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Any tangent to the hyperbola x²/a² - y²/b² = 1 forms, together with the two asymptotes, a triangle whose area has magnitude a²*tan(lambda). Find the eccentricity of the hyperbola.

1. sec(lambda)
2. cosec(lambda)
3. sec²(lambda)
4. cosec²(lambda)

Correct answer: sec(lambda)

Solution

Since the tangent-asymptote triangle area is a*b = a²*tan(lambda), we get b = a*tan(lambda); then e² = 1 + tan²(lambda) = sec²(lambda), so e = sec(lambda).

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