Conic sections: Intro to ellipse발음듣기
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Conic sections: Intro to ellipse
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인도네시아어 번역 1명 참여
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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.발음듣기
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You see here, we're really, if we're on this point on the ellipse, we're really close to the origin.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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And of course this point right here this will be a, so this would be the point minus a comma 0.발음듣기
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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.발음듣기
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But in the short and fat ellipse, the direction that you're short in that's called your minor axis.발음듣기
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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.발음듣기
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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.발음듣기
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That's b in this example, just because as I drew this ellipse it just happens to be that b is smaller then a.발음듣기
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In which case, all of a sudden b would be the semi-major axis, because b would be greater than a.발음듣기
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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.발음듣기
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And I think when I've done an example with actual numbers, it'll make it all a little bit clearer.발음듣기
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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.발음듣기
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And if we were just map it we'd say this is b squared than this tells us that b is equal to 5.발음듣기
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So, first of all, we have our radius in the y-direction is larger than our radius in the x-direction.발음듣기
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And actually just to kind of hit the point home that the circle is a special case of an ellipse.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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So let me give you a slightly - It'll look a lot more complicated, and this is something you might see on exam.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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And once again, the reason I know this is because now when y is minus 2, this whole term is 0.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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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.발음듣기
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A little bit fatter than it is tall, and that's because your x-radius is larger than your y-radius.발음듣기
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This distance right here is 3, this distance right here is 3, this distance right here is 2, this distance right here is 2.발음듣기
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So if you go three more than that - so if you add 3 to the x-direction this is the point 1 comma 1.발음듣기
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