Let $H$ be the hyperbola,whose foci are $(1 \pm \sqrt{2}, 0)$ and eccentricity is $e = \sqrt{2}$. Then the length of its latus rectum is

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sec \alpha$,then the area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is

If the straight line $(a-1)x - by + 4 = 0$ is normal to the hyperbola $xy = 1$, then which of the following does not hold?

Find the coordinates of the foci and the vertices,the eccentricity,and the length of the latus rectum of the hyperbola $5y^{2} - 9x^{2} = 36$.

The equations of the asymptotes of a hyperbola are $x+y+3=0$ and $2x-y+1=0$. If $(1,-2)$ is a point on this hyperbola,find the equation of its conjugate hyperbola.

The length of the latus rectum of the hyperbola $9x^2 - 16y^2 - 18x - 32y - 151 = 0$ is