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The ellipse x²/25 + y²/b² = 1 (where b < 5) has eccentricity e1, and the hyperbola x²/16 - y²/b² = 1 has eccentricity e2. Given that e1 * e2 = 1, let alpha be the distance between the two foci of the ellipse and beta be the distance between the two foci of the hyperbola. Find the ordered pair (alpha, beta).

1. (8, 10)
2. (8, 12)
3. (20/3, 12)
4. (24/5, 10)

Correct answer: (8, 10)

Solution

Setting e1*e2 = 1 and squaring gives (1 - b²/25)(1 + b²/16) = 1, which simplifies to b² = 9. The ellipse then has c² = 25 - 9 = 16 so alpha = 2*4 = 8, and the hyperbola has c² = 16 + 9 = 25 so beta = 2*5 = 10.

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