Find the standard form of the equation of each ellipse and give the location of its foci.

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Identify the center of the ellipse. Since the vertices and foci are symmetric about the origin, the center is at (0, 0).

Determine the orientation of the ellipse. The vertices are at (-5, 0) and (5, 0), which lie on the x-axis, so the major axis is horizontal.

Find the lengths of the semi-major axis (a) and semi-minor axis (b). The distance from the center to a vertex is the semi-major axis length, so a = 5. The distance from the center to a focus is c, so c = 3.

Use the relationship between a, b, and c for ellipses: \(c^2 = a^2 - b^2\). Substitute the known values to find \(b^2\).

Write the standard form of the ellipse equation with a horizontal major axis centered at the origin: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Also, list the foci coordinates as \((\pm c, 0)\).

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주요 개념

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Standard Form of an Ellipse

The standard form of an ellipse equation depends on the orientation of its major axis. For a horizontal major axis, the equation is (x^2/a^2) + (y^2/b^2) = 1, where 'a' is the semi-major axis and 'b' is the semi-minor axis. For a vertical major axis, the roles of 'a' and 'b' switch accordingly.

Vertices and Axes of an Ellipse

Vertices are the endpoints of the major axis of the ellipse and determine the length of the major axis (2a). The minor axis is perpendicular to the major axis and has length 2b. Identifying vertices helps in determining 'a' and 'b' values for the ellipse equation.

Foci and the Relationship Between a, b, and c

The foci are two fixed points inside the ellipse, located along the major axis, and satisfy the relationship c^2 = a^2 - b^2, where 'c' is the distance from the center to each focus. Knowing the foci helps in verifying the ellipse parameters and understanding its geometric properties.