The product of the lengths of the perpendiculars from any point on the hyperbola $x^2 - y^2 = 8$ to its asymptotes is

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

Let $C$ be the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $P$ be a point on it. If the tangent at $P$ to the hyperbola meets the straight lines $bx-ay=0$ and $bx+ay=0$ respectively in $Q$ and $R$,then $CQ \cdot CR=$

The product of the perpendicular distances drawn from any point on the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ to its asymptotes is

Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If the normal at the point $P$ intersects the $x$-axis at $(9,0)$,then the eccentricity of the hyperbola is

The intersection of two perpendicular tangents to the hyperbola $\frac{x^2}{4} - \frac{y^2}{2} = 1$ lies on the circle $x^2 + y^2 = \dots \dots \dots$