Problem

Complete the squares and normalize \(4x^2~+~9y^2~-~8x~+~36y~=~104\). Give the ellipse's standard form, center, semiaxis lengths, major-axis orientation, and major and minor axis lengths.

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What this problem is really about

Factor each quadratic coefficient before completing the square inside its variable group, then move the compensating constants carefully. Normalize the completed equation so the right side is one. The binomial shifts give the center, square roots of the denominators give semiaxis lengths, and the larger denominator identifies the major-axis direction; double the semiaxes only when full axis lengths are requested.

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The standard form is (x - 1)²/36 + (y + 2)²/16 = 1; the center is (1, -2); the semiaxis lengths are 6 and 4; the major axis is horizontal; and the major and minor axis lengths are 12, 8.

The standard form is (x - 1)²/4 + (y + 2)²/9 = 1; the center is (1, -2); the semiaxis lengths are 3 and 2; the major axis is vertical; and the major and minor axis lengths are 6 and 4.

The standard form is (x + 1)²/36 + (y - 2)²/16 = 1; the center is (-1, 2); the semiaxis lengths are 6 and 4; the major axis is horizontal; and the major and minor axis lengths are 12 and 8.

The standard form is (x - 1)²/36 + (y + 2)²/16 = 1; the center is (1, -2); the semiaxis lengths are 6 and 4; the major axis is vertical; and the major and minor axis lengths are 12 and 8.

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Complete the squares and normalize \(4x^2~+~9y^2~-~8x~+~36y~=~104\). Give the ellipse's standard form, center, semiaxis lengths, major-axis orientation, and major and minor axis lengths.

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