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For the ellipse x²/a + y²/b = 1 (with a > b), let A1 be the area of the ellipse and A2 the area of its director circle. Which statement is correct?

1. A2 >= 2A1
2. A2 < 2A1
3. if a = 4b, then A1 = (2/5) A2
4. if a = 4b, then A2 = (2/5) A1

Correct answer: A2 >= 2A1

Solution

By AM-GM, a + b >= 2 sqrt(ab), so the director-circle area pi(a+b) is at least twice the ellipse area pi sqrt(ab); thus A2 >= 2 A1.

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