If the center,vertex,and focus of a hyperbola are $(0, 0)$,$(4, 0)$,and $(6, 0)$ respectively,then the equation of the hyperbola is:

Consider a hyperbola $H$ having its centre at the origin and foci on the $x$-axis. Let $C_1$ be a circle touching the hyperbola $H$ and having its centre at the origin. Let $C_2$ be a circle touching the hyperbola $H$ at its vertex and having its centre at one of its foci. If the areas (in sq. units) of $C_1$ and $C_2$ are $36 \pi$ and $4 \pi$,respectively,then the length (in units) of the latus rectum of $H$ is

If a hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm 2, 0)$, then the tangent to this hyperbola at $P$ is:

The equation of the tangent parallel to $y-x+5=0$ drawn to $\frac{x^{2}}{3}-\frac{y^{2}}{2}=1$ is

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

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