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 $\frac{1}{a} =$

If $y=x+\sqrt{2}$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{2}=1$,then the equations of its directrices are

If $PN$ is the perpendicular from a point $P$ on the rectangular hyperbola $x^2 - y^2 = a^2$ to any of its asymptotes,then the locus of the midpoint of $PN$ is:

One of the latus recta of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle $2 \operatorname{Tan}^{-1}\left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2=36$ and $e$ is the eccentricity of the given hyperbola,then $\sqrt{a^2+e^2}=$

If the line $x-1=0$ is a directrix of the hyperbola $kx^{2}-y^{2}=6$,then the hyperbola passes through which of the following points?

Consider a hyperbola $H : x^{2}-2y^{2}=4$. Let the tangent at a point $P(4, \sqrt{6})$ meet the $x$-axis at $Q$ and the latus rectum at $R(x_{1}, y_{1})$,where $x_{1}>0$. If $F$ is a focus of $H$ which is nearer to the point $P$,then the area of $\Delta QFR$ is equal to ....... .