The locus of a point of intersection of two lines $x \sqrt{3}-y=k \sqrt{3}$ and $\sqrt{3} k x+k y=\sqrt{3}$,where $k \in R$,describes:

If $c$ is a real number and $\frac{x^2}{c-12}+\frac{y^2}{7-c}=1$ represents a hyperbola,then

The distance between the directrices of the hyperbola $x = 8 \sec \theta, y = 8 \tan \theta$ is

The product of the lengths of the perpendiculars from any point on the hyperbola $x^2-y^2=16$ to its asymptotes is

On a rectangular hyperbola $x^2-y^2=a^2, a > 0$,three points $A, B, C$ are taken as follows: $A=(-a, 0)$; $B$ and $C$ are placed symmetrically with respect to the $X$-axis on the branch of the hyperbola not containing $A$. Suppose that the $\triangle ABC$ is equilateral. If the side length of the $\triangle ABC$ is $ka$,then $k$ lies in the interval

$A$ rectangular hyperbola passing through $(3,2)$ has its asymptotes parallel to the coordinate axes. If $(1,1)$ is the point of intersection of the two perpendicular tangents of that hyperbola,then its equation is