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The equation x²/cos²(alpha) - y²/sin²(alpha) = 1 represents a family of hyperbolas as alpha varies. Which quantity stays constant for all members of the family?

1. the distance between the foci
2. the distance between the two directrices
3. the distance between the vertices
4. the distance between a focus and its corresponding directrix

Correct answer: the distance between the foci

Solution

Since c² = a² + b² = cos²(alpha) + sin²(alpha) = 1, c = 1 always, so the distance between the foci (2c = 2) is constant.

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