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For the circle represented by (x + c)² + y² = a² and the ellipse given as (x - h)² / b² + y² / a² = 1 (where a, b, c, and h are all positive), if they share a tangent that is horizontal, which condition must be satisfied?

1. c > b + a - h
2. c < b + a - h
3. c > b + a
4. None of the above

Correct answer: c < b + a - h

Solution

The condition for the circle and the ellipse to share a horizontal tangent can be derived by examining the equations of the circle and the ellipse, and it can be found that the correct condition is c < b + a - h, which ensures that the tangent is horizontal.

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