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

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

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is

The equation of the tangents to the hyperbola $3{x^2} - 4{y^2} = 12$ which cuts equal intercepts from the axes, are

Let $a>0, b>0$. Let $e$ and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$. Let $e ^{\prime}$ and $\ell^{\prime}$ respectively the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e ^{2}=\frac{11}{14} \ell$ and $\left( e ^{\prime}\right)^{2}=\frac{11}{8} \ell^{\prime}$, then the value of $77 a+44 b$ is equal to

The locus of the point of instruction of the lines $\sqrt 3 x - y - 4 \sqrt 3 t = 0$  $\&$  $\sqrt 3tx + ty - 4\sqrt 3 = 0$  (where $ t$  is a parameter) is a hyperbola whose eccentricity is