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For an ellipse x²/a² + y²/b² = 1 (a > b) with a fixed major axis length 2a, the extremities of the latus rectum trace a curve as b varies. Find the locus of these extremities.

1. x² = a*(a - y)
2. x² = a*(a + y)
3. x² = a*(a + x)
4. y² = a*(a - x)
5. x² = a*(a - |y|)

Correct answer: x² = a*(a - |y|)

Solution

With x = ae and y = b²/a = a(1-e²), eliminating e gives y = a(1 - x²/a²) = a - x²/a, i.e. x² = a(a - y) for the upper endpoints (and x² = a(a-|y|) overall).

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