Conic sections: Intro to ellipse | Conic sections | Algebra II | Khan Academy
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Conic sections: Intro to ellipse | Conic sections | Algebra II | Khan Academy


In the last video, we learned a
little bit about the circle. And the circle is really just
a special case of an ellipse. It’s a special case because in
a circle you’re always an equal distance away from the center
of the circle, while in an ellipse, the distance from the
center of the circle is always changing. You know what an
ellipse looks like. Well, I showed you that
in the first video. It looks something like that. What I mean is that the radius
or the distance from the center is always changing. Let me say this is
centered at the origin. So that’s the origin
right there. You see here, we’re really, if
we’re on this point on the ellipse, we’re really
close to the origin. This is actually the closest
we’ll ever get, just as close as well get down here. And when we’re out here we’re
really far away from the origin and that’s about as far as
we’re going to get right there. So a circle is a special case
of this, because in a circle’s case, the furthest we get from
the origin is the same distance as the closest we get, or, in
other words, we are always the exact same distance
away from the origin. Well, with that said, let’s
actually go a little bit into the math. So the general or the standard
form for an ellipse centered at the origin is x squared over a
squared plus y squared over b squared is equal to 1. Where a and b are just
any two numbers. I could have written this as
c squared and d squared. I mean, they’re just
place holders. Just to give you an idea of
what this means, if this was our ellipse in question right
now, a is the length of the radius in the x-direction. Remember, we’re going to
have a squared down here. So if you took the square
root of whatever is in the denominator, a is the x-radius. So this distance in our little
chart right here, in our little graph here, that distance is a,
or that this point right here, since we’re centered at the
origin, will be the point x is equal to a y is equal to 0. And of course this point right
here this will be a, so this would be the point
minus a comma 0. And then the radius in the
y-direction would be this radius right here and is b. So this point would be x is
equal to 0, y is equal to b. Likewise this point right here
would be x is equal to 0, y is equal to minus b. And the way I drew this, we
have kind of a short and fat ellipse you can also have kind
of a tall and skinny ellipse. But in the short and fat
ellipse, the direction that you’re short in that’s
called your minor axis. And so b, I always forget the
exact terminology, but b you can call it your semi or the
length of your semi-minor axis. And where did that
word come from? Well if this whole thing is
your minor axis or maybe you could call your minor diameter
if this whole thing is your minor diameter it’s called
minor, because it’s the shortest of all of the
diameters of this ellipse. And then the semi
is half of that. b is the length of
the semi-minor axis. That’s b in this example, just
because as I drew this ellipse it just happens to be that
b is smaller then a. If b was larger than a, I would
have a tall and skinny ellipse. Let me actually draw one. It could have been like this. I could have an ellipse that
looks something like that. In which case, all of a sudden
b would be the semi-major axis, because b would be
greater than a. That this would be
taller than it is wide. But let me not confuse
the graph too much. And in this case, a is the
length of– I think you’ve guessed it– a is the length of
the semi-major axis or you can even call it the length
of the major radius. I think that makes more sense. And you can call this
the minor radius. So let’s just do an example. And I think when I’ve done an
example with actual numbers, it’ll make it all a
little bit clearer. So let’s say I were to show up
at your door with the following: If I were to say x
squared over 9 plus y squared over 25 is equal to 1. So what is your radius
in the x-direction? This is your radius in
the x-direction squared. So your radius in the
x-direction if we just map it, we would say that
a is equal to 3. Because this is a squared. And if we were just map it we’d
say this is b squared than this tells us that b is equal to 5. So if we wanted to graph
this, and once again this is centered at the origin. Let me draw the ellipse first. So, first of all, we have our
radius in the y-direction is larger than our radius
in the x-direction. The ellipse is going to
be taller and skinnier. It’s going to look
something like that. Draw some axes, so that could
be your x-axis, your y-axis. This is your radius
in the y-direction. So this distance right here
is going to be 5, and so will this distance. And this is your radius
in the x-direction. So this will be 3,
and this will be 3. That’s it. You have now plotted
this ellipse. Nothing too fancy about it. And actually just to kind of
hit the point home that the circle is a special
case of an ellipse. We learned in the last video
that the equation of a circle is x squared and a circle
centered at the origin. x squared plus y squared
is equal to r squared. So if we were to divide both
sides of this by r squared, we would get– and this is just
little algebraic manipulation– x squared over r squared plus y
squared over r squared is equal to 1. Now in this case, your a
is r and so is your b. So your semi-minor axis
is r and so is your semi-major axis of r. Or, in other words, this
distance is the same as that distance, and so it will
neither be short and fat nor tall and skinny. It’ll be perfectly round. And so that’s why the circle is
a special case of an ellipse. So let me give you a slightly–
It’ll look a lot more complicated, and this is
something you might see on exam. But I just want to show you
that this is just a shifting. Let’s say we wanted to
shift this ellipse. Let’s say we wanted to shift
it to the right by 5. So instead of the origin being
at x is equal to 0, the origin will now be at x is equal 5. So a way to think about that is
what does this term have to be so that at 5 this term
ends up being 0. Well I’ll actually draw it
for you, because I think that might be confusing. So if we shift that over the
right by 5, the new equation of this ellipse will be x minus 5
squared over 9 plus y squared over 25 is equal to 1. So if I were to just draw
this ellipse right now, it would look like this. I want to make it look
fairly similar to the ellipse I had before. It would look just like that. Just shifted it over by five. And the intuition we learned a
little bit in the circle video where I said, oh well, you
know, if you have x minus something that means that the
new origin is now at positive 5. And you could memorize that. You could always say, oh, if I
have a minus here, that the origin is at the negative of
whatever this number is, so it would be a positive five. You know, if you had a positive
it would be the opposite that. But the way to really think
about it is now if you go to x is equal to 5, when x is equal
to 5, this whole term, x minus 5, will behave just like
this x term will here. When x is equal to 5 this
term is 0, just like when x was 0 here. So when x is equal to 5, this
term is 0, and then y squared over 25 is equal 1, so y
has to be equal five. Just like over here when x is
equal is 0, y squared over 25 had to be equal to 1,
y is equal to either positive or minus 5. And I really want to give
you that intuition. And then, let’s say we
wanted to shift this equation down by two. So our new ellipse looks
something like this. A lot of times you learned
this in conic sections. But this is true any function. When you shift things,
you shift it this way. If you shift this graph to the
right by five, you replace all of the x’s with x minus 5. And if you were to shift it
down by two, you would replace all the y’s with y plus 2. So let me draw our new
ellipse first, just to show you what I’m doing. So our new ellipse is going
to look something like that. I’m shifting the yellow
ellipse down by two. So this equation, if I shift it
down, well, the x is still where it was before. x minus 5
squared over 9 plus y plus 2 squared over 25 is equal to 1. And once again, the reason I
know this is because now when y is minus 2, this
whole term is 0. 0 when y equals minus 2. And when this term is 0, it
behaves the same way as when this term was 0. So when y is equal to minus 2,
you get the same behavior, you’re at the same point in the
curve, right here actually, as you are when y equaled 0
in this one, so here. So it’s not the same point. You can kind of view it as the
same part of the ellipse. You’re at kind of the maximum
width point on the ellipse here and here when y is equal to 2,
and you were here at y equal to 0– sorry, when y
equals minus 2. This is minus 2. And that’s because when you
put y equals minus 2 here this whole term is 0. Just like when y was 0 here. I don’t want to make
it too confusing. But just to kind of wrap it all
up, sometimes you might see something like graph the
following: y minus 1 squared over 4 plus x plus 2 squared
over 9 is equal to 1. And so the first thing you
could say is OK this is just like the standard ellipse y
squared over 4 plus x squared over 9 is equal to one. It’s just like this,
but it’s shifted over. This ones origin is 0,0, while
this ones origin would be what? It would be the point x
is minus 2 and y is 1. So if you were to graph
this, your radius in your y-direction is 2. 2 squared is equal to 4. Your radius in your
x-direction is 3. 3 squared is equal to 9. So your x-radius is actually
larger than your y-radius. So, it’s going to be a little
bit of a fat ellipse. Actually, let me draw
the axes first. Let me draw it like this. That’s my vertical axis,
this is my x-axis. And so my center is
now at minus 2, 1. That’s minus 2,
and I go up one. That’s the center
of my ellipse. And now in the x-direction,
this is the x term, my x-radius is 3. So the ellipse will go
three– in that direction. This is it’s widest point
will be 3 in that direction. And then in the y-direction,
it’ll go 2, so it’ll go up 1, 2 so that’s there and
then 1, 2 and it’s there. So if I were to draw that
ellipse it would look something like this through my best shot. A little bit fatter than it
is tall, and that’s because your x-radius is larger
than your y-radius. This distance right here is 3,
this distance right here is 3, this distance right here is 2,
this distance right here is 2. You could figure out
what these points are. I won’t do all of them right
now just for the sake of time. But this right here is
the point minus 2, 1. So if you go three more than
that– so if you add 3 to the x-direction this is
the point 1 comma 1. If you would take three away
from the x-direction, this would be minus 5 comma 1. And you could figure
out the other points. That might be good
exercise for you. Anyway that’s a little
bit on ellipses. In future videos we’ll do
really hairy problems where you have to simplify it into this
form so that we know that it definitely is an ellipse.

About James Carlton

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100 thoughts on “Conic sections: Intro to ellipse | Conic sections | Algebra II | Khan Academy

  1. I take algebra II online, and when i dont get what they teach(which is most of the time), the teachers themselves recommend you! You've actually been teaching me!

  2. i think it would be easier for the students to learn the complete formula (x-h)^2 /a^2 + (y-k)^2 /b^2 = 1 explaining that (h,k) are the coordinates of the center of the ellipse, i.e. if it's centered at the origin (h,k) would be (0,0). That way you could graph an ellipse anywhere on the plain without having to do much more thinking.

  3. In 14mins, I finally understood what my professor was trying to discuss for almost 2 hours. Mr. Khan, how I wish I had you as my professor. I'd surely understand every lesson well if you are. Thumbs up for this vid! Thank you so much! 😀

  4. Sal sir, u r just awesome! I am preparing for entrances back at india, and thanks to u ill get into a good college!
    I have been watching all your videos from ur phy maths and chem playlists and it have helped me a lot to understand and visualize things! i am getting intuition of everything now! 😀

    thank u sir! 🙂

  5. he's using Microsoft Paint, a program we all have defaulted on windows computers. It's just that he uses it so well that we think it's a top-notch program

  6. 10th or 11th. This was one of the last things we learned. I took an adavanced so I was able to learn this sophomore year which ended this year 😀

  7. When I hear you talking it seems like you might have a very stupid audience in general. Man, you repeat yourself so frequently…

  8. Thank you so so so so much! I have a huge test tomorrow, and all of your videos are helping me understand all of the lessons my teacher doesn't explain well!

  9. No, he's just an amazing teacher. Sal wants you to understand what it's all about. Clearly you have no respect for good teachers. You should be exiled from this planet.

  10. The video won't play. l also tried watching it at khanacademy but the video doesn't work. 🙁

  11. This may be useful.

    The circumfrance on an ellipse is aprox equal to 2 times pi times the square root of a squared plus b squared divided by 2.

  12. Love your videos. "a" will switch places with "b" and end up under the y^2 sometimes though. "a" is always the larger term.. not certain it matters but I've seen a breakdown proof where "a^2 – b^2 = c^2".

  13. Well, to be fair it is slightly easier for a teacher to make sense in instructional videos – a school teacher often has to work under pressure in order to cover a whole chapter or something during, for example, a seminar.
    But of course, a teacher should always try to explain the core of everything, and exactly WHY it works a certain way.

  14. I'm confused on the part where you shift it down 2 (y+2)^2. In base functions, I always think you shift it to the left two spaces. Please help.

  15. They shouldnt have teacher teach anymore… they should just have students watch your videos and teachers be there to facilitate.

  16. I just purchased into the ellipse (intotheellipse) and would like to make it a center for logical philosophical discussions from various viewpoints. My days of mathematics are behind me but I view the ellipse as an interesting model for what I am trying to achieve. I came across this page and I want to compliment you on an incredible presentation.

  17. thank you very much it was such an accommodating lesson that has extremely helped me i have really understood it thank you once again

  18. I'm learning about ellipses in math right now and your video really cleared things up for me. Just a small tip though: a always has to be the largest number on the bottom. So in your sample problem a=5 and y=3, not the other way around. That's how you tell whether it's a horizontal or vertical ellipse from looking at the equation. If the bigger number is under y, it's vertical 🙂

  19. You are amazing man. You are literally saving my life right now in math. My prof is terrible and I've learned more from 3 of your videos than I've learned my whole semester! Thank you for existing and making these videos lol

  20. this guy is better than my professor; I've learned more in one video than the week we spent on this stuff

  21. Khan academy is honestly better than my teachers at school. Spent an entire unit not understanding this stuff and now i completely understand it within like 10 minutes

  22. sorry if it was in the video and i missed it, in a bit of a rush. Given x^2/a^2 + y^2/b^2 as position 1, and x^2/b^2 + y^2/a^2 as position 2, how do you diffrientiate the two and determine whether the ellipse is in position 1 or 2 when values are in place of "a" and "b"? Thank you in advance

  23. Thank you so much for this… <3 :') OMG my eyes is watering.. im just so thankful for your vids, cause lately im starting to doubt myself if i can pass my math classes. it's really troubling, and i dont understand much of what my teacher is talking about, he's a really fast pace teacher and he just tells us that this is easy and dont explain anymore. but how i understand a little bout this thing here. thanks so much. 😀 <3 keep it up.

  24. Can any one explain to me what determines the maximum width and height of the ellipse?
    I'm trying to complete a free online course and it just jumped right into this topic with virtually no explanation.
    Thanks!

  25. This helps so much as an 8th grader doing this right now in A2, wish I could have used this at the start of the marking period.

  26. What about ehlipse equations that dont have fractions as well as hyperbolas, and a circle equation that doesn't have parenthesis!

  27. One correction: a will always be the length of the semi major axis. If the ellipse is "short and fat", a would be along the x-axis. If it is "tall and skinny", a would be along y-axis. By convention, a is never shorter than b.

  28. What if it's x^2 -y^2 instead of the x^2 + y^2 and what if instead of it equaling 1 it equaled like seven or negative 5

  29. there are reasons for why you can't fully understand certain topics in class. no. 1 reason = Distractions, talking to classmates a lot. that crush you keep looking at, and actually forgetting to pull your earphones out.

  30. when we shift to right by a, we subtract = x-a // when we shift to the left, we add: x+a………………. for y, it was y+a meant shifting up by a and y-a meant shifting down by a. In the equation of ellipse why have we inverted this principle for y but not for x

  31. Ok this might be a stupid question but… Say I want to find the coordinates on the circumference individually using the equation but!!! you cant just put in an absurd value like say x = 100, and get the corresponding y value and plot it. It wouldn't be a part of the circumference! Soooooo there must be some constraints right????

  32. Conic sections used to be the cause of my panic attacks back in 10th grade and also ebcaus eof the teacher that made me feel anxious during the lesson. Khan Academy is such a life saver!

  33. Why X_5 for x radious? If it is plus 5 towards the right the new origin? Please help me to see it clear?

  34. Wait a minute, on the part where you shifted the ellipse, I was taught that you aren't supposed to keep the y sign the same! (Ex: Y-2 means it goes down, not Y+2)

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