An equation of an ellipse is given. (a) Find the vertices,...

An equation of an ellipse is given.
(a) Find the vertices, foci, and eccentricity of the ellipse.
(b) Determine the lengths of the major and minor axes.
(c) Sketch a graph of the ellipse.
$$x^{2}+4 y^{2}=1$$

Key Concepts

Ellipse Standard Form

The equation of an ellipse can be written in standard form as ((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1, where (h, k) is the center of the ellipse, and a and b are the lengths of the semi-axes along the horizontal and vertical directions respectively. This form makes it easier to identify key features of the ellipse such as the vertices, foci, and the lengths of the major and minor axes.

Vertices of an Ellipse

The vertices of an ellipse are the points on the ellipse that lie along its major axis, representing the farthest distances from the center in the horizontal or vertical direction. They are determined by adding and subtracting the values of a or b from the center, depending on the orientation of the ellipse.

Foci of an Ellipse

The foci are two fixed points located along the major axis of an ellipse. The sum of the distances from any point on the ellipse to these foci remains constant. The distance from the center to each focus is calculated using the relationship c^2 = a^2 - b^2, where a is the semi-major axis and b is the semi-minor axis.

Eccentricity

Eccentricity is a measure that describes the shape of an ellipse. Defined as e = c/a, where c is the distance from the center to a focus and a is the length of the semi-major axis, the eccentricity indicates how elongated the ellipse is. A lower eccentricity means the ellipse is more circular, while a higher value indicates a more stretched shape.

Major and Minor Axes Lengths

The major axis is the longest diameter of the ellipse, passing through its foci, and its total length is 2a. The minor axis is the shortest diameter, perpendicular to the major axis at the center, with a total length of 2b. These axes are essential for describing the overall geometry and dimensions of the ellipse.

Sketching a Graph of an Ellipse

Graphing an ellipse involves plotting its center, vertices, foci, and drawing an outline that is symmetric about both the major and minor axes. The identification of these key points from the standard form of the ellipse helps in accurately representing the ellipse on a coordinate plane.

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