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A hyperbola is centred at the origin and passes through the point (4, -2*sqrt(3)). One of its directrices is 5x = 4*sqrt(5), and its eccentricity is e. Which equation does e satisfy?

1. 4e⁴ - 24e² + 35 = 0
2. 4e⁴ + 8e² - 35 = 0
3. 4e⁴ - 12e² - 27 = 0
4. 4e⁴ - 24e² + 27 = 0

Correct answer: 4e⁴ - 24e² + 35 = 0

Solution

From the directrix a = (4*sqrt(5)/5)e, and plugging the given point into the hyperbola yields a quartic 4e⁴ - 24e² + 35 = 0.

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