If the point $(1,1)$ and the origin lie in the same region with respect to the hyperbola…
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
$\left(\frac{1}{\sqrt{2}}, \infty\right)$
$\left(0, \frac{1}{\sqrt{2}}\right)$
$(0,1)$
$(0, \sqrt{2})$
Solution
Since centre of given hyperbola is $(0,0)$ Given that origin and $(1,1)$ lie in same region therefore $(1,1)$ lie inside the hyperbola.
$\begin{aligned}
& \therefore \frac{1}{a^2}-1-1 < 0 \Rightarrow \frac{2 a^2-1}{a^2} < 0 \Rightarrow a>\frac{1}{\sqrt{2}} \\
& \therefore a \in\left(\frac{1}{\sqrt{2}}, \infty\right)
\end{aligned}$