University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
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University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
An ellipse is an important conic section and is formed by intersecting a cone with a plane that does not go through the vertex of a cone. The ellipse is defined by two points, each called a focus. From any point on the ellipse, the sum of the distances to the focus points is constant. The position of the foci determine the shape of the ellipse. The ellipse is related to the other conic sections and a circle is actually a special case of an ellipse.
So an ellipse is basically a oval okay? And if you think about a circle every point on a circle is equidistant from the center and an ellipse is a similar type of concept except instead of being equidistant from one point where we actually have is every point on the circle, the sum of the distances from two points are equal okay?
I have a little demo that I'm going to try I make no promises to see if it's actually going to work but we're going to try this out, so what I actually have is basically just a piece of paper on a box and I have a little string here that is around two cuts two pins and basically what this is doing is securing the distance between these two pins and then I have this extra little flap where if I extend it out every single point if I draw this around the sum from the distance from one of these points which is called the focus and the distance from the other focus are added together it's always going to be the same okay, so no matter where that is it's always going to be the same sum this distance plus this distance. I'm going to try it out with a marker see if I can actually draw this out we'll see if it works okay? So basically I want to try to extend this out as far as I can little extra tick and then just go round and hopefully if everything goes according to plan we will end up with something that looks fairly elliptical, not too too bad, not great could be better but it'll do. Okay, so that's basically what is an ellipse, every single point if we take the distance from one of these four sides to the other we'll get the same distance okay? So that's basically the concept of what an ellipse is.
Now let's go look at the board and get some language about ellipses. Okay so what we have is basically two different arrangements an ellipse can be. We can either have them horizontally or we can have them vertically and I apologize for my drawings I'm not good at circles or ellipses or anything really curved or straight for that matter, so what we actually have is some language. The major axis is what we call the longer diameter okay? So if we're dealing with a horizontal one, our major axis is going to be the diameter from this endpoint to this endpoint. If we are dealing with a vertical ellipse our major axis is switched it's now the longer one from top to bottom okay? And the endpoint to that major axis is called the vertices okay you've heard vertices before so hopefully that's not a new concept, so here a vertice here is a vertice here is a vertice here is a vertice okay?
The other word word we use is a minor axis and that's going to be the smaller diameter okay? So if we're dealing with the horizontal ellipse our minor axis is going to be from top to bottom and a vertical one obviously side to side and the endpoints of those are called co-vertices okay? So you'll know that you're talk if you hear vertices you're talking about the endpoints of the major axis the longer diameter. If you here co-vertices you're talking about the endpoints of the minor axis or the shorter diameter okay?
I often add in a little bit of a different language, I often call things x radius and y radius just because to me it tends to make a little bit more sense. I say x radius I know that I'm dealing with this dimension I say y radius I know I'm dealing with this dimension it loses a little bit of the major minor bit but its still being able to talk about what's going on and have it understood okay?
So let's go to a equation of ellipses. And basically they are barely similar to circles and what I mean by that is we have an x squared plus y squared and we're equal to something the difference here is we're having these over fractions and a is always, so here we have the equation for a horizontal, here we have the equation for a vertical, you see that they're fairly similar and the only difference is I have switched my a's and my b's and the reason being is that a is the major axis portion okay? So a is actually the distance from the center to that vertice the top of the axis or the side of the axis and so what ends up happening is our major axis are always 2a and our minor axis are always 2b, b is the distance from the center to the co-vertices so therefore the whole axis is twice of that okay?
In addition, a lot of information I'm throwing at you hopefully you're following along, we also have what are called foci which are those two points that I secured my little string with and there's a relationship between our major, major vertice our vertices are co-vertices and a focus and that is basically a squared minus b squared is equal to c squared. It's almost like the Pythagorean Theorem but instead of adding we're subtracting okay? So a squared is our bigger number b squared is our major axis major radius, b squared is our minor axis our smaller radius and c squared is the distance from the center to a foci and it goes in either direction on that major axis okay? Lot of information I'm throwing at you but hopefully it is all making sense, the main major axis is referring to the wider or the taller portion though basically the larger dimension end points are vertices, minor axis is the smaller diameter and points are co-vertices, foci are where you're two strings would attach and there is a relationship large radius squared minus smaller radius squared is equal to the distance from the center to the focus.
Unit
Conic Sections