Let A ( sec θ, 2 tan θ) and B ( sec φ, 2 tan φ), where θ+φ=π / 2, be two points on the hyperbola 2 x^2-y^2=2. If (α, β) is the point of the…

Let $\displaystyle \mathrm{A}(\sec \theta, 2 \tan \theta)$ and $\displaystyle \mathrm{B}(\sec \phi, 2 \tan \phi)$, where $\displaystyle \theta+\phi=\pi / 2$, be two points on the hyperbola $\displaystyle 2 x^2-y^2=2$. If $\displaystyle (\alpha, \beta)$ is the point of the intersection of the normals to the hyperbola at $\displaystyle A$ and B, then $\displaystyle (2 \beta)^2$ is equal to \_\_\_\_.