Hey, everyone, and welcome back. So in the last video, we were introduced to the concept of an ellipse and how you can graph this shape. In this video, we're going to learn about new elements of the ellipse, specifically the vertices and foci. There are two vertices and two foci for each ellipse. Sometimes, calculating these can be a bit tedious because you will need to know different variables and equations. But don't worry, we're going to find out that the vertices and foci are just various points that you can find along the major axis of the ellipse, depending on how the ellipse is stretched.
Without further ado, let's get right into this. When it comes to the vertices, these are the points on the ellipse that are furthest away from the center. If you have a horizontally stretched ellipse, the vertices are going to be the two points that are furthest from the center along the major axis. Now, you'll also need to know how to find the foci, which are the points that determine the general symmetry of the ellipse. The foci could be right here, for example. They indicate that the sum of any distance from the foci to a single point on the ellipse is always constant. For instance, if we have one distance to a point on the ellipse being 1 and another being 3, then 3 plus 1 equals 4. This tells us that for any point on the ellipse, the sum of the distances from the foci to that point will always sum to 4.
In order to calculate the vertices, you will need the distance for the semi-major axis, denoted as a. This makes sense because if you travel along the semi-major axis in either direction, it will lead you to the vertex points. To find the foci, you will need a distance, c, which relates a and b as follows: c2 = a2 − b2. This will allow you to calculate the distance to the two foci.
Let's take a look at an example where we are asked to find the vertices and foci of an ellipse. Given a2 = 25, we find a = 25 = 5, which is our distance for the semi-major axis. With a calculated, we can determine our vertices. Moving 5 units in each direction from the center will locate our two vertices at 50 and -50. We are also asked to find the foci. Identifying b using b2 = 9, we find b = 9 = 3. From here, we calculate the c value, which reveals the distance to the two foci as follows: If c2 = a2 − b2, then 16 = 4, marking the locations of the foci at 40 and -40.
The arrangement changes if dealing with a vertical ellipse, where the coordinates for the vertices and foci would be on the y-axis. In a horizontal ellipse, as we saw, the vertices and foci are along the x-axis. For a vertical ellipse, you would find the vertices and foci vertically aligned along the y-axis at positions 0, a and 0, −a for vertices, and 0, c and 0, −c for foci. That's the basic idea of finding the vertices and foci of an ellipse. I hope you found this video helpful, and thanks for watching.