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The line x - 2y + k = 0 is a common tangent to the parabola y² = 4x and the ellipse x²/a² + y²/3 = 1 (with a > sqrt(3)). Which of the following is correct?

1. The other common tangent is x + 2y + 4 = 0
2. a = 3
3. k = 2
4. The other common tangent is x - 2y + 4 = 0

Correct answer: The other common tangent is x + 2y + 4 = 0

Solution

With slope 1/2, the parabola tangent fixes k = 4 and the ellipse condition gives a² = 4; by symmetry the other common tangent is x + 2y + 4 = 0.

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