CRYPT

Mummies Crypt

Down in the crypt.

There's blood everywhere in the crypt, now call the police.

There's an opening. A hole. In the crypt, behind the fountain.

You have the key to the family crypt. Are you suggesting I...

And in future videos I'll show you the foci of a hyperbola or the the foci of a well, it only has one focus of a parabola.

Tell them. We're the first people to set foot in this crypt in centuries.

And we could use that information to actually figure out where the foci lie.

Now in the hyperbola, what is the difference of the distances to the two foci?

But the whole point of this video is to discuss foci, and you might have guessed and I did touch on it in the last video that hyperbola also have foci, but they open.

Yeah, so why didn't he die of a heart attack? They're going down into a crypt, Werner. It's dark.

Now, we said that we have these two foci that are symmetric around the center of the ellipse.

So the entire time, the contents, the sum of the distances to the two foci was equal to 2a.

Now, the next thing, now that we've realized that, is how do we figure out where these foci stand.

For a given messaging crypt, the Bombe could scan through all possible rotor positions and orders, in order to find possible key settings in a matter of minutes.

In the last video, we learned that an ellipse can be defined as the locus of all points where the sum of the distances to two special points, called foci and let me draw this all out, so that's my x axis the sum of the distance to these two special points, called focuses or foci, is a constant.

So, anyway, this is the really neat thing about conic sections, is they have these interesting properties in relation to these foci or in relation to these focus points.

Now I did all of that to kind of compare it to what we're going to cover in this video, which is the focus points or the foci of a hyperbola.

And then I have this distance over here, so I'm taking any point on that ellipse, or this particular point, and I'm measuring the distance to each of these two foci.

Say, y minus 1 is equal to 2 times x minus 3 squared, we can graph this and draw the focus fairly and actually it's foci, if there was more than one focus.

But anyway, let's draw a graphical representation of all this just to make sure we understand, or we re understand, or better understand what the foci points are and where they sit on the hyperbola So those are my axis.

Now the definition of a hyperbola, one of the definitions of a hyperbola, can be the locus of all points where you take the difference not the sum, you take the difference of the distances between the two foci, so if

In fact this is equation of the hyperbola but instead set of writing b squared, since we wrote it, we essentially said, what is the locus of all points where the difference of the distances to those two foci is equal to 2a?

So let's take this point right here on the hyperbola, and we're saying so what is let me do a good color what is the magenta distance that's the distance to that foci minus this let me pick another color minus this

Now we learned in the last video that one of the definitions of a hyperbola is the locus of all points, or the set of all points, where if I take the difference of the distances to the two foci, that difference will be a constant number.

An ellipse, one of the definitions of an ellipse, was the locus of all points or the set of all points where the distance from each of those points, the sum of the distance from each of those points to the two foci is a constant.

They're going to be to the right and the left of these two points, so this is a let me do them in a bright color, because I want you see them let's say that those are the two foci and a hyperbola, this is and notice the difference.

So if this is the point x comma y, and it could be any point that satisfies this equation, it's any point on the byperbola, we know, or we are told, that if we take this distance right here let's call that d1 and subtract from that the distance to the high other foci call that d2

Now, another super interesting, and perhaps the most interesting property of an ellipse, is that if you take any point on the an ellipse, and measure the distance from that point to two special points which we, for the sake of this discussion, and not just for the sake of this discussion, for pretty much forever, we will call the focuses, or the foci, of this ellipse.