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The hyperbola x² - (y - 1)² = 1 has exactly one tangent line of positive slope passing through the origin, touching the curve at the point (a, b). If sin⁻¹(a/b) = lambda*pi, find lambda.

1. 1/4
2. 1/6
3. 1/3
4. 1/2

Correct answer: 1/4

Solution

The tangency condition gives slope sqrt(2) and contact point (sqrt(2), 2), so a/b = sqrt(2)/2 = 1/sqrt(2) and sin⁻¹(1/sqrt(2)) = pi/4, giving lambda = 1/4.

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