In Exercises 25–36, find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (0, -4), (0, 4); vertices: (0, −7), (0, 7)
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Identify the center of the ellipse. Since the foci are at (0, -4) and (0, 4), and the vertices are at (0, -7) and (0, 7), the center is at the midpoint of the vertices, which is (0, 0).
Determine the orientation of the ellipse. The foci and vertices are aligned along the y-axis, indicating a vertical ellipse.
Calculate the distance from the center to a vertex, which is the semi-major axis length, \(a\). The distance from (0, 0) to (0, 7) is 7, so \(a = 7\).
Calculate the distance from the center to a focus, which is the value \(c\). The distance from (0, 0) to (0, 4) is 4, so \(c = 4\).
Use the relationship \(c^2 = a^2 - b^2\) to find \(b\), the semi-minor axis. Substitute \(c = 4\) and \(a = 7\) into the equation to solve for \(b^2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Ellipse Definition
An ellipse is a set of points in a plane where the sum of the distances from two fixed points, called foci, is constant. The standard form of an ellipse's equation varies based on its orientation, either horizontal or vertical, which is determined by the placement of its foci and vertices.
The standard form of the equation of an ellipse is given by (x-h)²/a² + (y-k)²/b² = 1 for a horizontal ellipse, and (x-h)²/b² + (y-k)²/a² = 1 for a vertical ellipse. Here, (h, k) is the center of the ellipse, 'a' is the distance from the center to the vertices, and 'b' is the distance from the center to the co-vertices.
In an ellipse, the distance from the center to each focus is denoted as 'c', and it relates to 'a' and 'b' through the equation c² = a² - b². The vertices are located at a distance 'a' from the center along the major axis, while the foci are located at a distance 'c' from the center along the same axis, which helps in determining the overall shape and dimensions of the ellipse.