Hey everyone and welcome back. So, up to this point, we've been talking about the various shapes that you get from conic sections, and in the last video, we looked at hyperbolas. Now in this video, what we're going to do is see how we can find the vertices and foci of a hyperbola. Now, I will warn you that this process is pretty tedious because there are some new equations that you'll need to know in order to solve these types of problems, and you are going to be asked to solve these at some point in this course. But don't sweat it because in this video, we're going to be going over some examples and scenarios that I think will make this process seem a lot more clear. And I'll also mention that I think you'll find there are a lot of similarities in the problem-solving with hyperbolas that we did with ellipses in previous videos. So, without further ado, let's get right into this.

Now, just like an ellipse, every hyperbola has 2 vertices and 2 foci, and they are on the major axis. Now the vertices are the points of the hyperbola that are closest to the center. And this distance from the center to either vertex is a distance of \( a \). Now to understand this a bit better, let's actually take a look at this example we have down here. Where we're asked to find the vertices and foci of the hyperbola in the graph. Now, in order to find the vertices, we said they are a distance \( a \) from the center, and we can see the center is at the origin here 0. Now recall that \( a^2 \) is going to be the first thing that we see in our denominator assuming we have the standard form of this fraction minus that fraction is equal to 1. And we can see that we do have the standard form here, so that means \( a^2 \) is going to be equal to this first thing in the denominator which is 4. That means to calculate \( a \) we just need to take the square root of 4 which is 2. So if \( a \) is equal to 2, then we can start here at the center of our hyperbola and we can go 1, 2 units to the left and 1, 2 units to the right starting from the center. This puts our vertices at negative two zero and positive two zero, and that is how you can calculate the vertices of a hyperbola.

Now, you will also need to know how to find the foci. And the thing that makes the foci or the focus points unique is that these are going to be a situation where for any point on the hyperbola the difference of the distances from each focus point to any point of the hyperbola is going to be constant. Now to make sense of this, let's say that we have this hyperbola down here and let's say that the foci end up right about there and right about here. If these are the 2 foci, what you can do is find a point, any point somewhere on the curve of the hyperbola. And if you take this distance and you subtract off that distance, the number that you get will be constant for any point on the hyperbola. So let's say this distance is 5 and that distance is 2. 5 minus 2 is 3. And that means any point that you look at anywhere on the hyperbola, the distances from the foci to there, if you find their difference, it will always come out to 3. So, that's what makes the foci unique.

Now, in order to calculate the foci, you will need the distance \( c \). And \( c \) can be calculated from this equation. Now notice that this equation looks a lot like the equation we had for the ellipse where \( c^2 \) was \( a^2 - b^2 \). The only difference is for the foci of a hyperbola, \( c^2 \) is equal to \( a^2 + b^2 \). Now, you might notice at this point, there is a lot of similarities for the equations between the hyperbola and the ellipse. The main differences are the fact that we have a plus sign here rather than a minus sign for the hyperbola, and then we have a minus sign rather than a plus sign for the standard equation of a hyperbola. So just keep in mind that there are going to be differences in the signs and how you use the equations generally, but a lot of the problem-solving ends up looking pretty similar.

Now, let's actually see if using this equation we can calculate the foci on this hyperbola. So \( c^2 \) is equal to \( a^2 + b^2 \) And recall that \( b^2 \) is going to be equal to the second thing we see in the denominator in the standard form. So that means that \( b^2 \) is equal to 9, meaning that \( b \) is going to be the square root of 9, which is 3. So in order to calculate \( c \), what we can do is we can recognize that \( a^2 \) is going to be 2 squared, which is 4, and \( b^2 \) is going to be 3 squared which is 9. So \( c^2 \) is equal to 4 plus 9 which is 13. If I take the square root on both sides of this equation, we'll get that \( c \) is equal to the square root of 13. And the square root of 13 is approximately equal to 3.6. So what that means is that if we start at the center of our hyperbola here I can go approximately 3.6 units to the right which will get me to this point at square root 13, and I can go approximately 3.6 units to the left getting me at this point which is negative square root 13. And this is how you can find the vertices as well as the foci for a hyperbola.

Now, one more thing that I want to mention is how the vertices and foci will look different depending on the hyperbola's orientation. Because we just looked at an example of a horizontal hyperbola and notice how the vertices ended up on the x-axis at \(a0 \, \text{and} \, -a0\), and notice how the foci also ended up on the x-axis at \(c0 \, \text{and} \, -c0\). So we can conclude is that when we have a horizontal hyperbola, the vertices and foci end up on the x-axis. Now if we instead had a vertical hyperbola, the vertices and foci would actually end up on the y-axis. So you'd end up seeing the vertices end up around here and there at \(0, a\) and \(0, -a\), and you find the foci would end up just outside of these two curves just like we had with the horizontal hyperbola, except these would be at \(0, c\) and \(0, -c\). So when it's a vertical hyperbola, the vertices and foci are on the y-axis. As for a horizontal hyperbola, they're on the x-axis. So that is how you can find the vertices and foci of a hyperbola. Hope you found this video helpful. Thanks for watching.