The product of the lengths of perpendiculars drawn from any point on the hyperbola ${x^2} - 2{y^2} - 2 = 0$ to its asymptotes is

The distance between the directrices of a rectangular hyperbola is $10$ units, then distance between its foci is

Tangents are drawn to the hyperbola $4{x^2} - {y^2} = 36$ at the points $P$ and $Q.$ If these tangents intersect at the point $T(0,3)$ then the area (in sq. units) of $\Delta PTQ$ is :

An ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ passes through the vertices of the hyperbola $H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113\,l$ is equal to $....$

The locus of the mid points of the chords of the hyperbola $\mathrm{x}^{2}-\mathrm{y}^{2}=4$, which touch the parabola $\mathrm{y}^{2}=8 \mathrm{x}$, is :

The equation to the hyperbola having its eccentricity $2$ and the distance between its foci is $8$