Understanding the Ellipse: Center, Foci, and Vertex

Ellipses are a fascinating and fundamental part of conic sections, playing a significant role in both mathematics and practical applications. This article aims to provide a comprehensive understanding of an ellipse's center, foci, and vertex, alongside its graph representation. By the end, you will understand how to identify and plot these key features on an ellipse.

What is the Ellipse?

An ellipse can be defined as the set of all points in a plane such that the sum of their distances from two fixed points, called foci, is constant. The standard form of an ellipse is often expressed as an equation that includes both its center and dimensions.

The Equation of an Ellipse

Consider the given equation:

left( frac{y-2}{2.5} right)^2 left( frac{x-3}{sqrt{5}} right)^2 1

This is the standard form of an ellipse centered at (3, 2), with a major axis parallel to the y-axis and a minor axis parallel to the x-axis.

Identifying the Center

The center of the ellipse is given by the coordinates (3, 2). This point is the midpoint of the line segment connecting the two foci and the midpoint of the line segment connecting the two vertices.

Calculating the Foci

First, let's identify the lengths of the semi-major and semi-minor axes. From the equation, we can see:

a 2.5 (semi-major axis)

b sqrt{5} (semi-minor axis)

The distance from the center to each focus (denoted as 'd') can be calculated using the formula:

d sqrt{a^2 - b^2}

Substituting the values, we get:

d sqrt{(2.5)^2 - (sqrt{5})^2} sqrt{6.25 - 5} sqrt{1.25} 1.118

The foci are located along the major axis, which is parallel to the y-axis. Therefore, the foci are:

The upper focus: (3, 2 1.118) (3, 3.118)

The lower focus: (3, 2 - 1.118) (3, 0.882)

Locating the Vertices

The vertices of the ellipse are the endpoints of the major and minor axes. The major axis, being parallel to the y-axis, is located at y 2 2.5 and y 2 - 2.5. The minor axis, being parallel to the x-axis, is located at x 3 sqrt{5} and x 3 - sqrt{5}.

The vertices are thus:

The upper vertex: (3, 4.5)

The lower vertex: (3, -0.5)

The leftmost vertex: (3 - sqrt{5}, 2)

The rightmost vertex: (3 sqrt{5}, 2)

Graphing the Ellipse

To graph the ellipse, plot the center at (3, 2). Then, draw the major axis (vertical) from y 4.5 to y -0.5 and the minor axis (horizontal) from x 3 - sqrt{5} to x 3 sqrt{5}. The foci will be plotted at (3, 3.118) and (3, 0.882).

Conclusion

This article has provided a detailed guide on understanding the center, foci, and vertices of an ellipse, along with graphing it. By applying the principles discussed, you can easily analyze and interpret ellipses in mathematical and practical contexts.

Related Keywords

Ellipse, center, foci, vertex, graph