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The line 2x + y = 1 is tangent to the hyperbola x²/a² - y²/b² = 1 and passes through the point where the nearer directrix meets the x-axis. Find the eccentricity of the hyperbola.

1. 2
2. sqrt(2)
3. sqrt(3)
4. 2/sqrt(3)

Correct answer: 2

Solution

From tangency 1 = 4a² - b² and the directrix point a/e = 1/2, combined with b² = a²(e²-1), solving e⁴ - 5e² + 4 = 0 gives e = 2.

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