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

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be the reciprocal of the eccentricity of the ellipse $x^2+4y^2=4$. If the hyperbola passes through a focus of the ellipse,then:

The equation of the normal at the point $(a \sec \theta, b \tan \theta)$ of the curve $b^2 x^2 - a^2 y^2 = a^2 b^2$ is

The line segment joining the foci of the hyperbola $x^{2}-y^{2}+1=0$ is one of the diameters of a circle. The equation of the circle is

The area of the triangle formed by the tangent to the hyperbola $xy = a^2$ and the coordinate axes is:

The circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ intersect at points $A$ and $B$. The line $2x + y = 1$ is a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If this line passes through the point of intersection of the nearest directrix and the $x$-axis,find the eccentricity of the hyperbola.