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Consider the ellipse x²/tan²(alpha) + y²/sec²(alpha) = 1 with alpha in (0, pi/2). Which of the following quantities VARIES as alpha changes?

1. The eccentricity
2. The distance between the foci
3. The coordinates of the foci
4. None of these vary

Correct answer: The eccentricity

Solution

Here c² = a² - b² = sec² - tan² = 1, so the foci stay fixed at (0, +/-1) and the focal distance is constant; but a = sec(alpha) changes, so the eccentricity e = c/a = cos(alpha) varies.

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