On a rectangular hy perbola $x^2-y^2= a ^2, a >0$, 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 $\triangle A B C$ is equilateral. If the side length of the $\triangle A B C$ is $k a$, then $k$ lies in the interval

The eccentricity of the hyperbola $\frac{{\sqrt {1999} }}{3}({x^2} - {y^2}) = 1$ is

If $\frac{{{{\left( {3x - 4y - z} \right)}^2}}}{{100}} - {\frac{{\left( {4x + 3y - 1} \right)}}{{225}}^2} = 1$ then
length of latusrectum of hyperbola is

The tangent to the hyperbola $xy = c^2$  at the point $P$  intersects the $x-$ axis at $T$ and the $y-$ axis at $T'$. The normal to the hyperbola at $P$ intersects the $ x-$ axis at $N$  and the $y-$ axis at $N'$. The areas of the triangles $PNT$  and $PN'T' $ are $ \Delta$  and $ \Delta ' $ respectively, then $\frac{1}{\Delta }\,\, + \,\,\frac{1}{{\Delta '}}\,$ is

If $e$ and $e’$ are the eccentricities of the ellipse $5{x^2} + 9{y^2} = 45$ and the hyperbola $5{x^2} - 4{y^2} = 45$ respectively, then $ee' = $

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