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Consider a family of ellipses, all centred with axes along the coordinate axes and all sharing the same major axis length 2a, but with varying minor axis. The tangents drawn at the ends of the latus rectum of these ellipses all pass through certain fixed points. Identify those fixed points.
- (0, a) and (0, -a)
- (a, 0) and (-a, 0)
- (0, 0) only
- (a, a) and (-a, -a)
Correct answer: (0, a) and (0, -a)
Solution
The tangent at a latus-rectum end simplifies to (e x + y)/a = 1; using b² = a²(1-e²) one finds it always passes through (0, a) (and by symmetry (0, -a)).
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