The curve $xy = c^2$ is known as:

The equation of a hyperbola,whose foci are $(5, 0)$ and $(-5, 0)$ and the length of whose conjugate axis is $8$,is

The locus of the point of intersection of the lines $ax \sec \theta + by \tan \theta = a$ and $ax \tan \theta + by \sec \theta = b$,where $\theta$ is the parameter,is

If $(0, \pm 4)$ and $(0, \pm 2)$ are the foci and vertices of a hyperbola,respectively,then its equation is:

If the length of the latus rectum of a hyperbola is $8$ and its eccentricity is $\frac{3}{\sqrt{5}}$,then the equation of the hyperbola is:

If $\sqrt{5} y - \sqrt{8} = 0$ is the equation of the directrix of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} + 1 = 0$ and $\frac{\sqrt{5}}{2}$ is its eccentricity,then $\frac{1}{a} =$