The distance between the directrices of a rectangular hyperbola is $10$ units. What is the distance between its foci?

What is the distance between the directrices of the hyperbola $x = 8 \sec \theta, y = 8 \tan \theta$ (in $\sqrt{2}$)?

If two tangents are drawn to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ such that the product of their gradients is $c^2$,then they intersect on the curve:

The sound of a cannon firing is heard one second later at position $B$ than at position $A$. If the speed of sound is uniform,then

$A$ point moves in such a way that the difference of its distances from two points $(8,0)$ and $(-8,0)$ always remains $4$. Then,the locus of the point is

Let the equation of two diameters of a circle $x^{2} + y^{2} - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0, \infty)$ of the tangent to the hyperbola $3x^{2} - y^{2} = 3$ passing through the centre of the circle is equal to $......$