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Two thumbtacks are fixed at distinct points A and B. A string of fixed length has its ends fastened to the tacks; pulling it taut with a pencil traces an ellipse. The area enclosed by this ellipse is maximised when:

1. the two points A and B coincide
2. the distance between A and B is the maximum possible
3. A and B are placed vertically
4. the area is the same regardless of where A and B are

Correct answer: the two points A and B coincide

Solution

Since 2a is fixed and area = pi a b with b = sqrt(a² - c²), area grows as c decreases; c = 0 (foci coincide, a circle) gives maximum area.

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