Find the vertices and foci of the ellipse and sketch its...

Key Concepts

Standard Equation of an Ellipse

The standard equation of an ellipse is represented in the form x²/a² + y²/b² = 1. This form clearly shows the ellipse's orientation and size, with a and b being the denominators that correspond to the distances from the center to the ellipse along the x-axis and y-axis, respectively. It serves as the foundational description for analyzing the ellipse's properties.

Vertices

Vertices are the points where the ellipse intersects its major or minor axes, representing the farthest and closest points from the center along those axes. Determining the vertices involves understanding the values of a and b; the vertices along the major axis are located at distances corresponding to the larger of a or b from the center, while those along the minor axis are located at distances corresponding to the smaller value.

Foci

Foci are two fixed points located along the major axis of an ellipse, and every point on the ellipse has the property that the sum of its distances from these two foci is constant. The distance from the center to a focus, denoted as c, is calculated using the relationship c² = (square of the semi-major axis) - (square of the semi-minor axis). This relationship highlights how the ellipse's shape deviates from being a circle.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, marking the vertices along its axes, and identifying the positions of the foci. By drawing a smooth, closed curve that passes through the vertices and gives a visual representation of the ellipse’s stretching along its axes, one can clearly illustrate the shape, orientation, and key features, making the analytical properties more accessible.

Transcript

00:01 Okay, so first we can compare the equation of this ellipse to the two standard forms for ellipses.

00:07 So you can see there is no h and k with the x and y, which means the center of our ellipse is going to be just zero, zero, because there is no h and k with this x and y.

00:18 So now remember in the ellipse that the way you tell which one is a, the distance to the vertex, and which one is b, the difference to what some books call the co -vertex, is that a, has to be bigger than b in a elis.

00:34 So in this problem, i can tell that 25 is the a squared, because it's bigger than the 16, which is the b squared.

00:43 So a squared is 25, which means my a would have to be 5.

00:48 So i'm going to take the square root of both sides to get a is 5.

00:51 So that's going to be the distance to my vertex.

00:55 And since, okay, we're going to start with the center of 0 -0.

00:57 Since the 5 is under the y, that means i'm going to go up 5, 1, 2, 3, 4, 5, and down 5, 1, 2, 3, 4, 5.

01:08 Those are my two vertices.

01:11 Okay, so the name of this first point, the name of the first vertex would be 05, and the name of the second one would be 0, negative 5.

01:21 Okay? now we're going to find the, go ahead and get the graph going.

01:27 So we're going to use the 4.

01:30 B squared is equal to 16 and b is equal to 4.

01:36 And since that's underneath the x, i need to go 4 to the left and right.