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Let a, b, lambda be positive reals. P is an endpoint of the latus rectum of the parabola y² = 4*lambda*x, and the ellipse x²/a² + y²/b² = 1 passes through P. If the tangents to the parabola and the ellipse at P are perpendicular, find the eccentricity of the ellipse.

1. 1/sqrt(2)
2. 1/2
3. 1/3
4. 2/5

Correct answer: 1/sqrt(2)

Solution

At the latus rectum endpoint the parabola tangent has slope 1; requiring the ellipse tangent perpendicular (slope -1) at the same point forces e² = 1/2, so e = 1/sqrt(2).

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