Let’s see if we can learn

a thing or two about conic sections. So first of all, what are

they and why are they called conic sections? Actually, you probably

recognize a few of them already, and I’ll

write them out. They’re the circle, the

ellipse, the parabola, and the hyperbola. That’s a p. Hyperbola. And you know what

these are already. When I first learned conic

sections, I was like, oh, I know what a circle is. I know what a parabola is. And I even know a little bit

about ellipses and hyperbolas. Why on earth are they

called conic sections? So to put things simply because

they’re the intersection of a plane and a cone. And I draw you

that in a second. But just before I do that it

probably makes sense to just draw them by themselves. And I’ll switch colors. Circle, we all know

what that is. Actually let me see if

I can pick a thicker line for my circles. so a circle looks

something like that. It’s all the points that are

equidistant from some center, and that distance that they

all are that’s the radius. So if this is r, and this is

the center, the circle is all the points that are exactly

r away from this center. We learned that early in our

education what a circle is; it makes the world

go round, literally. Ellipse in layman’s terms is

kind of a squished circle. It could look

something like this. Let me do an ellipse

in another color. So an ellipse could

be like that. Could be like that. It’s harder to draw using the

tool I’m drawing, but it could also be tilted and

rotated around. But this is a general sense. And actually, circles are a

special case of an ellipse. It’s an ellipse where it’s not

stretched in one dimension more than the other. It’s kind of perfectly

symmetric in every way. Parabola. You’ve learned that if you’ve

taken algebra two and you probably have if you care

about conic sections. But a parabola– let me draw a

line here to separate things. A parabola looks something like

this, kind of a U shape and you know, the classic parabola. I won’t go into the

equations right now. Well, I will because you’re

probably familiar with it. y is equal to x squared. And then, you could shift it

around and then you can even have a parabola that

goes like this. That would be x is

equal to y squared. You could rotate these things

around, but I think you know the general shape

of a parabola. We’ll talk more about how do

you graph it or how do you know what the interesting points

on a parabola actually are. And then the last one,

you might have seen this before, is a hyperbola. It almost looks like two

parabolas, but not quite, because the curves look a

little less U-ish and a little more open. But I’ll explain what

I mean by that. So a hyperbola usually

looks something like this. So if these are the axes,

then if I were to draw– let me draw some asymptotes. I want to go right through

the– that’s pretty good. These are asymptotes. Those aren’t the

actual hyperbola. But a hyperbola would look

something like this. They get to be right here

and they get really close to the asymptote. They get closer and closer to

those blue lines like that and it happened on this side too. The graphs show up here and

then they pop over and they show up there. This magenta could be one

hyperbola; I haven’t done true justice to it. Or another hyperbola could be

on, you could kind of call it a vertical hyperbola. That’s not the exact word, but

it would look something like that where it’s below

the asymptote here. It’s above the asymptote there. So this blue one would be

one hyperbola and then the magenta one would be a

different hyperbola. So those are the

different graphs. So the one thing that I’m sure

you’re asking is why are they called conic sections? Why are they not called

bolas or variations of circles or whatever? And in fact, wasn’t

even the relationship. It’s pretty clear that

circles and ellipses are somehow related. That an ellipse is just

a squished circle. And maybe it even seems that

parabolas and hyperbolas are somewhat related. This is a P once again. They both have bola in their

name and they both kind of look like open U’s. Although a hyperbola has two of

these going and kind of opening in different directions,

but they look related. But what is the connection

behind all these? And that’s frankly where

the word conic comes from. So let me see if I can draw

a three-dimensional cone. So this is a cone. That’s the top. I could’ve used an

ellipse for the top. Looks like that. Actually, it has no top. It would actually keep going

on forever in that direction. I’m just kind of slicing it

so you see that it’s a cone. This could be the

bottom part of it. So let’s take different

intersections of a plane with this cone and see if we can at

least generate the different shapes that we talked

about just now. So if we have a plane that goes

directly– I guess if you call this the axis of this

three-dimensional cone, so this is the axis. So if we have a plane that’s

exactly perpendicular to that axis– let’s see if I can

draw it in three dimensions. The plane would look

something like this. So it would have a line. This is the front line that’s

closer to you and then they would have another

line back here. That’s close enough. And of course, you know these

are infinite planes, so it goes off in every direction. If this plane is directly

perpendicular to the axis of these and this is where

the plane goes behind it. The intersection of this

plane and this cone is going to look like this. We’re looking at it from an

angle, but if you were looking straight down, if you were

listening here and you look at this plane– if you were

looking at it right above. If I were to just flip this

over like this, so we’re looking straight down on this

plane, that intersection would be a circle. Now, if we take the plane and

we tilt it down a little bit, so if instead of that we

have a situation like this. Let me see if I can

do it justice. We have a situation

where it’s– whoops. Let me undo that. Edit. Undo. Where it’s like this and has

another side like this, and I connect them. So that’s the plane. Now the intersection of this

plane, which is now not orthogonal or it’s not

perpendicular to the axis of this three-dimensional cone. If you take the intersection of

that plane and that cone– and in future videos, and you

don’t do this in your algebra two class. But eventually we’ll kind of

do the three-dimensional intersection and prove that

this is definitely the case. You definitely do get the

equations, which I’ll show you in the not too far future. This intersection would

look something like this. I think you can

visualize it right now. It would look

something like this. And if you were to look

straight down on this plane, if you were to look right above

the plane, this would look something– this figure I

just drew in purple– would look something like this. Well, I didn’t draw

it that well. It’d be an ellipse. You know what an

ellipse looks like. And if I tilted it the other

way, the ellipse would squeeze the other way. But that just gives you a

general sense of why both of these are conic sections. Now something very interesting. If we keep tilting this plane,

so if we tilt the plane so it’s– so let’s say we’re

pivoting around that point. So now my plane– let me

see if I can do this. It’s a good exercise in

three-dimensional drawing. Let’s say it looks

something like this. I want to go through

that point. So this is my

three-dimensional plane. I’m drawing it in such a way

that it only intersects this bottom cone and the surface

of the plane is parallel to the side of this top cone. In this case the intersection

of the plane and the cone is going to intersect

right at that point. You can almost view that I’m

pivoting around this point, at the intersection of this point

and the plane and the cone. Well this now, the

intersection, would look something like this. It would look like that. And it would keep going down. So if I were to draw it,

it would look like this. If I was right above the

plane, if I were to just draw the plane. And there you get

your parabola. So that’s interesting. If you keep kind of tilting–

if you start with a circle, tilt a little bit,

you get an ellipse. You get kind of a more

and more skewed ellipse. And at some point, the ellipse

keeps getting more and more skewed like that. It kind of pops right when you

become exactly parallel to the side of this top cone. And I’m doing it all very

inexact right now, but I think I want to give

you the intuition. It pops and it turns

into a parabola. So you can kind of view

a parabola– there is this relationship. Parabola is what happens when

one side of an ellipse pops open and you get this parabola. And then, if you keep tilting

this plane, and I’ll do it another color– so

it intersects both sides of the cone. Let me see if I can draw that. So if this is my new

plane– whoops. That’s good enough. So if my plane looks like

this– I know it’s very hard to read now– and you wanted the

intersection of this plane, this green plane and the cone–

I should probably redraw it all, but hopefully you’re not

getting overwhelmingly confused– the intersection

would look like this. It would intersect the bottom

cone there and it would intersect the top

cone over there. And then you would have

something like this. This would be intersection of

the plane and the bottom cone. And then up here would be

the intersection of the plane and the top one. Remember, this plane goes off

in every direction infinitely. So that’s just a general sense

of what the conic sections are and why frankly they’re

called conic sections. And let me know if this got

confusing because maybe I’ll do another video while I redraw

it a little bit cleaner. Maybe I can find some kind of

neat 3D application that can do it better than I can do it. This is kind of just the reason

why they all are conic sections, and why they really

are related to each other. And will do that a little

more in depth mathematically in a few videos. But in the next video, now that

you know what they are and why they’re all called conic

sections, I’ll actually talk about the formulas about these

and how do you recognize the formulas. And given a formula, how do

you actually plot the graphs of these conic sections? See you in the next video.

wish me luck in my exam later

The funnest thing is i'm in pre-algebra and now i know and can look into the future :]

That's beautiful! It's sad that most math teachers in high school are too exhausted and the students themselves are too lazy to emphasize the beauty of mathematical concepts. I wish I knew this before so that i could have truely understood conics and be able to pass the tests. Instead, my math teacher was nice enough in 11th grade to drop this test and signed me up for AP calculus AB for the next year. Well, now that I'm a freshman in college calculus 2 leveled,Iwishiknewthisearlier

Thank you!!!!! Now that I have watched your videos I understand all of it!!!! I use your website too. 😀

Check this out! This video has a Download button in the description !!!

Thank you so much.

You can not understand how greatly thankful I am to for you, you're what we need in front of our classrooms and at our dining tables teaching kids things.

Just, thank you.

thank you

What if you put the plane completely vertical and in the centre of the cone? Is the result of that an absolute value graph…or am I just confusing myself now?

Nice video btw! 🙂

Haha, 4:17 there's a smily face 🙂

Awesome video, very clear understanding at the end of it's ten minutes. Audio quality was superb.

The line and point are both conic sections as well, yes?

That's a hyperbola 😛 x^2 – y^2 = 0

I hate my algrebra teacher for her lame way of teaching this simple stuff. And how she gave us that mocking sneer when we couldnt understand her.

That was a time when Cobain was ruling the airwaves, and so we just opted to skip class and play grunge music….

….and I admit the consequences of having LAME teachers are totally disastrous to a child's life.

Good thing theres youtube to rectify it all…oh wait, is that a Nirvana video thumbnail… m/

assholetotes

" A Parabola is what happen when one side of an ellipse POPS OPEN! " 😀

Thanks for the idea!

Exercise in 3-d drawing!! 😀 XD

thank you

I have a video presentation about Conics in my channel. I'd appreciate it if you guys gave it a look 😀

yeah, correct! us too.

why aren't they called "bolas" xD

10:52 … wow

2:14 happy face

thnx good explanation

Everyone from Teague and Anderson, this is pretty good stuff! Better hope we don't end up chasing this down with 8 sections of notes though. XD Have a great day everyone!

#p.4 teauge? nah #'s too lame

Baylife

Heater gonna heat

when he started the 3D drawing I kept moving my head right and left without even noticing lol XD

4:20

smiley face 🙂

I guess you never heard of Khan Academy,just search for khanacademy on google and visit their website and browse through the entire collection of video lectures.

studying for my upcoming midterm and then my final…i hope i do well! :/

Unquestionably World's best teacher!

I actually fell asleep your voice is so soothing.

ms paint

Someone buy Sal a lap dance!

I would never pass without you Sal, thank you very much!!!!!!!!

2:21 looks like a smiley..

Im currently learning this through BYU. this is much better

Thanks keep up the good work sir

2:40 looks like he's drawn a smiley face 😀

PARABOLA ANR CIRCLES

And*

Thanks thanks thanks

really great introduction 🙂

Great introduction

Whats wrong with the website??? On YouTube I have to search for each video of a Chapter individually!

excellent!

epic

At 2:30 you have a smiley face lol

define focus?

wtf I learningwhat I should've learned years ago, didn't have the resources till now. I would've skipped ahead.

thank u sir for excellent lecture

Hey the formulas for parabola are not y = x^2 and x = y^2…..But they are x^2 = 4ay and y^2 = 4ax…..

Can u link previous videos please also using all maths on khan webpage how long is that going to take as takeb 5 days and only learnt 20 skills

ATTACK OF THE COLORED LINES!!

Smiley face!

why 2 cones?

Very cool, orbits are great.

taking precalc II trying to read ahead is this the same content that would be taught in my class? anyone have an idea?

Sir I have a question: wasn't that parabola one intersecting the above cone (the one which was parallel ) so won't it will actually become a hyperbola?

I wish your videos were shorter, there's a lot of "wasted" time. Otherwise, good video!

Are you the same guy that does videos on other topics(e.g. vectors)? This one was terrible, I grasped nothing. But the other videos you're brilliant

I can sleep in class without any worry till the existence of Khan Academy 😛

does the bottom part of the cone go on forever too???

most excellent! good explanation.

KHANic sections.

Instead of Parabola, it's Ebola! XD

EMPOWERED'''''

Pss, don't worry this is very understandable

man you have done a mistake in 10:16 its x^2=y in first and its y^2=x in second one!!!!!!! just kidding. KEEP IT UP

you sound more like Khanic sections !😂

pioneering e learning!

Your comparison of the two dimensional plain representing a single cross section of a three dimensional object completely just changed my picture of geometry, thank you.

jajajj you drew a smiley face

mannn i dont know anything

i am watching it in 2017

ebola also has bola in its name

4 minutes in…still haven't learned anything

Loliconic anyone?

for simple 3d conic structures>>>>>>https://youtu.be/HO2zAU3Eppo

"you don't do this in your algebra 2 class"

I am actually using this video for a project in my algebra 2 class where we are doing exactly what you said we shouldn't XD

i wish you never spit out unnecessary words. it will make the video less complicated to understand.

I dont think this one would be useful lol. You can just proceed to those fvcking formulas and teach it to us because our teachers dont teach this fvcking lessons properly.

DRAW ANOTHER 3D CONES IN EACH CONICS SECTION YOU IDIOT!

How can I contact Salman ?

Idky but his voice annoys me

I like your humor khan academy :))

Thank you for this video. I am taking precalculus in college. It really helps. Thanks again, Sal Khan, Khan Academy

Prrrrraabola!!

That video was amazing! Great job Sal

Do you know how many tests I didn't fail because of khan academy

Now I'm less likely to fail my test. I literally made a 0 on a quiz to to a lack of understanding

🙏

The parabola should've been drawn as a line in the planar view, not a curve, since the plan you're cutting with is perpendicular to the plan view.

Still helpful for an old video and this is better than all the math videos from 2006 and 2007.

Dude this video is still useful even after 10 years of when it was published

Awe

i learn more from this channel than i do from my school teacher

5:40 what's that sound

if you understood this 10:02 ,you deserve a like:)

Did you realise that if we say the angle between axis and plane is alpha, and suppose orbital speed is a function of alpha orbit's shape will change as well as that conic section.

This is the proof that physics is the first derivative of math, in other words physics is nothing but applied mathematics.