The equation of a tangent to the hyperbola $4x^2 -5y^2 = 20$ parallel to the line $x -y = 2$ is

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

The latus rectum of the hyperbola $9{x^2} - 16{y^2} - 18x - 32y - 151 = 0$ is

Let $\mathrm{A}\,(\sec \theta, 2 \tan \theta)$ and $\mathrm{B}\,(\sec \phi, 2 \tan \phi)$, where $\theta+\phi=\pi / 2$, be two points on the hyperbola $2 \mathrm{x}^{2}-\mathrm{y}^{2}=2$. If $(\alpha, \beta)$ is the point of the intersection of the normals to the hyperbola at $\mathrm{A}$ and $\mathrm{B}$, then $(2 \beta)^{2}$ is equal to ..... .

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

The difference of the focal distance of any point on the hyperbola $9{x^2} - 16{y^2} = 144$, is