The eccentricity of the hyperbola $-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is:

Let $x+y+1=0$ and $x-y+4=0$ be the asymptotes of a hyperbola $H$. If $(1,1)$ is a point on $H$,then the length of the latus rectum of $H$ is

The eccentricity of the conjugate hyperbola of $16x^2 - 9y^2 - 32x - 36y - 164 = 0$ is:

If $\theta$ is the angle between the asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{(y-2)^2}{4}=1$ and $\cos \theta=\frac{5}{13}$,then $a^2=$

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$ be two points on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,where $\theta + \phi = \frac{\pi}{2}$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$,then $k = \dots$

Consider a branch of the hyperbola $x^2 - 2y^2 - 2\sqrt{2}x - 4\sqrt{2}y - 6 = 0$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$,then the area of the triangle $ABC$ is