Problem

For \((x~-~1)^2/25~+~(y~+~2)^2/9~=~1\), give the ellipse's center, \(a\), \(b\), major-axis orientation, \(c\), vertices, covertices, and foci.

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Read an ellipse outward from its center. The larger denominator gives the squared major semiaxis and its direction, while the smaller gives the squared minor semiaxis; use their difference to find the focal distance. Move the three distances from the center along the proper axes to locate vertices, covertices, and foci, checking that each focus lies inside its corresponding vertex.

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The center is (1, -2); a = 5, b = 3; the major axis is horizontal; c = 4; vertices are (-4, -2), (6, -2); covertices are (1, -5), (1, 1); and foci are (-3, -2), (5, -2).

The center is (-1, 2); a = 5, b = 3; the major axis is horizontal; c = 4; vertices are (-6, 2), (4, 2); covertices are (-1, -1), (-1, 5); and foci are (-5, 2), (3, 2).

The center is (1, -2); a = 5, b = 3; the major axis is vertical; c = 4; vertices are (1, -7), (1, 3); covertices are (-2, -2), (4, -2); and foci are (1, -6), (1, 2).

The center is (1, -2); a = 25, b = 9; the major axis is horizontal; c = 4sqrt(34); vertices are (-24, -2), (26, -2); covertices are (1, -11), (1, 7); and foci are (1 - 4sqrt(34), -2), (1 + 4sqrt(34), -2).

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For \((x~-~1)^2/25~+~(y~+~2)^2/9~=~1\), give the ellipse's center, \(a\), \(b\), major-axis orientation, \(c\), vertices, covertices, and foci.

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