For the hyperbola $\frac{x^2}{\cos^2\alpha} - \frac{y^2}{\sin^2\alpha} = 1$, which of the following remains constant as $\alpha$ varies?
Step-by-Step Solution
Key Concept: Here $a^2 = \cos^2 \alpha, b^2 = \sin^2 \alpha$, so $c^2 = a^2 + b^2 = 1$ for all $\alpha$. The foci are $(\pm 1, 0)$, so the abscissae of foci are $\pm 1$, which are constant. All other quantities ($e = 1/\cos \alpha$, vertices $\pm \cos \alpha$) change with $\alpha$.
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Correct Answer: (3)