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(A) Let the equation of the hyperbola be $S(x, y) = \frac{x^2}{a^2} - y^2 - 1 = 0$. Since the origin $(0,0)$ and the point $(1,1)$ lie in the same reg

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$\left(\frac{1}{\sqrt{2}}, \infty\right)$

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(A) Let the equation of the hyperbola be $S(x, y) = \frac{x^2}{a^2} - y^2 - 1 = 0$. Since the origin $(0,0)$ and the point $(1,1)$ lie in the same region,the value of $S(0,0)$ and $S(1,1)$ must have the same sign. $S(0,0) = \frac{0^2}{a^2} - 0^2 - 1 = -1$. Since $S(0,0) $S(1,1) = \frac{1^2}{a^2} - 1^2 - 1 = \frac{1}{a^2} - 2$. Setting $\frac{1}{a^2} - 2  \frac{1}{2}$. Since $a > 0$,we have $a > \frac{1}{\sqrt{2}}$. Thus,the range of $a$ is $\left(\frac{1}{\sqrt{2}}, \infty\right)$.

If the point $(1,1)$ and the origin lie in the same region with respect to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{1} = 1$ $(a > 0)$,then the range of $a$ is

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