Conic Sections: Intro to Circles Introduction to the Circle
Conic Sections: Intro to Circles
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- Now that you know what the conic sections are and why they're
- called conic sections, let's see if we can understand the
- actual equations of the conic sections a little bit better,
- and use that knowledge to be able to at least recognize them
- when we see the equation, and then if we see the equation,
- know how to plot them.
- So the first one we'll start off with is the circle.
- And you've probably seen this for many years already,
- but I'll review, if you don't know it already.
- The general form of an equation, or the general
- equation for a circle, is x squared plus y squared is equal
- to r squared, where r is the radius of the circle.
- And this would be a circle centered at the point 0,0.
- So I'll just draw this kind of a general graph right here.
- So that's the x-axis, that's the y-axis, the circle will
- look like this-- let me do it in a different color-- the
- circle would look something like-- make sure it's a
- circle-- well, it's supposed to be centered at 0,0, but that's
- close enough-- so its center will be right here, and the
- radius if you go from the center to any point along that
- circle will have a distance of r, so if you go from there to
- there it's r, from there to there it's r, from there to
- there it's r, and to some degree, this formula, all it is
- is an extension of the distance formula, which is really just
- the extension of the Pythagorean Theorem.
- So for example, the distance formula, if I want to know the
- distance between some point x,y and the point 0,0, what you do
- is, you take the difference of the x's-- so x minus 0-- you
- square that, and then you add that to the distance between
- the y's squared-- so that's one y point minus 0y-- y is equal
- to 0-- square that, and that is equal to the distance squared.
- So if you simplify this, x minus 0 squared, that's just
- x squared plus-- and this is just y squared is equal
- to distance squared.
- So essentially, this equation is the plot of all points that
- are exactly d away, a distance of d away, from the point 0,0,
- and that's just a circle.
- And I'll let you think about it, I think I actually showed
- this to you in a distance formula video, but the distance
- formula just comes out of the Pythagorean Theorem.
- And if that's not completely obvious for you, just think
- about a little bit, and it'll hopefully become a
- little bit more obvious.
- But anyway, this was probably-- you probably already knew this,
- and actually just to hit the point home, if we had the
- equation like this x squared plus y squared is equal to 9,
- the graph of this circle would look like this.
- So that's the x-axis, that's the y-axis, then draw the
- circle itself, circle looks like-- close enough, and then
- the distance or the radius from the center of the circle,
- that's going to be 3.
- There's a 9 here, why isn't the radius 9?
- Oh, that's because this is the radius squared.
- Remember the original formula I just showed you. x squared plus
- y squared is equal to r squared.
- So this right here is r squared, so if r squared
- is equal to 9, than r is equal to 3.
- It can't be minus 3.
- I mean, it could, you can't have a negative radius, or if
- you did you're just going in the other direction, but
- it's the same thing.
- So the radius is equal to 3.
- So that's a circle, and that's pretty straightforward, but in
- a lot of algebra classes, they complicate the issue a little
- bit by shifting the circle.
- So let's just shift this circle.
- Instead of having-- so let me just rewrite it.
- So the unshifted circle was x squared plus y squared is equal
- to-- let me write it this way-- is equal to 3 squared, that's
- the same thing as 9, and let's say that the new circle, the
- shifted circle, is x minus 1 squared plus y plus 2 squared
- is equal to 3 squared.
- Now all of a sudden this looks really complicated and daunting
- and all the rest, but all you have to recognize is, we just
- substituted an x minus 1 for the-- whoops, messed
- up my pointer.
- We just substituted an x minus 1 for the x, and
- we just substituted a y plus 2 for the y.
- So it has the same basic pattern as of this circle, and
- the fact that we added or subtracted numbers from the x's
- and the y's tells us that we shifted the circle, and so the
- next obvious question is, where did you shift it to?
- And your impulse might be, oh, well maybe I shifted it to,
- instead of the center being 0,0, your intuition might be
- to say that, well, now the center is at negative 1,2.
- And you'd be almost right, except you'd be exactly the
- opposite of the correct answer.
- The new center is now x is equal to positive 1 and
- y is equal to minus 2.
- And that might be unintuitive for you at first-- and you
- might want to watch some of the videos, I think I've done them
- already, or I've always intended to, on shifting
- functions-- but the way to think about it is the center
- here is x is equal to 0.
- So when x and y is equal to 0, x squared plus y squared is 0,
- you're exactly 0 away from the center, or we're at the center.
- So now if we want to be-- if we want x to be 0 away from our
- new center, this term has to be equal to 0.
- And if-- just like when x was equal to 0, this term equaled
- 0, so now for us to be at the center of our new circle, so to
- speak, this term has to be 0.
- So the new center has to be at x is equal to 1.
- Likewise, this has to be 0, and so the center
- is at y is equal to 2.
- Another way to think about it is this-- let's say when y,
- you know what happens here when y is equal to 2.
- Whatever part of the circle we're in when y is equal to 2.
- There's some part of the circle we're at, in fact I could draw
- it, when y is equal to 2.
- Let's say this radius is 3 when y is equal to 2, we're probably
- right around there on the circle.
- We could be there or we could be there.
- Now we're shifting the circle, so now instead of being at 0,0,
- we're going to be at 1 minus 2.
- So now we're going to be at the new center is x is equal to 1,
- y is equal to minus 2, the new center is there, and if I were
- to draw the new circle, it would look something like this.
- I'm going to try my best to draw it still as a
- circle and show you that it's been shifted.
- No, that's not good.
- Let me draw it like-- I pressed the wrong button.
- That's not good.
- Let me draw it right there.
- That's close enough.
- I don't have to keep doing it.
- So what we did is we shifted this circle down 2,
- and to the right 1.
- So if we take its center point, we went down 2
- and to the right 1.
- And so if you think about up here, when y was equal to 2, we
- could have been at this point or this point, the kind of
- equivalent points of the new circle are going to be here.
- Are going to be roughly here, where you're shifting it
- down and to the right.
- And in order to have that same behavior in the circle
- there, this whole thing should be equal to 2.
- So that same point on the circle, if this whole thing is
- going to be equal to 2-- because it's going to be the
- same kind of behavior in the equation, and I hope I'm not
- confusing you there-- then the new y has to be 0, and
- you see it there.
- At both of these points now, y is equal to 0.
- So I know that's a little unintuitive, but I want you to
- sit and think about that a lot.
- I mean, you could just memorize it, that it's the opposite,
- when you have x minus 1 and y plus 2 that it's actually you
- shifted to x is equal to-- the center is now 1, minus 2, or
- you could memorize, if you like, that what makes this 0
- and what makes this 0, and that's your new center.
- But I really want you to think about it, this
- is really a shift.
- And of course, if you were to graph it, you were to
- get this thing up there.
- Anyway, let me see how much time I have.
- Actually I didn't even keep track of the time.
- I'll leave you there, I'll continue this in the next video
- where I'll talk a little bit about ellipses.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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