yk, which is the same thing, actually let me just write that down. That’s going to be 2y– Do it in green, actually,

well, yeah, why not green? That’s going to be– Actually, let me start

a new color. (chuckles) That’s going to be 2yb minus 2yk. You can factor out a 2y, and it’s gonna be 2y times, b minus k. So let’s do that. So we could write this as 2 times, b minus k, y if you factor out a 2 and a y, so that’s the left-hand side, so that’s that piece right over there. 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 gonna be

b squared minus k squared, plus b squared minus k squared. Now, I said all I want is

a y on the left-hand side, so let’s divide everything

by two times, b minus k. So, let’s divide everything, two times, b minus k, so, two times, b minus k. And I’m actually gonna

divide this whole thing by two times, b minus k. 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 notice, 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. So if you knew what b minus k was, 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, (chuckles) if you do remember parabolas

from your childhood. Alright, 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 b plus k, times b minus k, so the b minus k’s cancel out, and we are just left with, and we deserve a little

bit of a drum roll, we are just left with 1/2 times, b plus k. So, there you go. Given a focus at a point (a,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, if I had a focus at the point, I don’t know, let’s say the point (1,2), and I had a directrix at y is equal to, I don’t know, let’s make it y is equal to -1, what would the equation

of this parabola be? Well, it would be y is equal to one over, two times, b minus k, so two minus -1, that’s the same thing as two plus one, so that’s just three, two minus -1 is three, times x minus one, squared, plus 1/2 times, b plus k. Two plus -1 is one, so one, and so what is this going to be? You’re gonna get y is equal to 1/6, x minus one, squared, plus 1/2. There you go. That is the parabola with a focus at (1,2) and a directrix at y equals -1. Fascinating.

Fascinating indeed

Thank you so much man ! You are fantastic

Thank you !!!

tnx alot !!!!!!!!!!

"Hairy algebra" he says.

That formula applies to ALL parabolas?

Which software do you use to write everything?

The link under (Missed the previous lesson?) is broken.

Thank you! I was in need of such a video for computing voronoi diagrams in my project

the point in last was the coordinate (x,y) or (a,b)

hey thanks !!

Oh Nice !! ………………………… Khanic Sections

how can we find the equation of a hyperbola given the focus and directrix

Everytime I struggle with my algebra, I come here. I do online school so it's hard to move forward when I'm stuck without my teacher getting back to me (especially since I'm a night owl. I normally have to wait until the following day). Thank you! I'm bad with numbers but when I watch these videos they (sort of) make sense.

When you're in high school, and you're asking yourself, "when am I ever gonna need to know this?" That time is grad school. I never throught I would have to revisit precalc to find the arc length of a cable on a suspension bridge

tnx

"hairy algebra"

can we just

crying and laughing at the same time. lol "hairy algebra" it is!

Because of this parabola I lost my childhood and teenage days

what if the equation of dirctrix includes 'x' also?

A parabola is defined as the locus of points that are equidistant from both the directrix and the focus in a coordinate plane. … the math (the derivation of the general form of the equation of a parabola) follows from that definition. It is simple IF you begin with the correct definition.

I like my algebra waxed !!

How do I find the equation of the ellipse given only that the the major axis is vertical and the graph passes through the vertex of a parabola?

not helpful

nice

Define childhood ….. do I remember parabolas from my childhood? ✨📚🍨🍉

when your teacher gives you a question on a test that he has never covered…

hola que tal, como les va

Hate math!!!!

love yall

y-k then y=k so it means its 0?

"It looks really hairy." HAHAHA

How do we find the focus or the directrix when the vertex is (4,0) and solving for y=? I can’t find anywhere that answers this. My question is to find if there is a minimum or maximum of the equation y=(x+4)squared.

Really fantastic Sal. Thank you so much.

for what grade is this mainly on? like 8th, 9th, 10th etc.

" Hairy algebra"

Shivers5:21

*Right side of the screen exists*Sal: it's free real estate

I’m a little stupid so I’d like to know how you got “-2yk”

i dare you to prove parabola opens left

1:29

What’s the purpose of using the distant formula here?

THE PAIN!!

5:20 "Let me give myself a little bit real estate over here" lol

The directrix cannot be y=k since k is the why coordinate of the vertex >_<

Thanks I will just memorize it

what if directrix is x = a

Thanks Salman khan to create this Educational Organization!

I watched another video regarding this. It demonstrated another formula (y-k)^2=4p(x-h) Is this formula applicable?