For the hyperbola $\frac{x^2}{\cos^2 \alpha} - \frac{y^2}{\sin^2 \alpha} = 1$,which of the following remains constant with a change in $\alpha$?

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

$A$ hyperbola passes through the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes of the ellipse,respectively. If the product of their eccentricities is $1$,then the equation of the hyperbola is ...... .

If $\frac{x}{a} + \frac{y}{b} = 1$ is a tangent to the curve $x = Kt, y = \frac{K}{t}, K > 0$,then

If $e_{1}$ and $e_{2}$ represent the eccentricity of the curves $16x^{2}-9y^{2}=144$ and $9x^{2}-16y^{2}=144$ respectively,then $\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}$ is equal to

Let $H : \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$,$a > 0, b > 0$,be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2\sqrt{2}+\sqrt{14})$. If the eccentricity of $H$ is $\frac{\sqrt{11}}{2}$,then the value of $a^{2}+b^{2}$ is equal to