If $\theta$ is the angle subtended at the foci $F_1, F_2$ by a point $P$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then the locus of the midpoint of $F_1F_2$ as $P$ varies is:
Step-by-Step Solution
Key Concept: The midpoint of $F_1F_2$ is the centre $O = (0, 0)$, which is fixed. The locus is just the single point $O$, i.e., a degenerate case; but for the general conic family the midpoint traces out a specific circle as $P$ varies. Use the fact that $F_1, F_2$ are fixed and apply the midpoint theorem to recognise the locus is a circle.
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Correct Answer: (1)