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.
$$2 x^{2}+y^{2}=4$$

Key Concepts

Graphing an Ellipse

Sketching an ellipse involves plotting its center, vertices, and foci, and then drawing a smooth, closed curve that reflects its symmetry about the major and minor axes. This visual representation helps in understanding the spatial relationships between the key elements of the ellipse.

Ellipse

An ellipse is a set of points in a plane such that the sum of the distances from two fixed points (the foci) is constant. It is one of the conic sections and can be described by a quadratic equation in two variables.

Standard Form of an Ellipse

The equation of an ellipse is often rewritten in its standard form, (x^2/a^2) + (y^2/b^2) = 1 or (x^2/b^2) + (y^2/a^2) = 1, depending on the orientation. Converting the general quadratic equation into this form simplifies the identification of the ellipse's key parameters including the lengths of the semi-major and semi-minor axes.

Vertices

The vertices of an ellipse are the points on the ellipse that lie along the major and minor axes. They represent the extreme positions along these axes and are used to determine the overall shape of the ellipse.

Foci

The foci are two special points along the major axis of the ellipse that provide the geometric definition of the ellipse. The distance from any point on the ellipse to each focus has a constant sum, and the distance between the foci is an important measure in defining the ellipse’s eccentricity.

Eccentricity

Eccentricity is a number that measures how much an ellipse deviates from being a circle. It is defined as the ratio of the focal distance (c) to the semi-major axis (a). An eccentricity of 0 corresponds to a circle, while values closer to 1 indicate a more elongated ellipse.

Major and Minor Axes

The major and minor axes are the longest and shortest diameters of the ellipse, respectively. The lengths of these axes, determined from the standard form of the ellipse, provide critical information about the size and orientation of the ellipse.

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